%------------CUT HERE------------- %Typeset with AMS-TeX2.1 %NOTE ABOUT PICTURE FILES %Their code are platform-dependent: % %For mac: % %\vskip0.95in\noindent\special{picture f.jpg scaled %0.4}\hskip3.9in\special{picture w.jpg scaled %0.4}\vskip-0.95in % %For pc: % %\vskip0.95in\noindent\special{bmp: f.jpg x= 3.0cm y= 3.4cm %}\hskip3.9in\special{bmp: w.jpg x= 2.1cm y= 3.2cm}\vskip-0.95in \input amstex \documentstyle {amsppt} \parindent=20pt \nopagenumbers \magnification=1200 \pagewidth{4.71in} \pageheight{7.8in} \hcorrection{0.2in} \vcorrection{-0.2in} \leftheadtext{ANDREW JOHN WILES}\rightheadtext{MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM} %For less than and greater than: \def\lt{\char'74 }\def\gt{\char'76 } %For commutative diagrams: \def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}} \def\mapdown#1{\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle\!\!\!#1$}}$}} %For title: \font\boldeighteenpoint=cmb10 scaled 1500 \loadbold \document \ \vskip-0.5in\centerline{\eightpoint{Annals of Mathematics, {\bf 141} (1995), 443-551}} \ %PICTURES: %for mac (delete comment symbol"%"): % %\vskip0.95in\noindent\special{picture f.jpg scaled %0.4}\hskip3.9in\special{picture w.jpg scaled %0.4}\vskip-0.95in % %for pc (delete comment symbol "%"): % %\vskip0.95in\noindent\special{bmp: f.jpg x= 3.0cm y= 3.4cm %}\hskip3.9in\special{bmp: w.jpg x= 2.1cm y= 3.2cm}\vskip-0.95in \vskip0.95in\noindent\special{bmp: f.jpg x= 3.0cm y= 3.4cm }\hskip3.9in\special{bmp: w.jpg x= 2.1cm y= 3.2cm}\vskip-0.95in \vskip0.85in {\fiverm\hskip-0.08in Pierre de Fermat\hskip3.04in Andrew John Wiles\vskip-1.49in} \vskip0.3in \centerline{\boldeighteenpoint{Modular elliptic curves}}\vskip4pt \centerline{\boldeighteenpoint{and}}\vskip4pt \centerline{\boldeighteenpoint{Fermat's Last Theorem}} \vskip4pt\centerline{By {\smc Andrew John Wiles}\footnote"*"{\eightpoint{The work on this paper was supported by an NSF grant.}}}\vskip4pt \centerline{\it For Nada, Claire, Kate and Olivia}\vskip6pt \ \hskip2pt{\it Cubum autem in duos cubos, aut quadratoquadratum in duos quadra- \hskip2pt toquadratos, \ et \ generaliter \ nullam \ in \ infinitum \ ultra \ quadratum \hskip2pt potestatum \ in \ duos \ ejusdem \ nominis \ fas \ est \ \!dividere$:$ \ cujes \ \!rei \hskip2pt demonstrationem \ mirabilem \ sane \ \!detexi. \ \!Hanc \ \!marginis \ \!exiguitas \hskip2pt non caperet. \ \hskip2pt - Pierre de Fermat $\sim$ 1637} \ \noindent{\eightpoint{\bf Abstract.} When Andrew John Wiles was 10 years old, he read Eric Temple Bell's {\it The Last Problem} and was so impressed by it that he decided that he would be the first person to prove Fermat's Last Theorem. This theorem states that there are no nonzero integers $a,b,c,n$ with $n\gt2$ such that $a^n+b^n=c^n$. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat's Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.} \ \centerline{\bf Introduction}\vskip6pt An elliptic curve over $\bold Q$ is said to be modular if it has a finite covering by a modular curve of the form $X_0(N).$ Any such elliptic curve has the property that its Hasse-Weil zeta function has an analytic continuation and satisfies a functional equation of the standard type. If an elliptic curve over $\bold Q$ with a given $j$-invariant is modular then it is easy to see that all elliptic curves with the same $j$-invariant are modular (in which case we say that the $j$-invariant\linebreak is modular). A well-known conjecture which grew out of the work of Shimura and Taniyama in the 1950's and 1960's asserts that every elliptic curve over $\bold Q$ is modular. However, it only became widely known through its publication in a paper of Weil in 1967 [We] (as an exercise for the interested reader!), in which,\linebreak moreover, Weil gave conceptual evidence for the conjecture. Although it had been numerically verified in many cases, prior to the results described in this paper it had only been known that finitely many $j$-invariants were modular. In 1985 Frey made the remarkable observation that this conjecture should imply Fermat's Last Theorem. The precise mechanism relating the two was formulated by Serre as the $\varepsilon$-conjecture and this was then proved by Ribet in the summer of 1986. Ribet's result only requires one to prove the conjecture for semistable elliptic curves in order to deduce Fermat's Last Theorem. \eject\pageno=444 Our approach to the study of elliptic curves is via their associated Galois representations. Suppose that $\rho_p$ is the representation of $\roman{Gal}(\bar{\bold Q}/\bold Q)$ on the $p$-division points of an elliptic curve over $\bold Q$, and suppose for the moment that $\rho_3$ is irreducible. The choice of 3 is critical because a crucial theorem of Lang-lands and Tunnell shows that if $\rho_3$ is irreducible then it is also modular. We then proceed by showing that under the hypothesis that $\rho_3$ is semistable at 3, together with some milder restrictions on the ramification of $\rho_3$ at the other primes, every suitable lifting of $\rho_3$ is modular. To do this we link the problem, via some novel arguments from commutative algebra, to a class number prob-lem of a well-known type. This we then solve with the help of the paper [TW]. This suffices to prove the modularity of $E$ as it is known that $E$ is modular if and only if the associated 3-adic representation is modular. The key development in the proof is a new and surprising link between two\linebreak strong but distinct traditions in number theory, the relationship between Galois\linebreak representations and modular forms on the one hand and the interpretation of special values of $L$-functions on the other. The former tradition is of course more recent. Following the original results of Eichler and Shimura in the\linebreak 1950's and 1960's the other main theorems were proved by Deligne, Serre and Langlands in the period up to 1980. This included the construction of Galois representations associated to modular forms, the refinements of Langlands and Deligne (later completed by Carayol), and the crucial application by Langlands of base change methods to give converse results in weight one. However with the exception of the rather special weight one case, including the extension by \linebreak Tunnell of Langlands' original theorem, there was no progress in the direction of associating modular forms to Galois representations. From the mid 1980's the main impetus to the field was given by the conjectures of Serre which\linebreak elaborated on the $\varepsilon$-conjecture alluded to before. Besides the work of Ribet and others on this problem we draw on some of the more specialized developments\linebreak of the 1980's, notably those of Hida and Mazur. The second tradition goes back to the famous analytic class number formula of Dirichlet, but owes its modern revival to the conjecture of Birch and Swinnerton-Dyer. In practice however, it is the ideas of Iwasawa in this field on which we attempt to draw, and which to a large extent we have to replace. The principles of Galois cohomology, and in particular the fundamental theorems of Poitou and Tate, also play an important role here. The restriction that $\rho_3$ be irreducible at 3 is bypassed by means of an intriguing argument with families of elliptic curves which share a common\linebreak $\rho_5$. Using this, we complete the proof that all semistable elliptic curves are modular. In particular, this finally yields a proof of Fermat's Last Theorem. In\linebreak addition, this method seems well suited to establishing that all elliptic curves over $\bold Q$ are modular and to generalization to other totally real number fields. Now we present our methods and results in more detail. \eject Let $f$ be an eigenform associated to the congruence subgroup $\Gamma_1(N)$ of $\roman{SL}_2(\bold Z)$ of weight $k\ge2$ and character $\chi.$ Thus if $T_n$ is the Hecke operator associated to an integer $n$ there is an algebraic integer $c(n,f)$ such that $T_nf=c(n,f)f$ for each $n$. We let $K_f$ be the number field generated over $\bold Q$ by the\linebreak $\{c(n,f)\}$ together with the values of $\chi$ and let $\Cal O_f$ be its ring of integers.\linebreak For any prime $\lambda$ of $\Cal O_f$ let $\Cal O_{f,\lambda}$ be the completion of $\Cal O_f$ at $\lambda$. The following theorem is due to Eichler and Shimura (for $k=2$) and Deligne (for $k\gt2$). The analogous result when $k=1$ is a celebrated theorem of Serre and Deligne but is more naturally stated in terms of complex representations. The image in that case is finite and a converse is known in many cases. \ {\smc Theorem} 0.1. {\it For each prime $p\in\bold Z$ and each prime $\lambda|p$ of $\Cal O_f$ there\linebreak is a continuous representation $$\rho_{f,\lambda}:\roman{Gal}(\bar\bold Q/\bold Q)\longrightarrow\roman{GL}_2(\Cal O_{f,\lambda})$$ which is unramified outside the primes dividing $Np$ and such that for all primes $q\nmid Np$, $$\roman{trace}\ \rho_{f,\lambda}(\roman{Frob}\ q)=c(q,f),\ \ \ \roman{det}\ \rho_{f,\lambda}(\roman{Frob}\ q)=\chi(q)q^{k-1}.$$} We will be concerned with trying to prove results in the opposite direction, that is to say, with establishing criteria under which a $\lambda$-adic representation arises in this way from a modular form. We have not found any advantage in assuming that the representation is part of a compatible system of $\lambda$-adic representations except that the proof may be easier for some $\lambda$ than for others. Assume $$\rho_0:\roman{Gal}(\bar\bold Q/\bold Q)\longrightarrow\roman{GL}_2(\bar\bold F_p)$$ is a continuous representation with values in the algebraic closure of a finite field of characteristic $p$ and that $\roman{det}\ \rho_0$ is odd. We say that $\rho_0$ is modular\linebreak if $\rho_0$ and $\rho_{f,\lambda}\mod\lambda$ are isomorphic over $\bar\bold F_p$ for some $f$ and $\lambda$ and some embedding of $\Cal O_f/\lambda$ in $\bar\bold F_p$. Serre has conjectured that every irreducible $\rho_0$ of odd determinant is modular. Very little is known about this conjecture except when the image of $\rho_0$ in $\roman{PGL}_2(\bar\bold F_p)$ is dihedral, $A_4$ or $S_4$. In the dihedral case it is true and due (essentially) to Hecke, and in the $A_4$ and $S_4$ cases it is again true and due primarily to Langlands, with one important case due to Tunnell (see Theorem 5.1 for a statement). More precisely these theorems actually associate a form of weight one to the corresponding complex representation but the versions we need are straightforward deductions from the complex case. Even in the reducible case not much is known about the problem in\linebreak the form we have described it, and in that case it should be observed that one must also choose the lattice carefully as only the semisimplification of $\overline{\rho_{f,\lambda}}=\rho_{f,\lambda}\mod\lambda$ is independent of the choice of lattice in $K^2_{f,\lambda}.$ \eject If $\Cal O$ is the ring of integers of a local field (containing $\bold Q_p$) we will say that $\rho:\roman{Gal}(\bar\bold Q/\bold Q)\longrightarrow\roman{GL}_2(\Cal O)$ is a lifting of $\rho_0$ if, for a specified embedding of the residue field of $\Cal O$ in $\bar\bold F_p,\bar\rho$ and $\rho_0$ are isomorphic over $\bar\bold F_p$. Our point of view\linebreak will be to assume that $\rho_0$ is modular and then to attempt to give conditions under which a representation $\rho$ lifting $\rho_0$ comes from a modular form in the sense that $\rho\simeq\rho_{f,\lambda}$ over $\overline{K_{f,\lambda}}$ for some $f,\lambda.$ We will restrict our attention to two cases: \ \hskip-12pt(I) $\rho_0$ is ordinary (at $p$) by which we mean that there is a one-dimensional\linebreak ${}$\hskip24pt subspace of $\bar\bold F^2_p,$ stable under a decomposition group at $p$ and such that\linebreak ${}$\hskip24pt the action on the quotient space is unramified and distinct from the\linebreak ${}$\hskip24pt action on the subspace. \ \hskip-15pt(II) $\rho_0$ is flat (at $p$), meaning that as a representation of a decomposition\linebreak ${}$\hskip24pt group at $p,\rho_0$ is equivalent to one that arises from a finite flat group\linebreak ${}$\hskip25pt scheme over $\bold Z_p$, and $\det\rho_0$ restricted to an inertia group at $p$ is the\linebreak ${}$\hskip25pt cyclotomic character. \ \noindent We say similarly that $\rho$ is ordinary (at $p$), if viewed as a representation to $\bar\bold Q^2_p$, there is a one-dimensional subspace of $\bar\bold Q^2_p$ stable under a decomposition group at $p$ and such that the action on the quotient space is unramified. Let $\varepsilon:\roman{Gal}(\bar\bold Q/\bold Q)\longrightarrow\bold Z^\times_p$ denote the cyclotomic character. Conjectural converses to Theorem 0.1 have been part of the folklore for many years but have hitherto lacked any evidence. The critical idea that one might dispense with compatible systems was already observed by Drinfield in the function field case [Dr]. The idea that one only needs to make a geometric condition on the restriction to the decomposition group at $p$ was first suggested by Fontaine and\linebreak Mazur. The following version is a natural extension of Serre's conjecture which is convenient for stating our results and is, in a slightly modified form, the one proposed by Fontaine and Mazur. (In the form stated this incorporates Serre's conjecture. We could instead have made the hypothesis that $\rho_0$ is modular.) \ {\smc Conjecture.} {\it Suppose that $\rho:\roman{Gal}(\bar\bold Q/\bold Q)\longrightarrow\roman{GL}_2(\Cal O)$ is an irreducible lifting of $\rho_0$ and that $\rho$ is unramified outside of a finite set of primes. There are two cases}: \ \hskip-12pt(i) {\it Assume that $\rho_0$ is ordinary. Then if $\rho$ is ordinary and $\det\rho=\varepsilon^{k-1}\chi$ for\linebreak ${}$\hskip22pt some integer $k\ge2$ and some $\chi$ of finite order}, {\it $\rho$ comes from a modular\linebreak ${}$\hskip22pt form.} \ \hskip-15pt(ii) {\it Assume that $\rho_0$ is flat and that $p$ is odd. Then if $\rho$ restricted to a de-\linebreak ${}$\hskip22pt composition group at $p$ is equivalent to a representation on a $p$-divisible\linebreak ${}$\hskip22pt group}, {\it again $\rho$ comes from a modular form.} \eject In case (ii) it is not hard to see that if the form exists it has to be of\linebreak weight 2; in (i) of course it would have weight $k$. One can of course enlarge this conjecture in several ways, by weakening the conditions in (i) and (ii), by\linebreak considering other number fields of $\bold Q$ and by considering groups other\linebreak than $\roman{GL}_2$. We prove two results concerning this conjecture. The first includes the hypothesis that $\rho_0$ is modular. Here and for the rest of this paper we will assume that $p$ is an odd prime. \ {\smc Theorem 0.2.} {\it Suppose that $\rho_0$ is irreducible and satisfies either} (I) {\it or}\linebreak (II) {\it above. Suppose also that $\rho_0$ is modular and that} \ \hskip-12pt (i) \hskip1pt$\rho_0$ {\it is absolutely irreducible when restricted to $\bold Q\bigg(\sqrt{(-1)^{p-1\over2}p}\bigg)$.} \ \hskip-15pt (ii) {\it If $q\equiv-1\mod p$ is ramified in $\rho_0$ then either $\rho_0|_{D_q}$ is reducible over\linebreak ${}$\hskip22pt the algebraic closure where $D_q$ is a decomposition group at $q$ or $\rho_0|_{I_q}$ is\linebreak ${}$\hskip22pt absolutely irreducible where $I_q$ is an inertia group at $q$. \ \noindent Then any representation $\rho$ as in the conjecture does indeed come from a mod-ular form.} \ The only condition which really seems essential to our method is the requirement that $\rho_0$ be modular. The most interesting case at the moment is when $p=3$ and $\rho_0$ can be de-\linebreak fined over $\bold F_3$. Then since $\roman{PGL}_2(\bold F_3)\simeq S_4$ every such representation is modular by the theorem of Langlands and Tunnell mentioned above. In particular, ev-ery representation into $\roman{GL}_2(\bold Z_3)$ whose reduction satisfies the given conditions is modular. We deduce: \ {\smc Theorem 0.3.} {\it Suppose that $E$ is an elliptic curve defined over $\bold Q$ and\linebreak that $\rho_0$ is the Galois action on the $3$-division points. Suppose that $E$ has the following properties}: \ \hskip-12pt(i) {\it $E$ has good or multiplicative reduction at $3$.} \ \hskip-14.5pt(ii) {\it $\rho_0$ is absolutely irreducible when restricted to $\bold Q\big(\sqrt{-3}\ \big)$.} \ \hskip-17pt(iii) {\it For any $q\equiv-1\mod3$ either $\rho_0|_{D_q}$ is reducible over the algebraic closure\linebreak ${}$\hskip22pt or $\rho_0|{I_q}$ is absolutely irreducible. \ \noindent Then $E$ should be modular.} \ We should point out that while the properties of the zeta function follow directly from Theorem 0.2 the stronger version that $E$ is covered by $X_0(N)$\linebreak \eject \noindent requires also the isogeny theorem proved by Faltings (and earlier by Serre when $E$ has nonintegral $j$-invariant, a case which includes the semistable curves).\linebreak We note that if $E$ is modular then so is any twist of $E$, so we could relax condition (i) somewhat. The important class of semistable curves, i.e., those with square-free conductor, satisfies (i) and (iii) but not necessarily (ii). If (ii) fails then in fact $\rho_0$ is reducible. Rather surprisingly, Theorem 0.2 can often be applied in this case\linebreak also by showing that the representation on the 5-division points also occurs for another elliptic curve which Theorem 0.3 has already proved modular. Thus Theorem 0.2 is applied this time with $p=5$. This argument, which is explained in Chapter 5, is the only part of the paper which really uses deformations of the elliptic curve rather than deformations of the Galois representation. The argument works more generally than the semistable case but in this setting\linebreak we obtain the following theorem: \ {\smc Theorem 0.4.} {\it Suppose that $E$ is a semistable elliptic curve defined over $\bold Q$. Then $E$ is modular.} \ \noindent More general families of elliptic curves which are modular are given in Chap-\linebreak ter 5. In 1986, stimulated by an ingenious idea of Frey [Fr], Serre conjectured and Ribet proved (in [Ri1]) a property of the Galois representation associated to modular forms which enabled Ribet to show that Theorem 0.4 implies `Fer-\linebreak mat's Last Theorem'. Frey's suggestion, in the notation of the following theorem, was to show that the (hypothetical) elliptic curve $y^2=x(x+u^p)(x-v^p)$ could not be modular. Such elliptic curves had already been studied in [He] but without the connection with modular forms. Serre made precise the idea of Frey by proposing a conjecture on modular forms which meant that the rep-\linebreak resentation on the $p$-division points of this particular elliptic curve, if modular, would be associated to a form of conductor 2. This, by a simple inspection, could not exist. Serre's conjecture was then proved by Ribet in the summer\linebreak of 1986. However, one still needed to know that the curve in question would have to be modular, and this is accomplished by Theorem 0.4. We have then (finally!): \ {\smc Theorem 0.5.} {\it Suppose that $u^p+v^p+w^p=0$ with $u,v,w\in\bold Q$ and $p\ge3,$ then $uvw=0$.} ({\it Equivalently - there are no nonzero integers $a,b,c,n$ with $n\gt2$ such that $a^n+b^n=c^n$.}) \ The second result we prove about the conjecture does not require the assumption that $\rho_0$ be modular (since it is already known in this case). \eject {\smc Theorem 0.6.} {\it Suppose that $\rho_0$ is irreducible and satisfies the hypothesis of the conjecture, including} (I) {\it above. Suppose further that} \ \hskip-12pt(i) $\rho_0=\roman{Ind}^\bold Q_L\kappa_0$ {\it for a character $\kappa_0$ of an imaginary quadratic extension $L$\linebreak${}$\hskip22pt of $\bold Q$ which is unramified at $p$.} \ \hskip-15pt(ii) $\det\rho_0|_{I_p}=\omega${\it . \ \noindent Then a representation $\rho$ as in the conjecture does indeed come from a modular form.} \ This theorem can also be used to prove that certain families of elliptic curves are modular. In this summary we have only described the principal theorems associated to Galois representations and elliptic curves. Our results concerning generalized class groups are described in Theorem 3.3. The following is an account of the origins of this work and of the more specialized developments of the 1980's that affected it. I began working on these problems in the late summer of 1986 immediately on learning of Ribet's result. For several years I had been working on the Iwasawa conjecture for totally real fields and some applications of it. In the process, I had been using and developing results on $\ell$-adic representations associated to Hilbert modular forms. It was therefore natural for me to consider the problem of modularity from the point of view of $\ell$-adic representations. I began with the assumption that the reduction of a given ordinary $\ell$-adic representation was reducible and tried to prove under this hypothesis that the representation itself would have to be modular. I hoped rather naively that in this situation I could apply the techniques of Iwasawa theory. Even more optimistically I hoped that the case $\ell=2$ would be tractable as this would suffice for the study of the curves used by Frey. From now on and in the main text, we write $p$ for $\ell$ because of the connections with Iwasawa theory. After several months studying the 2-adic representation, I made the first real breakthrough in realizing that I could use the 3-adic representation instead: the Langlands-Tunnell theorem meant that $\rho_3$, the mod 3 representation of any given elliptic curve over $\bold Q$, would necessarily be modular. This enabled me\linebreak to try inductively to prove that the $\roman{GL}_2(\bold Z/3^n\bold Z)$ representation would be\linebreak modular for each $n$. At this time I considered only the ordinary case. This led quickly to the study of $H^i(\roman{Gal}(F_\infty/\bold Q),W_f)$ for $i=1$ and 2, where $F_\infty$ is the\linebreak splitting field of the $\frak m$-adic torsion on the Jacobian of a suitable modular curve, $\frak m$ being the maximal ideal of a Hecke ring associated to $\rho_3$ and $W_f$ the module associated to a modular form $f$ described in Chapter 1. More specifically, I needed to compare this cohomology with the cohomology of $\roman{Gal}(\bold Q_\Sigma/\bold Q)$ acting on the same module. I tried to apply some ideas from Iwasawa theory to this problem. In my solution to the Iwasawa conjecture for totally real fields [Wi4], I had introduced \eject \noindent a new technique in order to deal with the trivial zeroes. It involved replacing the standard Iwasawa theory method of considering the fields in the cyclotomic $\bold Z_p$-extension by a similar analysis based on a choice of infinitely many distinct primes $q_i\equiv1\mod p^{n_i}$ with $n_i\rightarrow\infty$ as $i\rightarrow\infty.$ Some aspects of this method suggested that an alternative to the standard technique of Iwasawa theory, which seemed problematic in the study of $W_f$, might be to make a comparison between the cohomology groups as $\Sigma$ varies but with the field $\bold Q$ fixed. The new principle said roughly that the unramified cohomology classes are trapped by the tamely ramified ones. After reading the paper [Gre1]. I realized that the\linebreak duality theorems in Galois cohomology of Poitou and Tate would be useful for this. The crucial extract from this latter theory is in Section 2 of Chapter 1. In order to put ideas into practice I developed in a naive form the\linebreak techniques of the first two sections of Chapter 2. This drew in particular on\linebreak a detailed study of all the congruences between $f$ and other modular forms\linebreak of differing levels, a theory that had been initiated by Hida and Ribet. The outcome was that I could estimate the first cohomology group well under two assumptions, first that a certain subgroup of the second cohomology group vanished and second that the form $f$ was chosen at the minimal level for $\frak m$. These assumptions were much too restrictive to be really effective but at least they pointed in the right direction. Some of these arguments are to be found in the second section of Chapter 1 and some form the first weak approximation to the argument in Chapter 3. At that time, however, I used auxiliary primes $q\equiv-1\mod p$ when varying $\Sigma$ as the geometric techniques I worked with did not apply in general for primes $q\equiv1\mod p$. (This was for much the same\linebreak reason that the reduction of level argument in [Ri1] is much more difficult\linebreak when $q\equiv1\mod p.$) In all this work I used the more general assumption that $\rho_p$ was modular rather than the assumption that $p=-3$. In the late 1980's, I translated these ideas into ring-theoretic language. A few years previously Hida had constructed some explicit one-parameter fam-ilies of Galois representations. In an attempt to understand this, Mazur had been developing the language of deformations of Galois representations. Moreover, Mazur realized that the universal deformation rings he found should be given by Hecke ings, at least in certain special cases. This critical conjecture refined the expectation that all ordinary liftings of modular representations should be modular. In making the translation to this ring-theoretic language I realized that the vanishing assumption on the subgroup of $H^2$ which I had needed should be replaced by the stronger condition that the Hecke rings were complete intersections. This fitted well with their being deformation rings where one could estimate the number of generators and relations and so made the original assumption more plausible. To be of use, the deformation theory required some development. Apart from some special examples examined by Boston and Mazur there had been\linebreak \eject \noindent little work on it. I checked that one could make the appropriate adjustments to\linebreak the theory in order to describe deformation theories at the minimal level. In the\linebreak fall of 1989, I set Ramakrishna, then a student of mine at Princeton, the task of proving the existence of a deformation theory associated to representations arising from finite flat group schemes over $\bold Z_p.$ This was needed in order to remove the restriction to the ordinary case. These developments are described in the first section of Chapter 1 although the work of Ramakrishna was not completed until the fall of 1991. For a long time the ring-theoretic version\linebreak of the problem, although more natural, did not look any simpler. The usual methods of Iwasawa theory when translated into the ring-theoretic language seemed to require unknown principles of base change. One needed to know the exact relations between the Hecke rings for different fields in the cyclotomic $\bold Z_p$-extension of $\bold Q$, and not just the relations up to torsion. The turning point in this and indeed in the whole proof came in the\linebreak spring of 1991. In searching for a clue from commutative algebra I had been particularly struck some years earlier by a paper of Kunz [Ku2]. I had already needed to verify that the Hecke rings were Gorenstein in order to compute the congruences developed in Chapter 2. This property had first been proved by Mazur in the case of prime level and his argument had already been extended by other authors as the need arose. Kunz's paper suggested the use of an invariant (the $\eta$-invariant of the appendix) which I saw could be used to test for isomorphisms between Gorenstein rings. A different invariant (the $\frak p/\frak p^2$-invariant of the appendix) I had already observed could be used to test for isomorphisms between complete intersections. It was only on reading Section 6\linebreak of [Ti2] that I learned that it followed from Tate's account of Grothendieck duality theory for complete intersections that these two invariants were equal for such rings. Not long afterwards I realized that, unlike though it seemed at first, the equality of these invariants was actually a criterion for a Gorenstein ring to be a complete intersection. These arguments are given in the appendix. The impact of this result on the main problem was enormous. Firstly, the relationship between the Hecke rings and the deformation rings could be tested just using these two invariants. In particular I could provide the inductive ar-gument of section 3 of Chapter 2 to show that if all liftings with restricted ramification are modular then all liftings are modular. This I had been trying to do for a long time but without success until the breakthrough in commuta-tive algebra. Secondly, by means of a calculation of Hida summarized in [Hi2] the main problem could be transformed into a problem about class numbers of a type well-known in Iwasawa theory. In particular, I could check this in\linebreak the ordinary CM case using the recent theorems of Rubin and Kolyvagin. This is the content of Chapter 4. Thirdly, it meant that for the first time it could be verified that infinitely many $j$-invariants were modular. Finally, it meant that I could focus on the minimal level where the estimates given by me earlier\linebreak \eject \noindent Galois cohomology calculations looked more promising. Here I was also using the work of Ribet and others on Serre's conjecture (the same work of Ribet that had linked Fermat's Last Theorem to modular forms in the first place) to know that there was a minimal level. The class number problem was of a type well-known in Iwasawa theory and in the ordinary case had already been conjectured by Coates and Schmidt. However, the traditional methods of Iwasawa theory did not seem quite suf-ficient in this case and, as explained earlier, when translated into the ring-theoretic language seemed to require unknown principles of base change. So instead I developed further the idea of using auxiliary primes to replace the change of field that is used in Iwasawa theory. The Galois cohomology esti-mates described in Chapter 3 were now much stronger, although at that time I was still using primes $q\equiv-1\mod p$ for the argument. The main difficulty was that although I knew how the $\eta$-invariant changed as one passed to an auxiliary level from the results of Chapter 2, I did not know how to estimate the change in the $\frak p/\frak p^2$-invariant precisely. However, the method did give the right bound for the generalised class group, or Selmer group as it often called in this context, under the additional assumption that the minimal Hecke ring was a complete intersection. I had earlier realized that ideally what I needed in this method of auxiliary primes was a replacement for the power series ring construction one obtains in the more natural approach based on Iwasawa theory. In this more usual setting,\linebreak the projective limit of the Hecke rings for the varying fields in a cyclotomic tower would be expected to be a power series ring, at least if one assumed\linebreak the vanishing of the $\mu$-invariant. However, in the setting with auxiliary primes where one would change the level but not the field, the natural limiting process did not appear to be helpful, with the exception of the closely related and very important construction of Hida [Hi1]. This method of Hida often gave one step towards a power series ring in the ordinary case. There were also tenuous hints of a patching argument in Iwasawa theory ([Scho], [Wi4, \S10]), but I searched without success for the key. Then, in August, 1991, I learned of a new construction of Flach [Fl] and quickly became convinced that an extension of his method was more plausi-\linebreak ble. Flach's approach seemed to be the first step towards the construction of an Euler system, an approach which would give the precise upper bound for the size of the Selmer group if it could be completed. By the fall of 1992, I believed I had achieved this and begun then to consider the remaining case where the mod 3 representation was assumed reducible. For several months I tried simply to repeat the methods using deformation rings and Hecke rings. Then unexpectedly in May 1993, on reading of a construction of twisted forms of modular curves in a paper of Mazur [Ma3], I made a crucial and surprising\linebreak breakthrough: I found the argument using families of elliptic curves with a\linebreak \eject \noindent common $\rho_5$ which is given in Chapter 5. Believing now that the proof was complete, I sketched the whole theory in three lectures in Cambridge, England on June 21-23. However, it became clear to me in the fall of 1993 that the con-\linebreak struction of the Euler system used to extend Flach's method was incomplete and possibly flawed. Chapter 3 follows the original approach I had taken to the problem of bounding the Selmer group but had abandoned on learning of Flach's paper. Darmon encouraged me in February, 1994, to explain the reduction to the com-plete intersection property, as it gave a quick way to exhibit infinite families\linebreak of modular $j$-invariants. In presenting it in a lecture at Princeton, I made, almost unconsciously, critical switch to the special primes used in Chapter 3 as auxiliary primes. I had only observed the existence and importance of these primes in the fall of 1992 while trying to extend Flach's work. Previously, I had\linebreak only used primes $q\equiv-1\mod p$ as auxiliary primes. In hindsight this change was crucial because of a development due to de Shalit. As explained before, I had realized earlier that Hida's theory often provided one step towards a power series ring at least in the ordinary case. At the Cambridge conference de Shalit had explained to me that for primes $q\equiv1\mod p$ he had obtained a version of\linebreak Hida's results. But excerpt for explaining the complete intersection argument in the lecture at Princeton, I still did not give any thought to my initial approach, which I had put aside since the summer of 1991, since I continued to\linebreak believe that the Euler system approach was the correct one. Meanwhile in January, 1994, R. Taylor had joined me in the attempt to repair the Euler system argument. Then in the spring of 1994, frustrated in the efforts to repair the Euler system argument, I begun to work with Taylor on an attempt to devise a new argument using $p=2.$ The attempt to use $p=2$ reached an impasse at the end of August. As Taylor was still not convinced that\linebreak the Euler system argument was irreparable, I decided in September to take one last look at my attempt to generalise Flach, if only to formulate more precisely the obstruction. In doing this I came suddenly to a marvelous revelation: I saw in a flash on September 19th, 1994, that de Shalit's theory, if generalised, could be used together with duality to glue the Hecke rings at suitable auxiliary levels into a power series ring. I had unexpectedly found the missing key to my\linebreak old abandoned approach. It was the old idea of picking $q_i$'s with $q_i\equiv1$mod $p^{n_i}$\linebreak and $n_i\rightarrow\infty$ as $i\rightarrow\infty$ that I used to achieve the limiting process. The switch to the special primes of Chapter 3 had made all this possible. After I communicated the argument to Taylor, we spent the next few days making sure of the details. the full argument, together with the deduction of the complete intersection property, is given in [TW]. In conclusion the key breakthrough in the proof had been the realization in the spring of 1991 that the two invariants introduced in the appendix could be used to relate the deformation rings and the Hecke rings. In effect the $\eta$-\linebreak \eject \noindent invariant could be used to count Galois representations. The last step after the\linebreak June, 1993, announcement, though elusive, was but the conclusion of a long process whose purpose was to replace, in the ring-theoretic setting, the methods based on Iwasawa theory by methods based on the use of auxiliary primes. One improvement that I have not included but which might be used to simplify some of Chapter 2 is the observation of Lenstra that the criterion for Gorenstein rings to be complete intersections can be extended to more general rings which are finite and free as $\bold Z_p$-modules. Faltings has pointed out an improvement, also not included, which simplifies the argument in Chapter 3 and [TW]. This is however explained in the appendix to [TW]. It is a pleasure to thank those who read carefully a first draft of some of this\linebreak paper after the Cambridge conference and particularly N. Katz who patiently answered many questions in the course of my work on Euler systems, and together with Illusie read critically the Euler system argument. Their questions led to my discovery of the problem with it. Katz also listened critically to my first attempts to correct it in the fall of 1993. I am grateful also to Taylor for his assistance in analyzing in depth the Euler system argument. I am indebted to F. Diamond for his generous assistance in the preparation of the final version of this paper. In addition to his many valuable suggestions, several others also made helpful comments and suggestions especially Conrad, de Shalit, Faltings, Ribet, Rubin, Skinner and Taylor.I am most grateful to H. Darmon for his encouragement to reconsider my old argument. Although I paid no heed to his advice at the time, it surely left its mark. \ \centerline{\bf Table of Contents} \ \noindent Chapter 1\ \ 1.\ \ Deformations of Galois representations \hskip0.43in 2.\ \ Some computations of cohomology groups \hskip0.43in 3.\ \ Some results on subgroups of $\roman{GL}_2(k)$ \noindent Chapter 2\ \ 1.\ \ The Gorenstein property \hskip0.43in 2.\ \ Congruences between Hecke rings \hskip0.43in 3.\ \ The main conjectures \noindent Chapter 3\hskip0.3in Estimates for the Selmer group \noindent Chapter 4\ \ 1.\ \ The ordinary CM case \hskip0.43in 2.\ \ Calculation of $\eta$ \noindent Chapter 5\hskip0.3in Application to elliptic curves \noindent Appendix \noindent References \eject \centerline{\bf Chapter 1} \ This chapter is devoted to the study of certain Galois representations.\linebreak In the first section we introduce and study Mazur's deformation theory and discuss various refinements of it. These refinements will be needed later to make precise the correspondence between the universal deformation rings and the Hecke rings in Chapter 2. The main results needed are Proposition 1.2 which is used to interpret various generalized cotangent spaces as Selmer groups and (1.7) which later will be used to study them. At the end of the section we relate these Selmer groups to ones used in the Bloch-Kato conjecture, but this connection is not needed for the proofs of our main results. In the second section we extract from the results of Poitou and Tate on Galois cohomology certain general relations between Selmer groups as $\Sigma$ varies, as well as between Selmer groups and their duals. The most important observation of the third section is Lemma 1.10(i) which guarantees the existence of\linebreak the special primes used in Chapter 3 and [TW]. \ \centerline{\bf 1. Deformations of Galois representations} \ Let $p$ be an odd prime. Let $\Sigma$ be a finite set of primes including $p$ and\linebreak let $\bold Q_\Sigma$ be the maximal extension of $\bold Q$ unramified outside this set and $\infty$. Throughout we fix an embedding of $\overline{\bold Q}$, and so also of $\bold Q_\Sigma$, in $\bold C$. We will also fix a choice of decomposition group $D_q$ for all primes $q$ in $\bold Z$. Suppose that $k$ is a finite field characteristic $p$ and that $$\rho_0:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\roman{GL}_2(k)\leqno(1.1)$$ is an irreducible representation. In contrast to the introduction we will assume in the rest of the paper that $\rho_0$ comes with its field of definition $k$. Suppose further that $\det\rho_0$ is odd. In particular this implies that the smallest field of definition for $\rho_0$ is given by the field $k_0$ generated by the traces but we will not assume that $k=k_0$. It also implies that $\rho_0$ is absolutely irreducible. We con-\linebreak sider the deformation $[\rho]$ to $\roman{GL}_2(A)$ of $\rho_0$ in the sense of Mazur [Ma1]. Thus if $W(k)$ is the ring of Witt vectors of $k,A$ is to be a complete Noeterian local $W(k)$-algebra with residue field $k$ and maximal ideal $m,$ and a deformation $[\rho]$ is just a strict equivalence class of homomorphisms $\rho:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\roman{GL}_2(A)$ such that $\rho\mod m=\rho_0,$ two such homomorphisms being called strictly equiv-\linebreak alent if one can be brought to the other by conjugation by an element of $\ker:\roman{GL}_2(A)\rightarrow\roman{GL}_2(k).$ We often simply write $\rho$ instead of $[\rho]$ for the\linebreak equivalent class. \eject We will restrict our choice of $\rho_0$ further by assuming that either: \hskip-12pt(i) $\rho_0$ is {\it ordinary}; viz., the restriction of $\rho_0$ to the decomposition group $D_p$\linebreak ${}$\hskip22pt has (for a suitable choice of basis) the form $$\rho_0|_{D_p}\approx\pmatrix\chi_1&*\\ 0&\chi_2\endpmatrix\leqno(1.2)$$ ${}$\hskip22pt where $\chi_1$ and $\chi_2$ are homomorphisms from $D_p$ to $k^*$ with $\chi_2$ unramified.\linebreak ${}$\hskip22pt Moreover we require that $\chi_1\ne\chi_2$. We do allow here that $\rho_0|_{D_p}$ be\linebreak ${}$\hskip22pt semisimple. (If $\chi_1$ and $\chi_2$ are both unramified and $\rho_0|_{D_p}$ is semisimple\linebreak ${}$\hskip22pt then we fix our choices of $\chi_1$ and $\chi_2$ once and for all.) \ \hskip-15pt(ii) $\rho_0$ is {\it flat} at $p$ but not ordinary (cf. [Se1] where the terminology {\it finite} is\linebreak ${}$\hskip22pt used); viz., $\rho_0|_{D_p}$ is the representation associated to a finite flat group\linebreak ${}$\hskip23pt scheme over $\bold Z_p$ but is not ordinary in the sense of (i). (In general when we\linebreak ${}$\hskip23pt refer to the flat case we will mean that $\rho_0$ is assumed not to be ordinary\linebreak ${}$\hskip22pt unless we specify otherwise.) We will assume also that $\det\rho_0|_{I_p}=\omega$\linebreak ${}$\hskip22pt where $I_p$ is an inertia group at $p$ and $\omega$ is the Teichm\"uller character\linebreak ${}$\hskip23pt giving the action on $p^\roman{th}$ roots of unity. \ In case (ii) it follows from results of Raynaud that $\rho_0|_{D_p}$ is absolutely\linebreak irreducible and one can describe $\rho_0|_{I_p}$ explicitly. For extending a Jordan-H\"older series for the representation space (as an $I_p$-module) to one for finite flat group schemes (cf. \![Ray 1]) we observe first that the trivial character does not occur on\linebreak a subquotient, as otherwise (using the classification of Oort-Tate or Raynaud) the group scheme would be ordinary. So we find by Raynaud's results, that $\rho_0|_{I_p}\mathop{\otimes}\limits_k\bar k\simeq\psi_1\oplus\psi_2$ where $\psi_1$ and $\psi_2$ are the two fundamental characters of degree 2 (cf. Corollary 3.4.4 of [Ray1]). Since $\psi_1$ and $\psi_2$ do not extend to characters of $\roman{Gal}(\bar\bold Q_p/\bold Q_p),\rho_0|_{D_p}$ must be absolutely irreducible. We sometimes wish to make one of the following restrictions on the\linebreak deformations we allow: \ \noindent\hskip-5pt(i) (a) {\it Selmer deformations.} In this case we assume that $\rho_0$ is ordinary, with no-\linebreak ${}$\hskip24pt tion as above, and that the deformation has a representative\linebreak ${}$\hskip24pt$\rho:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\roman{GL}_2(A)$ with the property that (for a suitable choice\linebreak ${}$\hskip24pt of basis) $$\rho|_{D_p}\approx\pmatrix\tilde\chi_1&*\\0&\tilde\chi_2\endpmatrix$$ ${}$\hskip24pt with $\tilde\chi_2$ unramified, $\tilde\chi\equiv\chi_2\mod m$, and $\det\rho|_{I_p}=\varepsilon\omega^{-1}\chi_1\chi_2$ where\linebreak ${}$\hskip24pt$\varepsilon$ is the cyclotomic character, $\varepsilon:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\bold Z^*_p,$ giving the action\linebreak ${}$\hskip24pt on all $p$-power roots of unity, $\omega$ is of order prime to $p$ satisfying $\omega\equiv\varepsilon\mathbreak{}\hskip17pt\mod p,$ and $\chi_1$ and $\chi_2$ are the characters of (i) viewed as taking values in\linebreak ${}$\hskip24pt $k^*\hookrightarrow A^*.$ \eject \noindent\hskip-5pt(i) (b) {\it Ordinary deformations.} The same as in (i)(a) but with no condition on\linebreak ${}$\hskip28pt the determinant. \ \noindent\hskip-5pt(i) (c) \hskip2pt{\it Strict deformations.} This is a variant on (i) (a) which we only use when\linebreak ${}$\hskip26pt$\rho_0|_{D_p}$ is not semisimple and not flat (i.e. not associated to a finite flat\linebreak ${}$\hskip26pt group scheme). We also assume that $\chi_1\chi_2^{-1}=\omega$ in this case. Then a\linebreak ${}$\hskip26pt strict deformation is as in (i)(a) except that we assume in addition that\linebreak ${}$\hskip26pt$(\tilde\chi_1/\tilde\chi_2)|_{D_p}=\varepsilon.$ \ \hskip-11pt(ii) {\it Flat} ({\it at $p$}) {\it deformations.} We assume that each deformation $\rho$ to $\roman{GL}_2(A)$\linebreak ${}$\hskip26pt has the property that for any quotient $A/\frak a$ of finite order $\rho|_{D_p}\mod\frak a$\linebreak ${}$\hskip26pt is the Galois representation associated to the $\bar\bold Q_p$-points of a finite flat\linebreak ${}$\hskip26pt group scheme over $\bold Z_p.$ \ In each of these four cases, as well as in the unrestricted case (in which we\linebreak impose \!no \!local \!restriction \!at \!$p$) \!one \!can \!verify \!that \!Mazur's \!use \!of \!Schlessinger's\linebreak criteria [Sch] proves the existence of a universal deformation $$\rho:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\roman{GL}_2(R).$$ In the ordinary and restricted case this was proved by Mazur and in the\linebreak flat case by Ramakrishna [Ram]. The other cases require minor modifications of Mazur's argument. We denote the universal ring $R_\Sigma$ in the unrestricted case and $R^\roman{se}_\Sigma,R^\roman{ord}_\Sigma,R^\roman{str}_\Sigma,R^\roman f_\Sigma$ in the other four cases. We often omit the $\Sigma$ if the context makes it clear. There are certain generalizations to all of the above which we will also need. The first is that instead of considering $W(k)$-algebras $A$ we may consider $\Cal O$-algebras for $\Cal O$ the ring of integers of any local field with residue field $k$. If we need to record which $\Cal O$ we are using we will write $R_{\Sigma,\Cal O}$ etc. It is easy to see that the natural local map of local $\Cal O$-algebras $$R_{\Sigma,\Cal O}\rightarrow R_\Sigma\mathop{\otimes}\limits_{W(k)}\Cal O$$ is an isomorphism because for functorial reasons the map has a natural section which induces an isomorphism on Zariski tangent spaces at closed points, and one can then use Nakayama's lemma. Note, however, hat if we change the residue field via $i:\hookrightarrow k'$ then we have a new deformation problem associated to the representation $\rho'_0=i\circ\rho_0.$ There is again a natural map of $W(k')$-\linebreak algebras $$R(\rho'_0)\rightarrow R\mathop{\otimes}\limits_{W(k)}W(k')$$ which is an isomorphism on Zariski tangent spaces. One can check that this\linebreak is again an isomorphism by considering the subring $R_1$ of $R(\rho'_0)$ defined as the\linebreak subring of all elements whose reduction modulo the maximal ideal lies in $k$. Since $R(\rho'_0)$ is a finite $R_1$-module, $R_1$ is also a complete local Noetherian ring\linebreak \eject \noindent with residue field $k$. The universal representation associated to $\rho'_0$ is defined over $R_1$ and the universal property of $R$ then defines a map $R\rightarrow R_1.$ So we obtain a section to the map $R(\rho'_0)\rightarrow R\mathop{\otimes}\limits_{W(k)}W(k')$ and the map is therefore an isomorphism. (I am grateful to Faltings for this observation.) We will also need to extend the consideration of $\Cal O$-algebras tp the restricted cases. In each case we can require $A$ to be an $\Cal O$-algebra and again it is easy to see that $R^\cdot_{\Sigma,\Cal O}\simeq R^\cdot_\Sigma\mathop{\otimes}\limits_{W(k)}\Cal O$ in each case. The second generalization concerns primes $q\ne p$ which are ramified in $\rho_0$. We distinguish three special cases (types (A) and (C) need not be disjoint): \vskip6pt\hskip-12pt(A) $\rho_0|_{D_q}=({\chi_1\atop{}}{*\atop\chi_2})$ for a suitable choice of basis, with $\chi_1$ and $\chi_2$ unramified,\linebreak ${}$\hskip26pt$\chi_1\chi_2^{-1}=\omega$ and the fixed space of $I_q$ of dimension 1, \vskip6pt\hskip-12pt(B) $\rho_0|_{I_q}=({\chi_q\atop0}{0\atop1}),\chi_q\ne1,$ for a suitable choice of basis, \vskip6pt\hskip-12pt(C) $H^1(\bold Q_q,W_\lambda)=0$ where $W_\lambda$ is as defined in (1.6). \vskip6pt Then in each case we can define a suitable deformation theory by imposing additional restrictions on those we have already considered, namely: \vskip6pt \hskip-12pt(A) $\rho|_{D_q}=({\psi_1\atop{}}{*\atop\psi_2})$ for a suitable choice of basis of $A^2$ with $\psi_1$ and $\psi_2$ un-\linebreak ${}$\hskip27pt ramified and $\psi_1\psi_2^{-1}=\varepsilon;$ \vskip6pt\hskip-11pt(B) $\rho|_{I_q}=({\chi_q\atop0}{0\atop1})$ for a suitable choice of basis ($\chi_q$ of order prime to $p$, so the\linebreak ${}$\hskip26pt {\it same} character as above); \vskip6pt \hskip-12pt(C) $\det\rho|_{I_q}=\det\rho_0|_{I_q},$ i.e., of order prime to $p.$ \vskip6pt \noindent Thus if $\Cal M$ is a set of primes in $\Sigma$ distinct from $p$ and each satisfying one of (A), (B) or (C) for $\rho_0$, we will impose the corresponding restriction at each prime in $\Cal M$. Thus to each set of data $\Cal D=\{\cdot,\Sigma,\Cal O,\Cal M\}$ where $\cdot$ is Se, str, ord, flat or unrestricted, we can associate a deformation theory to $\rho_0$ provided $$\rho_0:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\roman{GL}_2(k)\leqno(1.3)$$ is itself of type $\Cal D$ and $\Cal O$ is the ring of integers of a totally ramified extension of $W(k);\rho_0$ is ordinary if $\cdot$ is Se or ord, strict if $\cdot$ is strict and flat if $\cdot$ is fl (meaning flat); $\rho_0$ is of type $\Cal M$, i.e., of type (A), (B) or (C) at each ramified primes $q\ne p,q\in\Cal M.$ We allow different types at different $q$'s. We will refer\linebreak to these as the standard deformation theories and write $R_\Cal D$ for the universal ring associated to $\Cal D$ and $\rho_\Cal D$ for the universal deformation (or even $\rho$ if $\Cal D$ is clear from the context). We note here that if $\Cal D=(\roman{ord},\Sigma,\Cal O,\Cal M)$ and $\Cal D'=(\roman{Se},\Sigma,\Cal O,\Cal M)$ then there is a simple relation between $R_\Cal D$ and $R_{\Cal D'}.$ Indeed there is a natural map\linebreak \eject \noindent$R_\Cal D\rightarrow R_{\Cal D'}$ by the universal property of $R_\Cal D$, and its kernel is a principal ideal generated by $T=\varepsilon^{-1}(\gamma)\det\rho_\Cal D(\gamma)-1$ where $\gamma\in\roman{Gal}(\bold Q_\Sigma/\bold Q)$ is any element whose restriction to $\roman{Gal}(\bold Q_\infty/\bold Q)$ is a generator (where $\bold Q_\infty$ is the $\bold Z_p$-extension of $\bold Q$) and whose restriction to $\roman{Gal}(\bold Q(\zeta_{N_p})/\bold Q)$ is trivial for any $N$ prime to $p$ with $\zeta_N\in\bold Q_\Sigma,\zeta_N$ being a primitive $N^\roman{th}$ root of 1: $$R_\Cal D/T\simeq R_\Cal D'.\leqno(1.4)$$ It turns out that under the hypothesis that $\rho_0$ is strict, i.e. that $\rho_0|_{D_p}$ is not associated to a finite flat group scheme, the deformation problems in (i)(a) and (i)(c) are the same; i.e., every Selmer deformation is already a strict deformation. This was observed by Diamond. the argument is local, so the decomposition group $D_p$ could be replaced by $\roman{Gal}(\bar\bold Q_p/\bold Q).$ \ {\smc Proposition 1.1} (Diamond). {\it Suppose that $\pi:D_p\rightarrow\roman{GL}_2(A)$ is a continuous representation where $A$ is an Artinian local ring with residue field $k$, a finite field of characteristic $p.$ Suppose $\pi\approx({\chi_1\varepsilon\atop0}{*\atop\chi_2})$ with $\chi_1$ and $\chi_2$ unramified and $\chi_1\ne\chi_2$. Then the residual representation $\bar\pi$ is associated to a finite flat group scheme over $\bold Z_p$.\vskip6pt Proof} (taken from [Dia, Prop. 6.1]). We may replace $\pi$ by $\pi\otimes\chi_2^{-1}$ and we let $\varphi=\chi_1\chi_2^{-1}.$ Then $\pi\cong({\varphi\varepsilon\atop0}{t\atop1})$ determines a cocycle $t:D_p\rightarrow M(1)$ where $M$ is a free $A$-module of rank one on which $D_p$ acts via $\varphi$. Let $u$ denote the cohomology class in $H^1(D_p,M(1))$ defined by $t$, and let $u_0$ denote its image\linebreak in $H^1(D_p,M_0(1))$ where $M_0=M/\frak mM.$ Let $G=\ker\varphi$ and let $F$ be the fixed field of $G$ (so $F$ is a finite unramified extension of $\bold Q_p$). Choose $n$ so that $p^nA\mathbreak=0.$ Since $H^2(G,\mu_{p^r}\rightarrow H^2(G,\mu_{p^s})$ is injective for $r\le s,$ we see that the natural map of $A[D_p/G]$-modules $H^1(G,\mu_{p^n}\otimes_{\bold Z_p}M)\rightarrow H^1(G,M(1))$ is an isomorphism. By Kummer theory, we have $H^1(G,M(1))\cong F ^\times/(F^\times)^{p^n}\otimes_{\bold Z_p}M$ as $D_p$-modules. Now consider the commutative diagram $$\matrix \hskip-6pt H^1(G,M(1))^{D_p}&\hskip-10pt\mathop{\hbox to 32pt{\rightarrowfill}}\limits^\sim&\hskip-10pt((F^\times/(F^\times)^{p^n}\otimes_{\bold Z_p}M)^{D_p}&\hskip-10pt{\hbox to 32pt{\rightarrowfill}}&\hskip-10pt M^{D_p}\\ \hskip-6pt\Bigg\downarrow&&\hskip-10pt\Bigg\downarrow&&\hskip-10pt\ \ \Bigg\downarrow\ \ ,\\ \hskip-6pt H^1(G,M_0(1))&\hskip-10pt\mathop{\hbox to 32pt{\rightarrowfill}}\limits^\sim&\hskip-10pt(F^\times/(F^\times)^p)\otimes_{\bold F_p}M_0&\hskip-10pt{\hbox to 32pt{\rightarrowfill}}&\hskip-10pt M_0\endmatrix$$ where the right-hand horizontal maps are induced by $v_p:F^\times\rightarrow\bold Z.$ If $\varphi\ne1,$ then $M^{D_p}\subset\frak mM,$ so that the element res \!$u_0$ of $H^1(G,M_0(1))$ is in the image of $(\Cal O^\times_F/(\Cal O^\times_F)^p)\otimes_{\bold F_p}M_0.$ But this means that $\bar\pi$ is ``peu ramifi\'e'' in the sense of [Se] and therefore $\bar\pi$ comes from a finite flat group scheme. (See [E1, (8.20].) \ {\it Remark.} Diamond also observes that essentially the same proof shows that if $\pi:\roman{Gal}(\bar\bold Q_q/\bold Q_q)\rightarrow\roman{GL}_2(A),$ where $A$ is a complete local Noetherian\linebreak \eject \noindent ring with residue field $k$, has the form $\pi|_{I_q}\cong({1\atop0}{*\atop1})$ with $\bar\pi$ ramified then $\pi$ is of type (A). \ Globally, Proposition 1.1 says that if $\rho_0$ is strict and if $\Cal D=(\roman{Se},\Sigma,\Cal O,\Cal M)$ and $\Cal D'=(\roman{str}, \Sigma,\Cal O,\Cal M)$ then the natural map $R_\Cal D\rightarrow R_{\Cal D'}$ is an isomorphism. In each case the tangent space of $R_\Cal D$ may be computed as in [Ma1]. Let $\lambda$ be a uniformizer for $\Cal O$ and let $U_\lambda\simeq k^2$ be the representation space for $\rho_0.$ (The motivation for the subscript $\lambda$ will become apparent later.) Let $V_\lambda$ be the\linebreak representation space of $\roman{Gal}(\bold Q_\Sigma/\bold Q)$ on $\roman{Ad}\rho_0=\roman{Hom}_k(U_\lambda,U_\lambda)\simeq M_2(k).$ Then there is an isomorphism of $k$-vector spaces (cf. the proof of Prop. 1.2 below) $$\roman{Hom}_k(m_\Cal D/(m^2_\Cal D,\lambda),k)\simeq H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_\lambda)\leqno(1.5)$$ where $H^1_\Cal D(\roman{Q}_\Sigma/\roman{Q},V_\lambda)$ is a subspace of $H^1(\bold Q_\Sigma/\bold Q,V_\lambda)$ which we now describe and $m_\Cal D$ is the maximal ideal of $R_Cal D$. It consists of the cohomology classes which satisfy certain local restrictions at $p$ and at the primes in $\Cal M$. We call $m_\Cal D/(m^2_\Cal D,\lambda)$ the reduced cotangent space of $R_\Cal D$. We begin with $p$. First we may write (since $p\ne2$), as $k[\roman{Gal}(\bold Q_\Sigma/\bold Q)]$-modules, $$\leqalignno{V_\lambda=W_\lambda\oplus k,\ \roman{where}\ W_\lambda&=\{f\in\roman{Hom}_k(U_\lambda,U_\lambda):\roman{trace}f=0\}&(1.6)\cr&\simeq (\roman{Sym}^2\otimes\roman{det}^{-1})\rho_0}$$ and $k$ is the one-dimensional subspace of scalar multiplications. Then if $\rho_0$\linebreak is ordinary the action of $D_p$ on $U_\lambda$ induces a filtration of $U_\lambda$ and also on $W_\lambda$ and $V_\lambda$. Suppose we write these $0\subset U^0_\lambda\subset U_\lambda,\ 0\subset W^0_\lambda\subset W^1_\lambda\subset W_\lambda$ and $0\subset V^0_\lambda\subset V^1_\lambda\subset V_\lambda.$ Thus $U^0_\lambda$ is defined by the requirement that $D_p$ act on it via the character $\chi_1$ (cf. (1.2)) and on $U_\lambda/U^0_\lambda$ via $\chi_2$. For $W_\lambda$ the filtrations are defined by $$\aligned W^1_\lambda\ \ \ &=\ \ \ \{f\in W_\lambda:f(U^0_\lambda)\subset U^0_\lambda\},\\ W^0_\lambda\ \ \ &=\ \ \ \{f\in W^1_\lambda:f=0\ \roman{on}\ U^0_\lambda\},\endaligned$$ and the filtrations for $V_\lambda$ are obtained by replacing $W$ by $V$. We note that these filtrations are often characterized by the action of $D_p.$ Thus the action\linebreak of $D_p$ on $W^0_\lambda$ is via $\chi_1/\chi_2$; on $W^1_\lambda/W^0_\lambda$ it is trivial and on $Q_\lambda/W^1_\lambda$ it is via $\chi_2/\chi_1$. These determine the filtration if either $\chi_1/\chi_2$ is not quadratic or $\rho_0|_{D_p}$ is not semisimple. We define the $k$-vector spaces $$\aligned V^\roman{ord}_\lambda&=\{f\in V^1_\lambda:f=0\ \ \roman{in}\ \ \roman{Hom}(U_\lambda/U^0_\lambda,U_\lambda/U^0_\lambda)\},\\ H^1_\roman{Se}(\bold Q_p,V_\lambda)&=\roman{ker}\{H^1(\bold Q_p,V_\lambda)\rightarrow H^1(\bold Q^\roman{unr}_p,V_\lambda/W^0_\lambda)\},\\ H^1_\roman{ord}(\bold Q_p,V_\lambda)&=\roman{ker}\{H^1(\bold Q_p,V_\lambda)\rightarrow H^1(\bold Q^\roman{unr}_p,V_\lambda/V^\roman{ord}_\lambda)\},\\ H^1_\roman{str}(\bold Q_p,V_\lambda)&=\roman{ker}\{H^1(\bold Q_p,V_\lambda)\rightarrow H^1(\bold Q_p,W_\lambda/W^0_\lambda)\oplus H^1(\bold Q^\roman{unr}_p,k)\}. \endaligned$$ \eject In the Selmer case we make an analogous definition for $H^1_\roman{Se}(\bold Q_p,W_\lambda)$ by replacing $V_\lambda$ by $W_\lambda$, and similarly in the strict case. In the flat case we use the fact that there is a natural isomorphism of $k$-vector spaces $$H^1(\bold Q_p,V_\lambda)\rightarrow\roman{Ext}^1_{k[D_p]}(U_\lambda,U_\lambda)$$ where the extensions are computed in the category of $k$-vector spaces with local\linebreak Galois action. Then $H^1_\roman{f}(\bold Q_p,V_\lambda)$ is defined as the $k$-subspace of $H^1(\bold Q_p,V_\lambda)$ which is the inverse image of $\roman{Ext}^1_\roman{fl}(G,G),$ the group of extensions in the category of finite flat commutative group schemes over $\bold Z_p$ killed by $p,G$ being the (unique) finite flat group scheme over $\bold Z_p$ associated to $U_\lambda$. By [Ray1] all such extensions in the inverse image even correspond to $k$-vector space schemes. For more details and calculations see [Ram]. For $q$ different from $p$ and $q\in\Cal M$ we have three cases (A), (B), (C). In\linebreak case (A) there is a filtration by $D_q$ entirely analogous to the one for $p$. We write this $0\subset W^{0,q}_\lambda\subset W^{1,q}_\lambda\subset W_\lambda$ and we set $$H^1_{D_q}(\bold Q_q,V_\lambda)=\cases\roman{ker}:H^1(\bold Q_q,V_\lambda\\ \hskip0.18in\rightarrow H^1(\bold Q_q,W_\lambda/W^{0,q}_\lambda)\oplus H^1(\bold Q^\roman{unr}_q,k)\hskip0.06in \roman{in\ case\ (A)}\\ \ \\ \roman{ker}:H^1(\bold Q_q,V_\lambda)\\ \hskip0.18in\rightarrow H^1(\bold Q^\roman{unr}_q,V_\lambda)\hskip1.26668in\roman{in\ case\ (B)\ or\ (C).} \endcases$$ Again we make an analogous definition for $H^1_{D_q}(\bold Q_q,W_\lambda)$ by replacing $V_\lambda$\linebreak by $W_\lambda$ and deleting the last term in case (A). We now define the $k$-vector space $H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_\lambda)$ as $$\aligned H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_\lambda)=\{\alpha\in H^1(\bold Q_\Sigma/\bold Q,V_\lambda):&\ \ \alpha_q\in H^1_{D_q}(\bold Q_q, V_\lambda)\ \roman{for\ all}\ q\in\Cal M,\\ &\ \ \alpha_q\in H^1_*(\bold Q_p,V_\lambda)\}\endaligned$$ where $*$ is Se, str, ord, fl or unrestricted according to the type of $\Cal D$. A similar definition applies to $H^1_\Cal D(\bold Q_\Sigma/\bold Q, W_\lambda)$ if $\cdot$ is Selmer or strict. Now and for the rest of the section we are going to assume that $\rho_0$ arises from the reduction of the $\lambda$-adic representation associated to an eigenform. More precisely we assume that there is a normalized eigenform $f$ of weight 2 and level $N$, divisible only by the primes in $\Sigma$, and that there ia a prime $\lambda$ of $\Cal O_f$ such that $\rho_0=\rho_{f,\lambda}\mod\lambda.$ Here $\Cal O_f$ is the ring of integers of the field generated by the Fourier coefficients of $f$ so the fields of definition of the two representations need not be the same. However we assume that $k\supseteq\Cal O_{f,\lambda}/\lambda$ and we fix such an embedding so the comparison can be made over $k$. It will be convenient moreover to assume that if we are considering $\rho_0$ as being of type $\Cal D$ then $\Cal D$ is defined using $\Cal O$-algebras where $\Cal O\supseteq\Cal O_{f,\lambda}$ is an unramified extension whose residue field is $k$. (Although this condition is unnecessary, it is convenient to use $\lambda$ as the uniformizer for $\Cal O$.) Finally we assume that $\rho_{f,\lambda}$\linebreak \eject \noindent itself is of type $\Cal D$. Again this is a slight abuse of terminology as we are really considering the extension of scalars $\rho_{f,\lambda}\mathop{\otimes}\limits_{\Cal O_{f,\lambda}}\Cal O$ and not $\rho_{f,\lambda}$ itself, but we will do this without further mention if the context makes it clear. (The analysis of this section actually applies to any characteristic zero lifting of $\rho_0$ but in all our applications we will be in the more restrictive context we have described here.) With these hypotheses there is a unique local homomorphism $R_\Cal D\rightarrow\Cal O$ of $\Cal O$-algebras which takes the universal deformation to (the class of) $\rho_{f,\lambda}.$ Let $\frak p_\Cal D=\ker:R_\Cal D\rightarrow\Cal O.$ Let $K$ be the field of fractions of $\Cal O$ and let $U_f=(K/\Cal O)^2$ with the Galois action taken from $\rho_{f,\lambda}$. Similarly, let $V_f=\roman{Ad}\rho_{f,\lambda}\otimes_\Cal OK/\Cal O\simeq(K/\Cal O)^4$ with the adjoint representation so that $$V_f\simeq W_f\oplus K/\Cal O$$ where $W_f$ has Galois action via $\roman{Sym}^2\rho_{f,\lambda}\otimes\det\rho^{-1}_{f,\lambda}$ and the action on the second factor is trivial. Then if $\rho_0$ is ordinary the filtration of $U_f$ under the $\roman{Ad}\rho$ action of $D_p$ induces one on $W_f$ which we write $0\subset W^0_f\subset W^1_f\subset W_f.$ Often to simplify the notation we will drop the index $f$ from $W^1_f,V_f$ etc. There is also a filtration on $W_{\lambda^n}=\{\ker\lambda^n:W_f\rightarrow W_f\}$ given by $W^i_{\lambda^n}=W^{\lambda^n}\cap W^i$ (compatible with our previous description for $n=1$). Likewise we write $V_{\lambda^n}$ for $\{\ker\lambda^n:V_f\rightarrow V_f\}.$ We now explain how to extend the definition of $H^1_\Cal D$ to give meaning to $H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_{\lambda^n})$ and $H^1_\Cal D(\Cal Q_\Sigma/\bold Q,V)$ and these are $\Cal O/\lambda^n$ and $\Cal O$-modules, respectively. In the case where $\rho_0$ is ordinary the definitions are the same with $V_{\lambda^n}$ or $V$ replacing $V_\lambda$ and $\Cal O/\lambda^n$ or $K/\Cal O$ replacing $k$. One checks easily that as $\Cal O$-modules $$H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_{\lambda^n})\simeq H^1_\Cal D(\bold Q_\Sigma/\bold Q,V)_{\lambda^n},\leqno(1.7)$$ where as usual the subscript $\lambda^n$ denotes the kernel of multiplication by $\lambda^n$. This just uses the divisibility of $H^0(\bold Q_\Sigma/\bold Q,V)$ and $H^0(\bold Q_p,W/W^0)$ in the strict case. In the Selmer case one checks that for $m\gt n$ the kernel of $$H^1(\bold Q^\roman{unr}_p,V_{\lambda^n}/W^0_{\lambda^n})\rightarrow H^1(\bold Q^\roman{unr}_p,V_{\lambda^m}/W^0_{\lambda^m})$$ has only the zero element fixed under $\roman{Gal}(\bold Q^\roman{unr}_p/\bold Q_p)$ and the ord case is similar. Checking conditions at $q\in\Cal M$ is dome with similar arguments. In the Selmer and strict cases we make analogous definitions with $W_{\lambda^n}$ in place of $V_{\lambda^n}$ and $W$ in place of $V$ and the analogue of (1.7) still holds. We now consider the case where $\rho_0$ is flat (but not ordinary). We claim first that there is a natural map of $\Cal O$-modules $$H^1(\bold Q_p,V_{\lambda_n})\rightarrow\roman{Ext}^1_{\Cal O[D_p]}(U_{\lambda^m},U_{\lambda^n})\leqno(1.8)$$ for each $m\ge n$ where the extensions are of $\Cal O$-modules with local Galois\linebreak action. To describe this suppose that $\alpha\in H^1(\bold Q_p,V_{\lambda^n}).$ Then we can associate to $\alpha$ a representation $\rho_\alpha:\roman{Gal}(\bar\bold Q_p/\bold Q_p)\rightarrow\roman{GL}_2(\Cal O_n[\varepsilon])$ (where $\Cal O_n[\varepsilon]=$\linebreak \eject \noindent$\Cal O[\varepsilon]/(\lambda^n\varepsilon,\varepsilon^2))$ which is an $\Cal O$-algebra deformation of $\rho_0$ (see the proof of Proposition 1.1 below). Let $E=\Cal O_n[\varepsilon]^2$ where the Galois action is via $\rho_\alpha.$ Then there is an exact sequence $$\matrix0&\longrightarrow&\varepsilon E/\lambda^m&\longrightarrow& E/\lambda^m&\longrightarrow&(E/\varepsilon)/\lambda^m&\longrightarrow&0\\ \ \\ &&|\wr&&&&|\wr\\ \ \\ &&U_{\lambda^n}&&&&U_{\lambda^m}\endmatrix$$ and hence an extension class in $\roman{Ext}^1(U_{\lambda^m},U_{\lambda^n}).$ One checks now that (1.8) is a map of $\Cal O$-modules. We define $H^1_f(\bold Q_p,V_{\lambda^n})$ to be the inverse image of $\roman{Ext}^1_\roman{fl}(U_{\lambda^n},U_{\lambda_n})$ under (1.8), i.e., those extensions which are already extensions in the category of finite flat group schemes $\bold Z_p.$ Observe that $\roman{Ext}^1_\roman{fl}(U_{\lambda^n},U_{\lambda^n})\cap\roman{Ext}^1_{\Cal O[D_p]}(U_{\lambda^n},U_{\lambda^n})$ is an $\Cal O$-module, so $H^1_\roman f(\bold Q_p,V_{\lambda^n})$ is seen to be an $\Cal O$-sub-module of $H^1(\bold Q_p,V_{\lambda_n}).$ We observe that our definition is equivalent to requiring that the classes in $H^1_\roman f(\bold Q_p,V_{\lambda^n})$ map under (1.8) to $\roman{Ext}^1_\roman{fl}(U_{\lambda^m},U_{\lambda^n})$ for all $m\ge n.$ For if $e_m$ is the extension class in $\roman{Ext}^1(U_{\lambda^m},U_{\lambda^n})$ then $e_m\hookrightarrow e_n\oplus U_{\lambda^m}$ as Galois-modules and we can apply results of [Ray1] to see that $e_m$ comes from a finite flat group scheme over $\bold Z_p$ if $e_n$ does. In the flat (non-ordinary) case $\rho_0|_{I_p}$ is determined by Raynaud's results as mentioned at the beginning of the chapter. It follows in particular that, since $\rho_0|_{D_p}$ is absolutely irreducible, $V(\bold Q_p=H^0(\bold Q_p,V)$ is divisible in this case\linebreak (in fact $V(\bold Q_p)\simeq KT/\Cal O).$ This $H^1(\bold Q_p,V_{\lambda^n})\simeq H^1(\bold Q_p,V)_{\lambda^n}$ and hence we can\linebreak define $$H^1_\roman f(\bold Q_p,V)=\bigcup_{n=1}^\infty H^1_\roman f(\bold Q_p,V_{\lambda^n}),$$ and we claim that $H^1_\roman f(\bold Q_p,V)_{\lambda^n}\simeq H^1_\roman f(\bold Q_p,V_{\lambda^n}).$ To see this we have to compare representations for $m\ge n,$ $$\matrix\rho_{n,m}:\roman{Gal}(\bar\bold Q_p/\bold Q_p)&\longrightarrow&\roman{GL}_2(\Cal O_n[\varepsilon]/\lambda^m)\\ \ \\ \Big\Vert&&\hskip0.25in\Big\downarrow{\scriptstyle\varphi_{m,n}}\\ \ \\ \hskip-0.03in\rho_{m,m}:\roman{Gal}(\bar\bold Q_p/\bold Q_p)&\longrightarrow&\roman{GL}_2(\Cal O_m[\varepsilon]/\lambda^m)\endmatrix$$ where $\rho_{n,m}$ and $\rho_{m,m}$ are obtained from $\alpha_n\in H^1(\bold Q_p,VX_{\lambda^n})$ and $\roman{im}(\alpha_n)\in H^1(\bold Q_p,V_{\lambda^m})$ and $\varphi_{m,n}:a+b\varepsilon\rightarrow a+\lambda^{m-n}b\varepsilon.$ By [Ram, Prop 1.1 and Lemma 2.1] if $\rho_{n,m}$ comes from a finite flat group scheme then so does $\rho_{m,m}$. Conversely $\varphi_{m,n}$ is injective and so $\rho_{n,m}$ comes from a finite flat group scheme if $\rho_{m,m}$ does; cf. [Ray1]. The definitions of $H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_{\lambda^n})$ and $H^1_\Cal D(\bold Q_\Sigma/\bold Q,V)$ now extend to the flat case and we note that (1.7) is also valid in the flat case. Still in the flat (non-ordinary) case we can again use the determination of $\rho_0|_{I_p}$ to see that $H^1(\bold Q_p,V)$ is divisible. For it is enough to check that $H^2(\bold Q_p,V_\lambda)=0$ and this follows by duality from the fact that $H^0(\bold Q_p,V^*_\lambda)=0$\linebreak \eject \noindent where $V^*_\lambda=\roman{Hom}(V_\lambda,\boldsymbol\mu_p)$ and $\boldsymbol\mu_p$ is the group of $p^\roman{th}$ roots of unity. (Again this follows from the explicit form of $\rho_0|_{{}_{\scriptstyle{D_p}}}$.) Much subtler is the fact that $H^1_\roman f(\bold Q_p,V)$ is divisible. This result is essentially due to Ramakrishna. For, using a local version of Proposition 1.1 below we have that $$\roman{Hom}_\Cal O(\frak p_R/\frak p^2_R,K/\Cal O)\simeq H^1_\roman f(\bold Q_p,V)$$ where $R$ is the universal local flat deformation ring for $\rho_0|_{D_p}$ and $\Cal O$-algebras. (This exists by Theorem 1.1 of [Ram] because $\rho_0|_{D_p}$ is absolutely irreducible.) Since $R\simeq R^\roman{fl}\mathop{\otimes}\limits_{W(k)}\Cal O$ where $R^\roman{fl}$ is the corresponding ring for $W(k)$-algebras the main theorem of [Ram, Th. 4.2] shows that $R$ is a power series ring and the divisibility of $H^1_\roman f(\bold Q_p,V)$ then follows. We refer to [Ram] for more details about $R^\roman{fl}.$ Next we need an analogue of (1.5) for $V$. Again this is a variant of standard results in deformation theory and is given (at least for $\Cal D=(\roman{ord},\Sigma,W(k),\phi)$ with some restriction on $\chi_1,\chi_2$ in i(a)) in [MT, Prop 25]. \ {\smc Proposition 1.2.} {\it Suppose that $\rho_{f,\lambda}$ is a deformation of $\rho_0$ of type\linebreak $\Cal D=(\cdot,\Sigma,\Cal O,\Cal M)$ with $\Cal O$ an unramified extension of $\Cal O_{f,\lambda}$. Then as $\Cal O$-modules $$\roman{Hom}_\Cal O(\frak p_\Cal D/\frak p^2_\Cal D,K/\Cal O)\simeq H^1_\Cal D(\bold Q_\Sigma/\bold Q,V).$$ Remark.} The isomorphism is functorial in an obvious way if one changes $\Cal D$ to a larger $\Cal D'$. \ {\it Proof.} We will just describe the Selmer case with $\Cal M=\phi$ as the other cases use similar arguments. Suppose that $\alpha$ is a cocycle which represents a cohomology class in $H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,V_{\lambda^n}).$ Let $\Cal O_n[\varepsilon]$ denote the ring $\Cal O[\varepsilon]/(\lambda^n\varepsilon,\varepsilon^2).$ We can associate to $\alpha$ a representation $$\rho_\alpha:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\roman{GL}_2(\Cal O_n[\varepsilon])$$ as follows: set $\rho_\alpha(g)=\alpha(g)\rho_{f,\lambda}(g)$ where $\rho_{f,\lambda}(g),$ {\it a priori} in $\roman{GL}_2(\Cal O)$, is viewed\linebreak in $\roman{GL}_2(\Cal O_n[\varepsilon])$ via the natural mapping $\Cal O\rightarrow\Cal O_n[\varepsilon].$ Here a basis for $\Cal O^2$ is chosen so that the representation $\rho_{f,\lambda}$ on the decomposition group $D_p\subset\roman{Gal}(\bold Q_\Sigma/\bold Q)$ has the upper triangular form of (i)(a), and then $\alpha(g)\in V_{\lambda^n}$ is\linebreak viewed in $\roman{GL}_2(\Cal O_n[\varepsilon])$ by identifying $$V_{\lambda_n}\simeq\bigg\{\pmatrix1+y\varepsilon&x\varepsilon\\ z\varepsilon&1-t\varepsilon\endpmatrix\bigg\}=\{\ker:\roman{GL}_2(\Cal O_n[\varepsilon])\rightarrow\roman{GL}_2(\Cal O)\}.$$ Then $$W^0_{\lambda^n}=\bigg\{\pmatrix1&x\varepsilon\\ &1\endpmatrix\bigg\},$$ \eject $$\aligned W^1_{\lambda^n}&=\bigg\{\pmatrix 1+y\varepsilon&x\varepsilon\\ &1-y\varepsilon\endpmatrix\bigg\},\\ W_{\lambda^n}&=\bigg\{\pmatrix1+y\varepsilon&x\varepsilon\\ z\varepsilon&1-y\varepsilon\endpmatrix\bigg\},\endaligned$$ and $$V^1_{\lambda^n}=\bigg\{\pmatrix1+y\varepsilon&x\varepsilon\\ &1-t\varepsilon\endpmatrix\bigg\}.$$ One checks readily that $\rho_\alpha$ is a continuous homomorphism and that the deformation $[\rho_\alpha]$ is unchanged if we add a coboundary to $\alpha.$ We need to check that $[\rho_\alpha]$ is a Selmer deformation. Let $\Cal H=\mathbreak\roman{Gal}(\bar\bold Q_p/\bold Q^\roman{unr}_p)$ and $\Cal G=\roman{Gal}(\bold Q^\roman{unr}_p/\bold Q_p).$ Consider the exact sequence of $\Cal O[\Cal G]$-modules $$0\rightarrow(V^1_{\lambda^n}/W^0_{\lambda^n})^\Cal H\rightarrow(V_{\lambda^n}/W^0_{\lambda^n})^\Cal H\rightarrow X\rightarrow0$$ where $X$ is a submodule of $(V_{\lambda^n}/V^1_{\lambda^n})^\Cal H.$ Since the action of $_p$ on $V_{\lambda^n}/V^1_{\lambda^n}$ is\linebreak via a character which is nontrivial mod $\lambda$ (it equals $\chi_2\chi_1^{-1}$ mod $\lambda$ and $\chi_1\not\equiv\chi_2),$ we see that $X^\Cal G=0$ and $H^1(\Cal G,X)=0.$ Then we have an exact diagram of $\Cal O$-modules $$\matrix0\\ \Bigg\downarrow\\ \hskip9pt H^1(\Cal G,(V^1_{\lambda^n}/W^0_{\lambda^n})^\Cal H)\simeq H^1(\Cal G,(V_{\lambda^n}/W^0_{\lambda^n})^\Cal H)\\ \Bigg\downarrow\\ H^1(\bold Q_p,V_{\lambda^n}/W^0_{\lambda^n})\\ \Bigg\downarrow\\ H^1(\bold Q^\roman{unr}_p,V_{\lambda^n}/W^0_{\lambda^n})^\Cal G.\endmatrix$$ By hypothesis the image of $\alpha$ is zero in $H^1(\bold Q^\roman{unr}_p,V_{\lambda^n}/W^0_{\lambda^n})^\Cal G.$ Hence it is in the image of $H^1(\Cal G,(V^1_{\lambda^n}/W^0_{\lambda^n})^\Cal H).$ Thus we can assume that it is represented in $H^1(\bold Q_p,V_{\lambda^n}/W^0_{\lambda^n})$ by a cocycle, which maps $\Cal G$ to $V^1_{\lambda^n}/W^0_{\lambda^n};$ i.e.,\linebreak $f(D_p)\subset V^1_{\lambda^n}/W^0_{\lambda^n},f(I_p)=0.$ The difference between $f$ and the image of $\alpha$ is a coboundary $\{\sigma\mapsto\sigma\bar\mu-\bar\mu\}$ for some $u\in V_{\lambda^n}.$ By subtracting the coboundary $\{\sigma\mapsto\sigma u-u\}$ from $\alpha$ globally we get a new $\alpha$ such that $\alpha= f$ as cocycles mapping $\Cal G$ to $V^1_{\lambda^n}/W^0_{\lambda^n}.$ Thus $\alpha(D_p)\subset V^1_{\lambda^n},\alpha(I_p)\subset W^0_{\lambda^n}$ and it is now easy to check that $[\rho_\alpha]$ is a Selmer deformation of $\rho_0$. Since $[\rho_\alpha]$ is a Selmer deformation there is a unique map of local $\Cal O$-\linebreak algebras $\varphi_\alpha:R_\Cal D\rightarrow\Cal O_n[\varepsilon]$ inducing it. (If $\Cal M\ne\phi$ we must check the\linebreak \eject \noindent other conditions also.) Since $\rho_\alpha\equiv\rho_{f,\lambda}$ mod $\varepsilon$ we see that restricting $\varphi_\alpha$ to $\frak p_\Cal D$ gives a homomorphism of $\Cal O$-modules, $$\varphi_\alpha:\frak p_\Cal D\rightarrow\varepsilon.\Cal O/\lambda^n$$ such that $\varphi_\alpha(\frak p^2_\Cal D)=0.$ Thus we have defined a map $\varphi:\alpha\rightarrow\varphi_\alpha,$ $$\varphi:H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,V_{\lambda^n})\rightarrow\roman{Hom}_\Cal O(\frak p_\Cal D/\frak p^2_\Cal D,\Cal O/\lambda^n).$$ It is straightforward to check that this is a map of $\Cal O$-modules. To check the injectivity of $\varphi$ suppose that $\varphi_\alpha(\frak p_\Cal D)=0.$ Then $\varphi_\alpha$ factors through $R_\Cal D/\frak p_\Cal D\simeq\Cal O$ and being an $\Cal O$-algebra homomorphism this determines $\varphi_\alpha.$ Thus $[\rho_{f,\lambda}]=[\rho_\alpha].$ If $A^{-1}\rho_\alpha A=\rho_{f,\lambda}$ then $A$ mod $\varepsilon$ is seen to be central by Schur's lemma and so may be taken to be $I$. A simple calculation now shows that $\alpha$ is a coboundary. To see that $\varphi$ is surjective choose $$\Psi\in\roman{Hom}_\Cal O(\frak p_\Cal D/\frak p^2_\Cal D,\Cal O/\lambda^n).$$ Then $\rho_\Psi:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\roman{GL}_2(R_\Cal D/(\frak p^2_\Cal D,\ker\Psi))$ is induced by a representative of the universal deformation (chosen to equal $\rho_{f,\lambda}$ when reduced mod $\frak p_\Cal D$) and we define a map $\alpha_\Psi:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow V_{\lambda^n}$ by \vskip12pt \noindent$\alpha_\Psi(g)=\rho_\Psi(g)\rho_{f,\lambda}(g)^{-1}\in\left\{\matrix1+\frak p_\Cal D/(\frak p^2_\Cal D,\ker\Psi)&\frak p_\Cal D/(\frak p^2_\Cal D,\ker\Psi)\\ \ \\ \frak p_\Cal D/(\frak p^2_\Cal D,\ker\Psi)&1+\frak p_\Cal D/(\frak p^2_\Cal D,\ker\Psi)\endmatrix\right\}\subseteq V_{\lambda^n}\mathbreak$ \vskip6pt \noindent where $\rho_{f,\lambda}(g)$ is viewed in $\roman{GL}_2(R_\Cal D/(\frak p^2_\Cal D,\ker\Psi))$ via the structural map $\Cal O\rightarrow R_\Cal D$ ($R_\Cal D$ being an $\Cal O$-algebra and the structural map being local because of\linebreak the existence of a section). The right-hand inclusion comes from $$\matrix\frak p_D/(\frak p^2_D,\ker\Psi)&\mathop{\hookrightarrow}\limits^\Psi&\Cal O/\lambda^n&\mathop{\rightarrow}\limits^\sim&(\Cal O/\lambda^n)\cdot\varepsilon\\ &&1&\mapsto&\varepsilon.\endmatrix$$ Then $\alpha_\Psi$ is really seen to be a continuous cocycle whose cohomology class lies in $H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,V_{\lambda^n}).$ Finally $\varphi(\alpha_\Psi)=\Psi.$ Moreover, the constructions are\linebreak compatible with change of $n$, i.e., for $V_{\lambda^n}\!\hookrightarrow\!V_{\lambda^{n+1}}$ and $\lambda\!:\!\Cal O/\lambda^n\hookrightarrow\Cal O/\lambda^{n+1}.\hskip0.125in\square$\linebreak \ We now relate the local cohomology groups we have defined to the theory of Fontaine and in particular to the groups of Bloch-Kato [BK]. We will distinguish these by writing $H^1_F$ for the cohomology groups of Bloch-Kato. None of the results described in the rest of this section are used in the rest of the paper. They serve only to relate the Selmer groups we have defined (and later compute) to the more standard versions. Using the lattice associated to $\rho_{f,\lambda}$ we obtain also a lattice $T\simeq\Cal O^4$ with Galois action via $\roman{Ad}\ \rho_{f,\lambda}.$ Let $\Cal V=T\otimes_{\bold Z_p}\bold Q_p$ be associated vector space and identify $V$ with $\Cal V/T.$ Let $\roman{pr}:\Cal V\rightarrow V$ be\linebreak \eject \noindent the natural projection and define cohomology modules by $$\aligned H^1_F(\bold Q_p,\Cal V)&=\ker:H^1(\bold Q_p,\Cal V)\rightarrow H^1(\bold Q_p,\Cal V\mathop{\otimes}\limits_{\bold Q_p}B_\roman{crys}),\\ H^1_F(\bold Q_p,V)&=\roman{pr}\Big(H^1_F(\bold Q_p,\Cal V)\Big)\subset H^1(\bold Q_p,V),\\ H^1_F(\bold Q_p,V_{\lambda^n})&=(j_n)^{-1}\Big(H^1_F(\bold Q_p,V)\Big)\subset H^1(\bold Q_p,V_{\lambda^n}),\endaligned$$ where $j_n:V_{\lambda^n}\rightarrow V$ is the natural map and the two groups in the definition\linebreak of $H^1_F(\bold Q_p,\Cal V)$ are defined using continuous cochains. Similar definitions apply to $\Cal V^*=\roman{Hom}_{\bold Q_p}(\Cal V,\bold Q_p(1))$ and indeed to any finite-dimensional continuous $p$-adic representation space. The reader is cautioned that the definition of $H^1_F(\bold Q_p,V_{\lambda^n})$ is dependent on the lattice $T$ (or equivalently on $V$). Under certainly conditions Bloch and Kato show, using the theory of Fontaine and Lafaille, that this is independent of the lattice (see [BK, Lemmas 4.4 and 4.5]). In any case we will consider in what follows a fixed lattice associated to $\rho=\rho_{f,\lambda},\roman{Ad}\ \rho,$ etc. Henceforth we will only use the notation $H^1_F(\bold Q_p,-)$ when the underlying vector space is crystalline. \ {\smc Proposition 1.3.} (i) {\it If $\rho_0$ is flat but ordinary and $\rho_{f,\lambda}$ is associated\linebreak to a $p$-divisible group then for all} $n$ $$H^1_\roman f(\bold Q_p,V_{\lambda^n})=H^1_F(\bold Q_p,V_{\lambda^n}).$$ (ii) {\it If $\rho_{f,\lambda}$ is ordinary, $\det\rho_{f,\lambda}\Big|_{I_p}=\varepsilon$ and $\rho_{f,\lambda}$ is associated to a $p$-divisible group, then for all $n$}, $$H^1_F(\bold Q_p,V_{\lambda^n})\subseteq H^1_\roman{Se}(\bold Q_p,V_{\lambda^n}.$$ {\it Proof.} Beginning with (i), we define $H^1_\roman f(\bold Q_p,\Cal V)=\{\alpha\in H^1(\bold Q_p,\Cal V):\kappa(\alpha/\lambda^n)\in H^1_\roman f(\bold Q_p,V)$ for all $n\}$ where $\kappa:H^1(\bold Q_p,\Cal V)\rightarrow H^1(\bold Q_p,V).$ Then\linebreak we see that in case (i), $H^1_\roman f(\bold Q_p,V)$ is divisible. So it is enough to how that $$H^1_F(\bold Q_p,\Cal V)=H^1_\roman f(\bold Q_p,\Cal V).$$ We have to compare two constructions associated to a nonzero element $\alpha$ of $H^1(\bold Q_p,\Cal V).$ The first is to associate an extension $$0\rightarrow\Cal V\rightarrow E\mathop{\rightarrow}\limits^\delta K\rightarrow0\leqno(1.9)$$ of $K$-vector spaces with commuting continuous Galois action. If we fix an $e$ with $\delta(e)=1$ the action on $e$ is defined by $\sigma e=e+\hat\alpha(\sigma)$ with $\hat\alpha$ a cocycle representing $\alpha$. The second construction begins with the image of the subspace $\langle\alpha\rangle$ in $H^1(\bold Q_p,V).$ By the analogue of Proposition 1.2 in the local case, there is an $\Cal O$-module isomorphism $$H^1(\bold Q_p,V)\simeq\roman{Hom}_\Cal O(\frak p_R/\frak p^2_R,K/\Cal O)$$ \eject \noindent where $R$ is the universal deformation ring of $\rho_0$ viewed as a representation\linebreak of $\roman{Gal}(\bar\bold Q_p/\bold Q)$ on $\Cal O$-algebras and $\frak p_R$ is the ideal of $R$ corresponding to $\frak p_\Cal D$ (i.e., its inverse image in $R$). Since $\alpha\ne0,$ associated to $\langle\alpha\rangle$ is a quotient $\frak p_R/(\frak p^2_R,\frak a)$ of $\frak p_R/\frak p^2_R$ which is a free $\Cal O$-module of rank one. We then obtain a homomorphism $$\rho_\alpha:\roman{Gal}(\bar\bold Q_p/\bold Q_p)\rightarrow\roman{GL}_2\Big(R/(\frak p^2_R,\frak a)\Big)$$ induced from the universal deformation (we pick a representation in the universal class). This is associated to an $\Cal O$-module of rank 4 which tensored with $K$ gives a $K$-vector space $E'\simeq(K)^4$ which is an extension $$0\rightarrow\Cal U\rightarrow E'\rightarrow\Cal U\rightarrow0\leqno(1.10)$$ where $\Cal U\simeq K^2$ has the Galis representation $\rho_{f,\lambda}$ (viewed locally). In the first construction $\alpha\in H^1_F(\bold Q_p,\Cal V)$ if and only if the extension (1.9) is crystalline, as the extension given in (1.9) is a sum of copies of the more usual extension where $\bold Q_p$ replaces $K$ in (1.9). On the other hand $\langle\alpha\rangle\subseteq H^1_\roman f(\bold Q_p,\Cal V)$ if\linebreak and only if the second construction can be made through $R^\roman{fl},$ or equivalently if\linebreak and only if $E'$ is the representation associated to a $p$-divisible group. {\it A priori}, the representation associated to $\rho_\alpha$ only has the property that on all finite quotients it comes from a finite flat group scheme. However a theorem of Raynaud [Ray1] says that then $\rho_\alpha$ comes from a $p$-divisible group. For more details on $R^\roman{fl}$, the universal flat deformation ring of the local representation $\rho_0$, see [Ram].) Now the extension $E'$ comes from a $p$-divisible group if and only if it is crystalline; cf. [Fo, \S6]. So we have to show that (1.9) is crystalline if and only if (1.10) is crystalline. One obtains (1.10) from (1.9) as follows. We view $\Cal V$ as $\roman{Hom}_K(\Cal U,\Cal U)$ and let $$X=\ker:\{\roman{Hom}_K(\Cal U,\Cal U)\otimes\Cal U\rightarrow\Cal U\}$$ where the map is the natural one $f\otimes w\mapsto f(w).$ (All tensor products in this proof will be as $K$-vector spaces.) Then as $K[D_p]$-modules $$E'\simeq(E\otimes\Cal U)/X.$$ To check this, one calculates explicitly with the definition of the action on $E$ (given above on $e$) and on $E'$ (given in the proof of Proposition 1.1). It follows\linebreak from standard properties of crystalline representations that if $E$ is crystalline, so is $E\otimes\Cal U$ and also $E'$. Conversely, we can recover $E$ from $E'$ as follows.\linebreak Consider $E'\otimes\Cal U\simeq(E\otimes\Cal U\otimes\Cal U)/(X\otimes\Cal U).$ Then there is a natural map $\varphi:E\otimes(\det)\rightarrow E'\otimes\Cal U$ induced by the direct sum decomposition $\Cal U\otimes\Cal U\simeq(\det)\oplus\roman{Sym}^2\Cal U$. Here det denotes a 1-dimensional vector space over $K$ with Galois action via $\det\rho_{f,\lambda}.$ Now we claim that $\varphi$ is injective on $\Cal V\otimes(\det).$ For\linebreak \eject \noindent if $f\in\Cal V$ then $\varphi(f)=f\otimes(w_1\otimes w_2-w_2\otimes w_1)$ where $w_1,w_2$ are a basis for $\Cal U$ for which $w_1\wedge w_2=1$ in $\det\simeq K.$ So if $\varphi(f)\in X\otimes\Cal U$ then $$f(w_1)\otimes w_2-f(w_2)\otimes w_1=0\ \roman{in}\ \Cal U\otimes\Cal U.$$ But this is false unless $f(w_1)=f(w_2)=0$ whence $f=0$. So $\varphi$ is injective\linebreak on $\Cal V\otimes\det$ and if $\varphi$ itself were not injective then $E$ would split contradicting $\alpha\ne0.$ So $\varphi$ is injective and we have exhibited $E\otimes(\det)$ as a subrepresentation of $E'\otimes\Cal U$ which is crystalline. We deduce that $E$ is crystalline if $E'$ is. This completes the proof of (i). To prove (ii) we check first that $H^1_\roman{Se}(\bold Q_p,V_{\lambda^n})=j^{-1}_n\Big(H^1_\roman{Se}(\bold Q_p,V)\Big)$ (this was already used in (1.7)). We next have to show that $H^1_F(\bold Q_p,\!\Cal V)\subseteq H^1_\roman{Se}(\bold Q_p,\!\Cal V)$ where the latter is defined by $$H^1_\roman{Se}(\bold Q_p,\Cal V)=\ker:H^1(\bold Q_p,\Cal V)\rightarrow H^1(\bold Q^\roman{unr}_p,\Cal V/\Cal V^0)$$ with $\Cal V^0$ the subspace of $\Cal V$ on which $I_p$ acts via $\varepsilon$. But this follows from the computations in Corollary 3.8.4 of [BK]. Finally we observe that $$\roman{pr}\Big(H^1_\roman{Se}(\bold Q_p,\Cal V)\big)\subseteq H^1_\roman{Se}(\bold Q_p,V)$$ although the inclusion may be strict, and $$\roman{pr}\Big(H^1_F(\bold Q_p,\Cal V)\Big)=H^1_F(\bold Q_p,V)$$ by definition. This completes the proof.\hfill$\square$ \ These groups have the property that for $s\ge r,$ $$H^1(\bold Q_p,V_\lambda^r)\cap j^{-1}_{r,s}\Big( H^1_F(\bold Q_p,V_{\lambda^s})\Big)=H^1_F(\bold Q_p,V_{\lambda^r})\leqno(1.11)$$ where $j_{r,s}:V_{\lambda^r}\rightarrow V_{\lambda^s}$ is the natural injection. The same holds for $V^*_{\lambda^r}$ and $V^*_{\lambda^s}$ in place of $V_{\lambda^r}$ and $V_{\lambda^s}$ where $V^*_{\lambda^r}$ is defined by $$V^*_{\lambda^r}=\roman{Hom}(V_{\lambda^r},\boldsymbol\mu_{p^r})$$ and similarly for $V^*_{\lambda^s}.$ Both results are immediate from the definition (and indeed were part of the motivation for the definition). We also give a finite level version of a result of Bloch-Kato which is easily deduced from the vector space version. As before let $T\subset\Cal V$ be a Galois stable lattice so that $T\simeq\Cal O^4.$ Define $$H^1_F(\bold Q_p,T)=i^{-1}\Big(H^1_F(\bold Q_p,\Cal V)\Big)$$ under the natural inclusion $i:T\hookrightarrow\Cal V,$ and likewise for the dual lattice $T^*=\mathbreak\roman{Hom}_{\bold Z_p}(V,(\bold Q_p/\bold Z_p)(1))$ in $\Cal V^*$. (Here $\Cal V^*=\roman{Hom}(\Cal V,\bold Q_p(1));$ throughout this paper we use $M^*$ to denote a dual of $M$ with a Cartier twist.) Also write\linebreak \eject \noindent$\roman{pr}_n:T\rightarrow T/\lambda^n$ for the natural projection map, and for the mapping it\linebreak induces on cohomology. \ {\smc Proposition 1.4.} {\it If $\rho_{f,\lambda}$ is associated to a $p$-divisible group} ({\it the ordi-nary case is allowed}) {\it then} \ \hskip-12pt(i) $\roman{pr}_n\Big(H^1_F(\bold Q_p,T)\Big)=H^1_F(\bold Q_p,T/\lambda^n)$ {\it and similarly for} $T^*,T^*/\lambda^n.$ \ \hskip-15pt(ii)\hskip3pt $H^1_F(\bold Q_p,V_{\lambda^n})$ {\it is the orthogonal complement of $H^1_F(\bold Q_p,V^*_{\lambda^n})$ under Tate\linebreak${}$\hskip22pt local duality between $H^1(\bold Q_p,V_{\lambda^n})$ and $H^1(\bold Q_p,V^*_{\lambda^n})$ and similarly for $W_{\lambda^n}$\linebreak${}$\hskip22pt and $W^*_{\lambda^n}$ replacing $V_{\lambda^n}$ and $V^*_{\lambda^n}$. \ More generally these results hold for any crystalline representation $\Cal V'$ in place of $\Cal V$ and $\lambda'$ a uniformizer in $K'$ where $K'$ is any finite extension of $\bold Q_p$ with $K'\subset\roman{End}_{\roman{Gal}(\overline{\bold Q}_p/\bold Q_p)}\Cal V'$. \ Proof.} We first observe that $\roman{pr}_n(H^1_F(\bold Q_p,T))\subset H^1_F(\bold Q_p,T/\lambda^n).$ Now\linebreak from the construction we may identify $T/\lambda^n$ with $V_{\lambda^n}.$ A result of Bloch-Kato ([BK, Prop. 3.8]) says that $H^1_F(\bold Q_p,\Cal V)$ and $H^1_F(\bold Q_p,\Cal V^*)$ are orthogonal\linebreak complements under Tate local duality. It follows formally that $H^1_F(\bold Q_p,V^*_{\lambda^n})$ and $\roman{pr}_n(H^1_F(\bold Q_p,T))$ are orthogonal complements, so to prove the proposition it is enough to show that $$\#H^1_F(\bold Q_p,V^*_{\lambda^n})\#H^1_F(\bold Q_p,V_{\lambda^n})=\#H^1(\bold Q_p,V_{\lambda^n}).\leqno(1.12)$$ Now if $r=\dim_KH^1_F(\bold Q_p,\Cal V)$ and $s=\dim_KH^1_F(\bold Q_p,\Cal V^*)$ then $$r+s=\dim_KH^0(\bold Q_p,\Cal V)+\dim_KH^0(\bold Q_p,\Cal V^*)+\dim_K\Cal V.\leqno(1.13)$$ >From the definition, $$\#H^1_F(\bold Q_p,V_{\lambda^n})=\#(\Cal O/\lambda^n)^r\cdot\#\ker\{H^1(\bold Q_p,V_{\lambda^n})\rightarrow H^1(\bold Q_p,V)\}.\leqno(1.14)$$ The second factor is equal to $\#\{V(\bold Q_p)/\lambda^nV(\bold Q_p)\}.$ When we write $V(\bold Q_p)^\roman{div}$ for the maximal divisible subgroup of $V(\bold Q_p)$ this is the same as $$\aligned\#(V(\bold Q_p)/V(\bold Q_p)^\roman{div})/\lambda^n&=\#(V(\bold Q_p)/V(\bold Q_p)^\roman{div})_{\lambda^n}\\ &=\#V(\bold Q_p)_{\lambda^n}/\#(V(\bold Q_p)^\roman{div})_{\lambda^n}.\endaligned$$ Combining this with (1.14) gives $$\leqalignno{\#H^1_F(\bold Q_p,V_{\lambda^n})&=\#(\Cal O/\lambda^n)^r&(1.15)\cr&\hskip0.15in\cdot\#H^0(\bold Q_p,V_{\lambda^n})/\#(\Cal O/\lambda^n)^{\roman{dim}_KH^0(\bold Q_p,\Cal V)}.}$$ This, together with an analogous formula for $\#H^1_F(\bold Q_p,V^*_{\lambda^n})$ and (1.13), gives \vskip6pt \noindent$\#H^1_F(\bold Q_p,V^{\lambda^n})\#H^1_F(\bold Q_p,V^*_{\lambda^n})=\#(\Cal O/\lambda^n)^4\cdot\#H^0(\bold Q_p,V_{\lambda^n})\#H^0(\bold Q_p,V^*_{\lambda^n}).$\linebreak \eject \noindent As $\#H^0(\bold Q_p,V^*{\lambda^n})=\#H^2(\bold Q_p,V_{\lambda^n})$ the assertion of (1.12) now follows from the formula for the Euler characteristic of $V_{\lambda^n}$. The proof for $W_{\lambda^n},$ or indeed more generally for any crystalline representation, is the same. \hfill$\square$ \ We also give a characterization of the orthogonal complements of\linebreak $H^1_\roman{Se}(\bold Q_p,W_{\lambda^n})$ and $H^1_\roman{Se}(\bold Q_p,V_{\lambda^n}),$ under Tate's local duality. We write these duals as $H^1_\roman{Se^*}(\bold Q_p,W^*_{\lambda^n})$ and $H^1_\roman{Se^*}(\bold Q_p,V^*_{\lambda^n})$ respectively. Let $$\varphi_w:H^1(\bold Q_p,W^*_{\lambda^n})\rightarrow(\bold Q_p,W^*_{\lambda^n}/(W^*_{\lambda^n})^0)$$ be the natural map where $(W^*_{\lambda^n})^i$ is the orthogonal complement of $W^{1-i}_{\lambda^n}$ in $W^*_{\lambda^n}$, and let $X_{n,i}$ be defined as the image under the composite map $$\aligned X_{n,i}=\roman{im}:\bold Z^\times_p/(\bold Z^\times_p)^{p^n}\otimes\Cal O/\lambda^n&\rightarrow H^1(\bold Q_p,\boldsymbol\mu_{p^n}\otimes\Cal O/\lambda^n)\\ &\rightarrow H^1(\bold Q_p,W^*_{\lambda^n}/(W^*_{\lambda^n})^0)\endaligned$$ where in the middle term $\boldsymbol\mu_{p^n}\otimes\Cal O/\lambda^n$ is to be identified with $(W^*_{\lambda^n})^1/(W^*_{\lambda^n})^0.$ Similarly if we replace $W^*_{\lambda^n}$ by $V^*_{\lambda^n}$ we let $Y_{n,i}$ be the image of $\bold Z^\times_p/(\bold Z^\times_p)^{p^n}\otimes(\Cal O/\lambda^n)^2$ in $H^1(\bold Q_p,V^*_{\lambda^n}/(W^*_{\lambda^n})^0),$ and we replace $\varphi_w$ by the analogous map $\varphi_v.$ \ {\smc Proposition 1.5.} \ $$\aligned H^1_\roman{Se^*}(\bold Q_p,W^*_{\lambda^n})&=\varphi^{-1}_w(X_{n,i}),\\ H^1_{\roman{Se}^*}(\bold Q_p,V^*_{\lambda^n})&=\varphi^{-1}_v(Y_{n,i}).\endaligned$$ \ {\it Proof.} This can be checked by dualizing the sequence $$\aligned0&\rightarrow H^1_\roman{Str}(\bold Q_p,W_{\lambda^n})\rightarrow H^1_\roman{Se}(\bold Q_p,W_{\lambda^n})\\ &\rightarrow\ker:\{H^1(\bold Q_p,W_{\lambda^n}/(W_{\lambda^n})^0)\rightarrow H^1(\bold Q^\roman{unr}_p,W_{\lambda^n}/(W_{\lambda^n})^0\},\endaligned$$ where $H^1_\roman{str}(\bold Q_p,W_{\lambda^n})=\ker:H^1(\bold Q_p,W_{\lambda^n})\rightarrow H^1(\bold Q_p,W_{\lambda^n}/(W_{\lambda^n})^0).$ The first term is orthogonal to $\ker:H^1(\bold Q_p,W^*_{\lambda^n})\rightarrow H^1(\bold Q_p,W^*_{\lambda^n}/(W^*_{\lambda^n})^1).$ By the naturality of the cup product pairing with respect to quotients and subgroups the claim then reduces to the well known fact that under the cup product pairing $$H^1(\bold Q_p,\boldsymbol\mu_{p^n})\times H^1(\bold Q_p,\bold Z/p^n)\rightarrow\bold Z/p^n$$ the orthogonal complement of the unramified homomorphisms is the image of the units $\bold Z^\times_p/(\bold Z^\times_p)^{p^n}\rightarrow H^1(\bold Q_p,\boldsymbol\mu_{p^n}).$ The proof for $V_{\lambda^n}$ is essentially the same.\hfill$\square$ \eject \centerline{\bf 2. Some computations of cohomology groups} \ We now make some comparisons of orders of cohomology groups using\linebreak the theorems of Poitou and Tate. We retain the notation and conventions of Section 1 though it will be convenient to state the first two propositions in a more general context. Suppose that $$L=\prod L_q\subseteq\prod_{p\in\Sigma}H^1(\bold Q_q,X)$$ is a subgroup, where $X$ is a finite module for $\roman{Gal}(\bold Q_\Sigma/\bold Q)$ of $p$-power order. We define $L^*$ to be the orthogonal complement of $L$ under the perfect pairing (local Tate duality) $$\prod_{q\in\Sigma}H^1(\bold Q_q,X)\times\prod_{q\in\Sigma}H^1(\bold Q_q,X^*)\rightarrow\bold Q_p/\bold Z_p$$ where $X^*=\roman{Hom}(X,\boldsymbol\mu_{p^\infty}).$ Let $$\lambda_X:H^1(\bold Q_\Sigma/\bold Q,X)\rightarrow\prod_{q\in\Sigma}H^1(\bold Q_q,X)$$ be the localization map and similarly $\lambda_{X^*}$ for $X^*$. Then we set $$H^1_L(\bold Q_\Sigma/\bold Q,X)=\lambda^{-1}_X(L),\ \ H^1_{L^*}(\bold Q_\Sigma/\bold Q,X^*)=\lambda^{-1}_{X^*}(L^*).$$ The following result was suggested by a result of Greenberg (cf. [Gre1]) and is a simple consequence of the theorems of Poitou and Tate. Recall that $p$ is always assumed odd and that $p\in\Sigma$. \ {\smc Proposition 1.6.} $$\#H^1_L(\bold Q_\Sigma/\bold Q,X)/\#H^1_{L^*}(\bold Q_\Sigma/\bold Q,X^*)=h_\infty\prod_{q\in\Sigma}h_q$$ {\it where} $$\cases h_q&=\#H^0(\bold Q_q,X^*)/[H^1(\bold Q_q,X):L_q]\\ h_\infty&=\#H^0(\bold R,X^*)\#H^0(\bold Q,X)/\#H^0(\bold Q,X^*).\endcases$$ \ {\it Proof.}Adapting \!the \!exact \!sequence \!proof \!of \!Poitou \!and \!Tate(\!cf.[Mi2,\!Th.4.20])\linebreak we get a seven term exact sequence $$\matrix0&\longrightarrow&H^1_L(\bold Q_\Sigma/\bold Q,X)&\longrightarrow&H^1(\bold Q_\Sigma/\bold Q,X)&\longrightarrow&\prod\limits_{q\in\Sigma}H^1(\bold Q_q,X)/L_q\\ &&&&&&\Big\downarrow\\ &&\prod\limits_{q\in\Sigma}H^2(\bold Q_q,X)&\longleftarrow&H^2(\bold Q_\Sigma/\bold Q,X)&\longleftarrow&H^1_{L^*}(\bold Q_\Sigma/\bold Q,X^*)^\wedge\endmatrix$$ \hskip64pt$|$\vskip-7.15pt\hskip64.8pt$\rightarrow H^0(\bold Q_\Sigma/\bold Q,X^*)^\wedge\longrightarrow0,$ \eject \noindent where $M^\wedge=\roman{Hom}(M,\bold Q_p/\bold Z_p).$ Now using local duality and global Euler characteristics (cf. [Mi2, Cor. 2.3 and Th. 5.1]) we easily obtain the formula in the\linebreak proposition. We repeat that in the above proposition $X$ can be arbitrary of $p$-power order.\hfill$\square$ \ We wish to apply the proposition to investigate $H^1_\Cal D.$ Let $\Cal D=(\cdot,\Sigma,\Cal O,\Cal M)$ be a standard deformation theory as in Section 1 and define a corresponding group $L_n=L_{\Cal D,n}$ by setting $$L_{n,q}=\cases H^1(\bold Q_q,V_{\lambda^n})&\roman{for}\ q\ne p\ \roman{and}\ q\not\in\Cal M\\ H^1_{D_q}(\bold Q_q,V_{\lambda^n})&\roman{for}\ q\ne p\ \roman{and}\ q\in\Cal M\\ {H.}^1(\bold Q_p,V_{\lambda^n})&\roman{for}\ q=p.\endcases$$ Then $H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_{\lambda^n})=H^1_{L^n}(\bold Q_\Sigma/\bold Q,V_{\lambda^n})$ and we also define $$H^1_{\Cal D^*}(\bold Q_\Sigma/\bold Q,V^*_{\lambda^n})=H^1_{L^*_n}(\bold Q_\Sigma/\bold Q,V^*_{\lambda^n}).$$ We will adopt the convention implicit in the above that if we consider $\Sigma'\supset\Sigma$ then $H^1_\Cal D(\bold Q_{\Sigma'}/\bold Q,V_{\lambda^n})$ places no local restriction on the cohomology classes at primes $q\in\Sigma'-\Sigma.$ Thus in $H^1_{\Cal D^*}(\bold Q_{\Sigma'}/\bold Q,V^*_{\lambda^n})$ we will require (by duality) that the cohomology class be locally trivial at $q\in\Sigma'-\Sigma.$ We need now some estimates for the local cohomology groups. First we consider an arbitrary finite $\roman{Gal}(\bold Q_\Sigma/\bold Q)$-module $X$: \ {\smc Proposition 1.7.} {\it If} $q\not\in\Sigma$, {\it and $X$ is an arbitrary finite $\roman{Gal}(\bold Q_\Sigma/\bold Q)$-module of $p$-power order}, $$\#H^1_{L'}(\bold Q_{\Sigma\cup q}/\bold Q,X)/\#H^1_L(\bold Q_\Sigma/\bold Q,X)\le\#H^0(\bold Q_q,X^*)$$ {\it where $L'_\ell=L_\ell$ for $\ell\in\Sigma$ and $L'_q=H'(\bold Q_q,X)$. \ Proof.} Consider the short exact sequence of inflation-restriction: \vskip6pt \noindent\ $0\!\rightarrow\!H^1_L(\bold Q_\Sigma/\bold Q,X)\!\rightarrow\!H^1_{L'}(\bold Q_{\Sigma\cup q}/\bold Q,X)\!\rightarrow\!\roman{Hom}(\roman{Gal}(\bold Q_{\Sigma\cup q}/\bold Q_\Sigma),X)^{\roman{Gal}(\bold Q_\Sigma/\bold Q)}$ \ \hskip1.63in$\Bigg\downarrow$\hskip1.44in$\Bigg\downarrow$\vskip-0.5in\hskip3.1965in$\cap$\vskip0.45in \hskip1.29in$H^1(\bold Q^\roman{unr}_q,X)^{\roman{Gal}(\bold Q^\roman{unr}_q/\bold Q_q)}\mathop{\rightarrow}\limits^{\sim}\!H^1(\bold Q^\roman{unr}_q,X)^{\roman{Gal}(\bold Q^\roman{unr}_q/\bold Q_q)}$ \ \noindent The proposition follows when we note that \vskip6pt \hskip0.6in$\#H^0(\bold Q_q,X^*)\ \ =\ \ \#H^1(\bold Q^\roman{unr}_q,X)^{\roman{Gal}(\bold Q^\roman{unr}_q/\bold Q_q)}.$\hfill$\square$ \ Now we return to the study of $V_{\lambda^n}$ and $W_{\lambda^n}.$ \ {\smc Proposition 1.8.} {\it If} $q\in\Cal M\ (q\ne p)$ {\it and} $X=V_{\lambda^n}$ {\it then} $h_q=1.$ \eject {\it Proof.} This is a straightforward calculation. For example if $q$ is of type (A) then we have $$L_{n,q}=\ker\{H^1(\bold Q_q,V_{\lambda^n})\rightarrow H^1(\bold Q_q,W_{\lambda^n}/W^0_{\lambda^n})\oplus H^1(\bold Q^\roman{unr}_q,\Cal O/\lambda^n)\}.$$ Using the long exact sequence of cohomology associated to $$0\rightarrow W^0_{\lambda^n}\rightarrow W_{\lambda^n}\rightarrow W_{\lambda^n}/W^0_{\lambda^n}\rightarrow0$$ one obtains a formula for the order of $L_{n,q}$ in terms of $\#H^1(\bold Q_q,W_{\lambda^n}),\mathbreak\#H^i(\bold Q_q,W_{\lambda^n}/W^0_{\lambda^n})$ etc. Using local Euler characteristics these are easily re-\linebreak duced to ones involving $H^0(\bold Q_q,W^*_{\lambda^n})$ etc. and the result follows easily.\hfill$\square$ \ The calculation of $h_p$ is more delicate. We content ourselves with an inequality in some cases. \ {\smc Proposition 1.9.} (i) {\it If $X=V_{\lambda^n}$ then $$h_ph_\infty=\#(\Cal O/\lambda)^{3n}\#H^0(\bold Q_p,V^*_{\lambda^n})/\#H^0(\bold Q,V^*_{\lambda^n})$$ in the unrestricted case.} (ii) {\it If $X=V_{\lambda^n}$ then $$h_ph_\infty\le\#(\Cal O/\lambda)^n\#H^0(\bold Q_p,(V^\roman{ord}_{\lambda^n})^*)/\#H^0(\bold Q,W^*_{\lambda^n})$$ in the ordinary case.} (iii) {\it If $X=V_{\lambda^n}$ or $W_{\lambda^n}$ {\it then} $h_ph_\infty\le\#H^0(\bold Q_p,(W^0_{\lambda^n})^*)/\#H^0(\bold Q,W^*_{\lambda^n})$ in the Selmer case.} (iv) {\it If $X=V_{\lambda^n}$ or $W_{\lambda^n}$ then $h_ph_\infty=1$ in the strict case.} (v) {\it If $X=V_{\lambda^n}$ then $h_ph_\infty=1$ in the flat case.} (vi) {\it If $X=V_{\lambda^n}$ or $W_{\lambda^n}$ then $h_ph_\infty=1/\#H^0(\bold Q,V^*_{\lambda^n})$ if $L_{n,p}=\mathbreak H^1_F(\bold Q_p,X)$ and $\rho_{f,\lambda}$ arises from an ordinary $p$-divisible group. \ Proof.} Case (i) is trivial. Consider then case (ii) with $X=V_{\lambda^n.}$ We have a long exact sequence of cohomology associated to the exact sequence: $$0\rightarrow W^0_{\lambda^n}\rightarrow V_{\lambda^n}\rightarrow V_{\lambda^n}/W^0_{\lambda^n}\rightarrow0.\leqno(1.16)$$ In particular this gives the map $u$ in the diagram \vskip6pt\hskip1.87in$H^1(\bold Q_p,V_{\lambda^n})$ \vskip0.26in\hskip2.12in$u$ \vskip-0.36in\hskip2.25in$\vert$\vskip-0.02in\hskip2.223in$\Bigg\downarrow$ \vskip-0.6in\hskip2.6in$\diagdown$ \vskip-0.047in\hskip2.715in$\diagdown$ $\delta$ \vskip-0.047in\hskip2.829in$\diagdown$ \vskip-0.047in\hskip2.944in$\diagdown$ \vskip-0.06in\hskip3.039in$\searrow$ \vskip8pt \noindent$1\!\!\rightarrow\!\!Z\!\!=\!\!H^1\!(\!\bold Q^\roman{unr}_p\!/\!\bold Q_p,\!(\!V_{\lambda^n}\!/W^0_{\lambda^n}\!)\!^\Cal H)\!\!\rightarrow\!\!H^1\!(\!\bold Q_p,\!\!V_{\lambda^n}\!/W^0_{\lambda^n}\!)\!\!\rightarrow\!\!H^1\!(\!\bold Q^\roman{unr}_p\!,\!V_{\lambda^n}\!/W^0_{\lambda^n}\!)^\Cal G\!\!\rightarrow\!\!1\mathbreak$ \ \noindent where $\Cal G=\roman{Gal}(\bold Q^\roman{unr}_p/\bold Q_p),\Cal H=\roman{Gal}(\bar\bold Q_p/\bold Q^\roman{unr}_p)$ and $\delta$ is defined to make the triangle commute. Then writing $h_i(M)$ for $\#H^1(\bold Q_p,M)$ we have that $\#Z=$\linebreak \eject \noindent$h_0(V_{\lambda^n}/W^0_{\lambda^n})$ and $\#\roman{im}\ \delta\ge(\#\roman{im}\ u)/(\#Z).$ A simple calculation using the\linebreak long exact sequence associated to (1.16) gives $$\#\roman{im}\ u={h_1(V_{\lambda^n}/W^0_{\lambda^n})h_2(V_{\lambda^n})\over h_2(W^0_{\lambda^n})h_2(V_{\lambda^n}/W^0_{\lambda^n})}.\leqno(1.17)$$ Hence $$[H^1(\bold Q_p,V_{\lambda^n}):L_{n,p}]=\#\roman{im}\delta\ge\#(\Cal O/\lambda)^{3n}h_0(V^*_{\lambda^n})/h_0(W^0_{\lambda^n})^*.$$ The inequality in (iii) follows for $X=V_{\lambda^n}$ and the case $X=W_{\lambda^n}$ is similar. Case (ii) is similar. In case (iv) we just need $\#\roman{im}\ u$ which is given by (1.17) with $W_{\lambda^n}$ replacing $V_{\lambda^n}.$ In case (v) we have already observed in Section 1 that Raynaud's results imply that $\#H^0(\bold Q_p,V^*_{\lambda^n})=1$ in the flat case. Moreover $\#H^1_\roman f(\bold Q_p,V_{\lambda^n})$ can be computed to be $\#(\Cal O/\lambda)^{2n}$ from $$H^1_\roman f(\bold Q_p,V_{\lambda^n})\simeq H^1_\roman f(\bold Q_p,V)_{\lambda^n}\simeq\roman{Hom}_\Cal O(\frak p_R/\frak p^2_R,K/\Cal O)_{\lambda^n}$$ where $R$ is the universal local flat deformation ring of $\rho_0$ for $\Cal O$-algebras. Using the relation $R\simeq R^\roman{fl}\mathop{\otimes}\limits_{W(k)}\Cal O$ where $R^\roman{fl}$ is the corresponding ring for $W(k)$-algebras, and the main theorem of [Ram] (Theorem 4.2) which computes $R^\roman{fl},$ we can deduce the result. We now prove (vi). From the definitions $$\#H^1_F(\bold Q_p,V_{\lambda^n})=\cases(\#\Cal O/\lambda^n)^r\#H^0(\bold Q_p,W_{\lambda^n})&\roman{if}\ \rho_{f,\lambda}|_{D_p}\ \roman{does\ not\ split}\\ (\#\Cal O/\lambda^n)^r&\roman{if}\ \rho_{f,\lambda}|_{D_p}\ \roman{splits}\endcases$$ where $r=\dim_KH^1_F(\bold Q_p,\Cal V).$ This we can compute using the calculations in [BK, Cor. 3.8.4]. We find that $r=2$ in the non-split case and $r=3$ in the split case and (vi) follows easily.\hfill$\square$ \ \ \centerline{\bf 3. Some results on subgroups of $\roman{GL}_2(k)$} \ We now give two group-theoretic results which will not be used until\linebreak Chapter 3. Although these could be phrased in purely group-theoretic terms it will be more convenient to continue to work in the setting of Section 1, i.e., with $\rho_0$ as in (1.1) so that $\roman{im}\ \rho_0$ is a subgroup of $\roman{GL}_2(k)$ and $\roman{det}\ \rho_0$ is assumed odd. \ {\smc Lemma 1.10.} {\it If $\roman{im}\ \rho_0$ has order divisible by $p$ then}: \ (i) {\it It contains an element $\gamma_0$ of order $m\ge3$ with $(m,p)=1$ and $\gamma_0$ trivial on any abelian quotient of $\roman{im}\ \rho_0$.} (ii) {\it It contains an element $\rho_0(\sigma)$ with any prescribed image in the Sylow $2$-subgroup of $(\roman{im}\ \rho_0)/(\roman{im}\ \rho_0)'$ and with the ratio of the eigenvalues not equal to $\omega(\sigma)$. $($Here $(\roman{im}\ \rho_0)'$ denotes the derived subgroup of $(\roman{im}\ \rho_ 0)$.$)$} \eject {\it The same results hold if the image of the projective representation $\tilde\rho_0$ associated to $\rho_0$ is isomorphic to $A_4,S_4$ or $A_5$. \ Proof.} (i) Let $G=\roman{im}\ \rho_0$ and let $Z$ denote the center of $G$. Then we\linebreak have a surjection $G'\rightarrow(G/Z)'$ where the ${}'$ denotes the derived group. By Dickson's classification of the subgroups of $\roman{GL}_2(k)$ containing an element of order $p,(G/Z)$ is isomorphic to $\roman{PGL}_2(k')$ or $\roman{PSL}_2(k')$ for some finite field $k'$ of\linebreak characteristic $p$ or possibly to $A_5$ when $p=3$, cf. [Di, \S260]. In each case we can\linebreak find, and then lift to $G'$, an element of order $m$ with $(m,p)=1$ and $m\ge3,$ except possibly in the case $p=3$ and $\roman{PSL}_2(\bold F_3)\simeq A_4$ or $\roman{PGL}_2(\bold F_3)\simeq S_4.$ However in these cases $(G/Z)'$ has order divisible by 4 so the 2-Sylow subgroup of $G'$ has order greater than 2. Since it has at most one element of exact order 2 (the eigenvalues would both be $-1$ since it is in the kernel of the determinant and hence the element would be $-I$) it must also have an element of order 4. The argument in the $A_4,S_4$ and $A_5$ cases is similar. (ii) Since $\rho_0$ is assumed absolutely irreducible, $G=\roman{im}\ \rho_0$ has no fixed line.\linebreak We claim that the same then holds for the derived group $G'$ For otherwise\linebreak since $G'\triangleleft G$ we could obtain a second fixed line by taking $\langle gv\rangle$ where $\langle v\rangle$ is the\linebreak original fixed line and $g$ is a suitable element of $G$. Thus $G'$ would be contained\linebreak in the group of diagonal matrices for a suitable basis and it would be\linebreak central in which case $G$ would be abelian or its normalizer in $\roman{GL}_2(k)$, and hence also $G$, would have order prime to $p$. Since neither of these possibilities is allowed, $G'$ has no fixed line. By Dickson's classification of the subgroups of $\roman{GL}_2(k)$ containing an element of order $p$ the image of $\roman{im}\ \rho_0$ in $\roman{PGL}_2(k)$ is isomorphic to $\roman{PGL}_2(k')$\linebreak or $\roman{PSL}_2(k')$ for some finite field $k'$ of characteristic $p$ or possibly to $A_5$ when $p=3.$ The only one of these with a quotient group of order $p$ is $\roman{PSL}_2(\bold F_3)$ when $p=3$. It follows that $p\nmid[G:G']$ except in this one case which we treat separately. So assuming now that $p\nmid[G:G']$ we see that $G'$ contains a non-\linebreak trivial unipotent element $u$. Since $G'$ has no fixed line there must be another noncommuting unipotent element $v$ in $G'$. Pick a basis for $\rho_0|_{G'}$ consisting of their fixed vectors. Then let $\tau$ be an element of $\roman{Gal}(\bold Q_\Sigma/\bold Q)$ for which the image of $\rho_0(\tau)$ in $G/G'$ is prescribed and let $\rho_0(\tau)=(\!{a\atop c}{b\atop d}\!)$. Then $$\delta=\pmatrix a&b\\ c&d\endpmatrix\pmatrix1&s\alpha\\ {}&1\endpmatrix\pmatrix1&{}\\ r\beta&1\endpmatrix$$ has $\det\ (\delta)=\det\rho_0(\tau)$ and $\roman{trace}\ \delta=s\alpha(ra\beta+c)+br\beta+a+d.$ Since $p\ge3$ we can choose this trace to avoid any two given values (by varying $s$) unless $ra\beta+c=0$ for all $r$. But $ra\beta+c$ cannot be zero for all $r$ as otherwise $a=c=0.$ So we can find a $\delta$ for which the ratio of the eigenvalues is not $\omega(\tau),\det(\delta)$ being, of course, fixed. \eject Now suppose that $\roman{im}\ \rho_0$ does not have order divisible by $p$ but that the associated projective representation $\widetilde{\hskip1pt\rho_0{}}$ has image isomorphic to $S_4$ or $A_5$, so necessarily $p\ne3.$ Pick an element $\tau$ such that the image of $\rho_0(\tau)$ in $G/G'$ is any prescribed class. Since this fixes both $\det\rho_0(\tau)$ and $\omega(\tau)$ we have to show that we can avoid at most two particular values of the trace for $\tau$. To achieve this we can adapt our first choice of $\tau$ by multiplying by any element og $G'$. So pick $\sigma\in G'$ as in (i) which we can assume in these two cases has order 3. Pick a basis for $\rho_0$, by expending scalars if necessary, so that $\sigma\mapsto({\alpha\atop{}}{{}\atop{\alpha^{-1}}}).$ Then one\linebreak checks easily that if $\rho_0(\tau)=(\!{a\atop c}{b\atop d}\!)$ we cannot have the traces of all of $\tau,\sigma\tau$ and $\sigma^2\tau$ lying in a set of the form $\{\mp t\}$ unless $a=d=0$. However we can ensure that $\rho_0(\tau)$ does not satisfy this by first multiplying $\tau$ by a suitable element of $G'$ since $G'$ is not contained in the diagonal matrices (it is not abelian). In the $A_4$ case, and in the $\roman{PSL}_2(\bold F_3)\simeq A_4$ case when $p=3,$ we use a different argument. In both cases we find that the 2-Sylow subgroup of $G/G'$ is generated by an element $z$ in the centre of $G$. Either a power of $z$ is a suitable candidate for $\rho_0(\sigma)$ or else we must multiply the power of $z$ by an element of $G'$, the ratio of whose eigenvalues is not equal to 1. Such an element exists because in $G'$ the only possible elements without this property are $\{\mp I\}$ (such elements necessary have determinant 1 and order prime to $p$) and we know that $\#G'\gt2$ as was noted in the proof of part (i).\hfill$\square$ \ {\it Remark.} By a well-known result on the finite subgroups of $\roman{PGL}_2(\overline{\bold F}_p)$ this lemma covers all $\rho_0$ whose images are absolutely irreducible and for which $\widetilde{\hskip1pt\rho_0{}}$ is not dihedral. \ Let $K_1$ be the splitting field of $\rho_0$. Then we can view $W_\lambda$ and $W^*_\lambda$ as $\roman{Gal}(K_1(\zeta_p)/\bold Q)$-modules. We need to analyze their cohomology. Recall that we are assuming that $\rho_0$ is absolutely irreducible. Let $\widetilde{\hskip1pt\rho_0{}}$ be the associated projective representation to $\roman{PGL}_2(k).$ \ The following proposition is based on the computations in [CPS]. \ {\smc Proposition 1.11.} {\it Suppose that $\rho_0$ is absolutely irreducible. Then $$H^1(K_1(\zeta_p)/\bold Q,W^*_\lambda)=0.$$ \ Proof.} If the image of $\rho_0$ has order prime to $p$ the lemma is trivial. The subgroups of $\roman{GL}_2(k)$ containing an element of order $p$ which are not contained in a Borel subgroup have been classified by Dickson [Di, \S260] or [Hu, II.8.27]. Their images inside $\roman{PGL}_2(k')$ where $k'$ is the quadratic extension of $k$ are conjugate to $\roman{PGL}_2(F)$ or $\roman{PSL}_2(F)$ for some subfield $F$ of $k'$, or they are isomorphic to one of the exceptional groups $A_4,S_4,A_5$. Assume then that the cohomology group $H^1(K_1(\zeta_p)/\bold Q,W^*_\lambda)\ne0.$ Then by considering the inflation-restriction sequence with respect to the normal\linebreak \eject \noindent subgroup $\roman{Gal}(K_1(\zeta_p)/K_1)$ we see that $\zeta_p\in K_1.$ Next, since the representation\linebreak is (absolutely) irreducible, the center $Z$ of $\roman{Gal}(K_1/\bold Q)$ is contained in the\linebreak diagonal matrices and so acts trivially on $W_\lambda$. So by considering the inflation-restriction sequence with respect to $Z$ we see that $Z$ acts trivially on $\zeta_p$ (and on $W^*_\lambda).$ So $\roman{Gal}(\bold Q(\zeta_p)/\bold Q)$ is a quotient of $\roman{Gal}(K_1/\bold Q)/Z.$ This rules out all cases when $p\ne3,$ and when $p=3$ we only have to consider the case where the image of the projective representation is isomporphic as a group to $\roman{PGL}_2(F)$ for some finite field of characteristic 3. (Note that $S_4\simeq\roman{PGL}_2(\bold F_3).)$ Extending scalars commutes with formation of duals and $H^1$, so we may assume without loss of generality $F\subseteq k$. If $p=3$ and $\#F\gt3$ then $H^1(\roman{PSL}_2(F),W_\lambda)=0$ by results of [CPS]. Then if $\widetilde{\hskip1pt\rho_0}$ is the projective\linebreak representation associated to $\rho_0$ suppose that $g^{-1}\roman{im}\ \widetilde{\hskip1pt\rho_0}g=\roman{PGL}_2(F)$ and let $H=g\roman{PSL}_2(F)g^{-1}.$ Then $W_\lambda\simeq W^*_\lambda$ over $H$ and $$H^1(H,W_\lambda)\mathop{\otimes}\limits_F\bar F\simeq H^1(g^{-1}Hg,g^{-1}(W_\lambda\mathop{\otimes}\limits_F\bar F))=0.\leqno(1.18)$$ We deduce also that $H^1(\roman{im}\ \rho_0,W^*_\lambda)=0.$ Finally we consider the case where $F=\bold F_3$. I am grateful to Taylor for the following argument. First we consider the action of $\roman{PSL}_2(\bold F_3)$ on $W_\lambda$ explicitly by considering the conjugation action on matrices $\{A\in M_2(\bold F_3):\roman{trace}\ A=0\}.$ One sees that no such matrix is fixed by all the elements of order 2, whence $$H^1(\roman{PSL}_2(\bold F_3),W_\lambda)\simeq H^1(\bold Z/3,(W_\lambda)^{C_2\times C_2})=0$$ where $C_2\times C_2$ denotes the normal subgroup of order 4 in $\roman{PSL}_2(\bold F_3)\simeq A_4.$ Next we verify that there is a unique copy of $A_4$ in $\roman{PGL}_2(\bar\bold F_3)$ up to conjugation.\linebreak For suppose that $A,B\in\roman{GL}_2(\bar\bold F_3)$ are such that $A^2=B^2=I$ with the images\linebreak of $A,B$ representing distinct nontrivial commuting elements of $\roman{PGL}_2(\bar\bold F_3).$ We can choose $A=(\!{1\atop0}{\hskip6.5pt0\atop-1}\!)$ by a suitable choice of basis, i.e., by a suitable conjugation. Then $B$ is diagonal or antidiagonal as it commutes with $A$ up to a scalar, and as $B,A$ are distinct in $\roman{PGL}_2(\overline{\bold F}_3)$ we have $B=(\!{0\atop a}{-a^{-1}\atop\hskip-3pt0}\!)$ for some $a$. By conjugating by a diagonal matrix (which does not change $A$) we can assume that $a=1$. The group generated by $\{A,B\}$ in $\roman{PGL}_2(\bold F_3)$ is its own centralizer so it has index at most 6 in its normalizer $N$. Since $N/\langle A,B\rangle\simeq S_3$ there is a unique subgroup of $N$ in which $\langle A,B\rangle$ has index 3 whence the image of the embedding of $A_4$ in $\roman{PGL}_2(\bar\bold F_3)$ is indeed unique (up to conjugation). So arguing as in (1.18) by extending scalars we see that $H^1(\roman{im}\ \rho_0,W^*_\lambda)=0$ when $F=\bold F_3$ also.\hfill$\square$ \ The following lemma was pointed out to me by Taylor. It permits most dihedral cases to be covered by the methods of Chapter 3 and [TW]. \ {\smc Lemma 1.12.} {\it Suppose that $\rho_0$ is absolutely irreducible and that} \ \hskip-12pt(a) $\tilde\rho_0$ {\it is dihedral}\ ({\it the case where the image is $\bold Z/2\times\bold Z/2$ is allowed}), \eject \hskip-12pt(b) $\rho_0|_L$ {\it is absolutely irreducible where $L=\bold Q\Big(\sqrt{(-1)^{(p-1)/2}p}\Big)$.} \ \noindent{\it Then for any positive integer $n$ and any irreducible Galois stable subspace $X$ of $W_\lambda\otimes\bar k$ there exists an element $\sigma\in\roman{Gal}(\bar\bold Q/\bold Q)$ such that} \ \hskip-12pt(i) $\tilde\rho_0(\sigma)\ne1,$ \ \hskip-15pt(ii) $\sigma$ {\it fixes} $\bold Q(\zeta_ {p^n}),$ \ \hskip-18pt(iii) $\sigma$ {\it has an eigenvalue $1$ on $X$. \ Proof.} If $\tilde\rho_0$ is dihedral then $\rho_0\otimes\bar k=\roman{Ind}^G_H\chi$ for some $H$ of index 2 in $G$,\linebreak where $G=\roman{Gal}(K_1/\bold Q).$ (As before, $K_1$ is the splitting field of $\rho_0$.) Here $H$ can be taken as the full inverse image of any of the normal subgroups of index 2 defining the dihedral group. Then $W_\lambda\otimes\bar k\simeq\delta\oplus\roman{Ind}^G_H(\chi/\chi')$ where $\delta$ is the quadratic character $G\rightarrow G/H$ and $\chi'$ is the conjugate of $\chi$ by any element of $G-H$. Note that $\chi\ne\chi'$ since $H$ has nontrivial image in $\roman{PGL}_2(\bar k).$ To find a $\sigma$ such that $\delta(\sigma)=1$ and conditions (i) and (ii) hold, observe that $M(\zeta_{p^n})$ is abelian where $M$ is the quadratic field associated to $\delta$. So conditions (i) and (ii) can be satisfied if $\tilde\rho_0$ is non-abelian. If $\tilde\rho_0$ is abelian (i.e., the image has the form $\bold Z/2\times\bold Z/2),$ then we use hypothesis (b). If $\roman{Ind}^G_H(\chi/\chi')$ is irreducible over $\bar k$ then $W_\lambda\otimes\bar k$ is a sum of three distinct quadratic characters, none of which is the quadratic character associated to $L$, and we can repeat the argument by changing the choice of $H$ for the other two characters. If $X=\roman{Ind}^G_H(\chi/\chi')\otimes\bar k$ is absolutely irreducible then pick any $\sigma\in G-H.$ This satisfies (i) and can be made to satisfy (ii) if (b) holds. Finally, since $\sigma\in G-H$ we see that $\sigma$ has trace zero and $\sigma^2=1$ in its action on $X$. Thus it has an eigenvalue equal to 1.\hfill$\square$ \ \ \centerline{\bf Chapter 2} \ In this chapter we study the Hecke rings. In the first section we recall some of the well-known properties of these rings and especially the Goren-\linebreak stein property whose proof is rather technical, depending on a characteristic\linebreak $p$ version of the $q$-expansion principle. In the second section we compute the relations between the Hecke rings as the level is augmented. The purpose is to find the change in the $\eta$-invariant as the level increases. In the third section we state the conjecture relating the deformation rings of Chapter 1 and the Hecke rings. Finally we end with the critical step of showing that if the conjecture is true at a minimal level then it is true at all levels. By the results of the appendix the conjecture is equivalent to the\linebreak \eject \noindent equality of the $\eta$-invariant for the Hecke rings and the $\frak p/\frak p^2$-invariant for the deformation rings. In Chapter 2, Section 2, we compute the change in the $\eta$-invariant and in Chapter 1, Section 1, we estimated the change in the $\frak p/\frak p^2$-invariant. \ \centerline{\bf 1. The Gorenstein property} \ For any positive integer $N$ let $X_1(N)=X_1(N)_{/\bold Q}$ be the modular curve over $\bold Q$ corresponding to the group $\Gamma_1(N)$ and let $J_1(N)$ be its Jacobian. Let $\bold T_1(N)$ be the ring of endomorphisms of $J_1(N)$ which is generated over $\bold Z$ by the standard Hecke operators $\{T_l=T_{l*}\ \roman{for}\ l\nmid N,U_q=U_{q*}\ \roman{for}\ q|N,\langle a\rangle=\langle a\rangle_*\mathbreak\ \roman{for}\ (a,N)=1\}.$ For precise definitions of these see [MW1, Ch. 2,\S5]. In particular if one identifies the cotangent space of $J_1(N)(\bold C)$ with the space of cusp forms of weight 2 on $\Gamma_1(N)$ then the action induced by $\bold T_1(N)$ is the usual one on cusp forms. We let $\Delta=\{\langle a\rangle:(a,N)=1\}$. The group $(\bold Z/N\bold Z)^*$ acts naturally on $X_1(N)$ via $\Delta$ and for any sub-\linebreak group $H\subseteq(\bold Z/N\bold Z)^*$ we let $X_H(N)=X_H(N)_{/\bold Q}$ be the quotient $X_1(N)/H$. Thus for $H=(\bold Z/N\bold Z)^*$ we have $X_H(N)=X_0(N)$ corresponding to the group $\Gamma_0(N)$. In Section 2 it will sometimes be convenient to assume that $H$ decomposes as a product $H=\prod H_q$ in $(\bold Z/N\bold Z)^*\simeq\prod(\bold Z/q^r\bold Z)^*$ where the product\linebreak is over the distinct prime powers dividing $N$. We let $J_H(N)$ denote the Ja-\linebreak cobian of $X_H(N)$ and note that the above Hecke operators act naturally on $J_H(N)$ also. The ring generated by these Hecke operators is denoted $\bold T_H(N)$ and sometimes, if $H$ and $N$ are clear from the context, we addreviate this\linebreak to $\bold T$. Let $p$ be a prime $\ge3.$ Let $\frak m$ be a maximal ideal of $\bold T=\bold T_H(N)$ with $p\in\frak m$. Then associated to $\frak m$ there is a continuous odd semisimple Galois representation $\rho_{\frak m}$,$$\rho_{\frak m}:\roman{Gal}(\overline{\bold Q}/\bold Q)\rightarrow\roman{GL}_2(\bold T/\frak m)\leqno(2.1)$$ unramified outside $Np$ which satisfies $$\roman{trace}\ \rho_{\frak m}(\roman{Frob}\ q)=T_q,\ \det\rho_{\frak m}(\roman{Frob}\ q)=\langle q\rangle q$$ for each prime $q\nmid Np.$ Here Frob $q$ denotes a Frobenius at $q$ in Gal$(\overline{\bold Q}/\bold Q)$.\linebreak The representation $\rho_{\frak m}$ is unique up to isomorphism. If $p\nmid N$ (resp. $p|N)$ we say that $\frak m$ is ordinary if $T_p\notin\frak m$ (resp. $U_p\notin\frak m$). This implies (cf., for example, theorem 2 of [Wi1]) that for our fixed decomposition group $D_p$ at $p$, $$\rho_{\frak m}\Big|_{D_p}\approx\pmatrix\chi_1&*\\ 0&\chi_2\endpmatrix$$ for a suitable choic of basis, with $\chi_2$ unramified and $\chi_2(\roman{Frob}\ p)=T_p$ mod\linebreak$\frak m$ (resp. equal to $U_p$). In particular $\rho_{\frak m}$ is ordinary in the sense of Chapter 1\linebreak \eject \noindent provided $\chi_1\ne\chi_2$. We will say that $\frak m$ is $D_p$-distinguished if $\frak m$ is ordinary and $\chi_1\ne\chi_2$. (In practice $\chi_1$ is usually ramified so this imposes no extra condition.) We caution the reader that if $\rho_{\frak m}$ is ordinary in the sense of Chapter 1 then we can only conclude that $\frak m$ is $D_p$-distinguished if $p\nmid N$. Let $\bold T_{\frak m}$ denote the completion of $\bold T$ at $\frak m$ so that $\bold T_{\frak m}$ is a direct factor of the complete semi-local ring $\bold T_p=\bold T\otimes\bold Z_p.$ Let $\Cal D$ be the points of the associated $\frak m$-divisible group $$\Cal D=J_H(N)(\overline{\bold Q})_{\frak m}\simeq J_H(N)(\overline{\bold Q})_{p^\infty}\mathop{\otimes}\limits_{\bold T_p}\bold T_{\frak m}.$$ It is known that $\hat\Cal D=\roman{Hom}_{\bold Z_p}(\Cal D,\bold Q_p/\bold Z_p)$ is a rank 2 $\bold T_{\frak m}$-module, i.e., that $\hat\Cal D\mathop{\otimes}\limits_{\bold Z_p}\bold Q_p\simeq(\bold T_{\frak m}\mathop{\otimes}\limits_{\bold Z_p}\bold Q_p)^2.$ Briefly it is enough to show that $H^1(X_H(N),\bold C)$ is free of rank 2 over $\bold T\otimes\bold C$ and this reduces to showing that $S_2(\Gamma_H(N),\bold C),$\linebreak the space of cusp forms of weight 2 on $\Gamma_H(N)$, is free of rank 1 over $\bold T\otimes\bold C$. One shows then that if $\{f_1,\dots,f_r\}$ is a complete set of normalized newforms in $S_2(\Gamma_H(N),\bold C)$ of levels $m_1,\dots,m_r$ then if we set $d_i=N/m_i$, the form $f=\Sigma f_i(d_iz)$ is a basis vector of $S_2(\Gamma_H(N),\bold C)$ as a $\bold T\otimes\bold C$-module. If $\frak m$ is ordinary then Theorem 2 of [Wi1], itself a straightforward generalization of Proposition 2 and (11) of [MW2], shows that (for our fixed decomposition group $D_p$) there is a filtration of $\Cal D$ by Pontrjagin duals of rank 1 $\bold T_{\frak m}$-modules (in the sense explained above) $$0\rightarrow\Cal D^0\rightarrow\Cal D\rightarrow\Cal D^E\rightarrow 0\leqno(2.2)$$ where $\Cal D^0$ is stable under $D_p$ and the induced action on $\Cal D^E$ is unramified with Frob $p=U_p$ on it if $p|N$ and Frob $p$ equal to the unit root of $x^2-T_px+p\langle p\rangle\mathbreak=0$ in $\bold T_{\frak m}$ if $p\nmid N$. We can describe $\Cal D^0$ and $\Cal D^E$ as follows. Pick a $\sigma\in\mathbreak I_p$ which induces a generator of Gal$(\bold Q_p(\zeta_{Np^\infty})/\bold Q_p(\zeta_{Np})).$ Let $\varepsilon:D_p\rightarrow\bold Z_p^\times$ be the cyclotomic character. Then $\Cal D^0=\ker(\sigma-\varepsilon(\sigma))^\roman{div}$, the kernel being\linebreak taken inside $\Cal D$ and `div' meaning the maximal divisible subgroup. Although\linebreak in [Wi1] this filtration is given only for a factor $A_f$ of $J_1(N)$ it is easy to\linebreak deduce the result for $J_H(N)$ itself. We note that this filtration is defined without reference to characteristic $p$ and also that if $\frak m$ is $D_p$-distinguished, $\Cal D^0$ (resp. $\Cal D^E$) can be described as the maximal submodule on which $\sigma-\tilde\chi_1(\sigma)$ is topologically nilpotent for all $\sigma\in\roman{Gal}(\overline{\bold Q}_p/\bold Q_p)$ (resp. quotient on which $\sigma-\tilde\chi_2(\sigma)$ is topologically nilpotent for all $\sigma\in\roman{Gal}(\overline{\bold Q}_p/\bold Q_p))$, where $\tilde\chi_i(\sigma)$ is any lifting of $\chi_i(\sigma)$ to $\bold T_{\frak m}$. The Weil pairing $\langle\ ,\ \rangle$ on $J_H(N)(\overline{\bold Q})_{p^M}$ satisfies the relation $\langle t_*x,y\rangle=\langle x,t^*y\rangle$ for any Hecke operator $t$. It is more convenient to use an adapted pairing defined as follows. Let $w_\zeta$, for $\zeta$ a primitive $N^\roman{th}$ root of 1, be the\linebreak involution of $X_1(N)_{/\bold Q(\zeta)}$ defined in [MW1, p. 235]. This induces an involution\linebreak of $X_H(N)_{/\bold Q(\zeta)}$ also. Then we can define a new pairing [ , ] by setting (for a\linebreak \eject \noindent fixed choice of $\zeta$) $$[x,y]=\langle x,w_\zeta y\rangle.\leqno(2.3)$$ Then $[t_*x,y]=[x,t_*y]$ for all Hecke operators $t$. In particular we obtain an induced pairing on $\Cal D_{p^M}$. The following theorem is the crucial result of this section. It was first proved by Mazur in the case of prime level [Ma2]. It has since been generalized in [Ti1], [Ri1] [M Ri], [Gro] and [E1], but the fundamental argument remains that of [Ma2]. For a summary see [E1, \S9]. However some of the cases we need are not covered in these accounts and we will present these here. \vskip6pt {\smc Theorem} 2.1. (i) {\it If $p\nmid N$ and $\rho_{\frak m}$ is irreducible then $$J_H(N)(\overline{\bold Q})[\frak m]\simeq(\bold T/\frak m)^2.$$} (ii) {\it If $p\nmid N$ and $\rho_{\frak m}$ is irreducible and $\frak m$ is $D_p$-distinguished then $$J_H(Np)(\overline{\bold Q})[\frak m]\simeq(\bold T/\frak m)^2.$$} ({\it In case} (ii) $\frak m$ {\it is a maximal ideal of $\bold T=\bold T_H(Np).)$} \vskip6pt {\smc Corollary} 1. {\it In case} (i), $J_H\widehat{(N)(\overline{\bold Q})}_{\frak m}\simeq\bold T_{\frak m}^2$ {\it and} $\roman{Ta}_{\frak m}\Big(J_H(N)(\overline{\bold Q})\Big)\simeq\bold T_{\frak m}^2$. {\it In case} (ii), $J_H\widehat{(Np)(\overline{\bold Q})}_{\frak m}\simeq\bold T_{\frak m}^2$ {\it and} $\roman{Ta}_{\frak m}\Big(J_H(Np)(\overline{\bold Q})\Big)\simeq\bold T_{\frak m}^2$ ({\it where} $\bold T_{\frak m}=\bold T_H(Np)_{\frak m}$). \vskip6pt {\smc Corollary} 2. {\it In either of cases} (i) {\it or} (ii) $\bold T_{\frak m}$ {\it is a Gorenstein ring.} \vskip6pt In each case the first isomorphisms of Corollary 1 follow from the theorem together with the rank 2 result alluded to previously. Corrollary 2 and the second isomorphisms of corollory 1 then follow on applying duality (2.4). (In the proof and in all applications we will only use the notion of a Gorenstein $\bold Z_p$-algebra as defined in the appendix. For finite flat local $\bold Z_p$-algebras the notions of Gorenstein ring and Gorenstein $\bold Z_p$-algebra are the same.) Here $\roman{Ta}_{\frak m}\Big(J_H(N)(\overline{\bold Q})\Big)=\roman{Ta}_p\Big(J_H(N)(\overline{\bold Q})\Big)\mathop{\otimes}\limits_{\bold T_p}{\bold T_{\frak m}}$ is the $\frak m$-adic Tate module of $J_H(N).$ We should also point out that although Corollary 1 gives a representation from the $\frak m$-adic Tate module $$\rho=\rho_{\bold T_{\frak m}}:\roman{Gal}(\overline{\bold Q}/\bold Q)\rightarrow\roman{GL}_2(\bold T_{\frak m})$$ this can be constructed in a much more elementary way. (See [Ca3] for another argument.) For, the representation exists with $\bold T_{\frak m}\otimes\bold Q$ replacing $\bold T_{\frak m}$ when we use the fact that Hom$(\bold Q_p/\bold Z_p,\Cal D)\otimes\bold Q$ was free of rank 2. A standard argument\linebreak \eject \noindent using the Eichler-Shimura relations implies that this representation $\rho'$ with values in $\roman{GL}_2(\bold T_{\frak m}\otimes\bold Q)$ has the property that $$\roman{trace}\ \rho'(\roman{Frob}\ \ell)=T_\ell,\ \ \det\ \rho'(\roman{Frob}\ \ell)=\ell\langle\ell\rangle$$ for all $\ell\nmid Np.$ We can normalize this representation by picking a complex\linebreak conjugation $c$ and choosing a basis such that $\rho'(c)\!=\!\big({1\atop0}\ {\ \ 0\atop-1}\big)$, and then by picking\linebreak a $\tau$ for which $\rho'(\tau)=\big({a_\tau\atop c_\tau}\ {b_\tau\atop d_\tau}\big)$ with $b_\tau c_\tau\not\equiv0(\frak m)$ and by rescaling the basis so that $b_\tau=1.$ (Note that the explicit description of the traces shows that if $\rho_{\frak m}$ is also normalized so that $\rho_{\frak m}(c)=\big({1\atop0}\ {\ \ 0\atop-1}\big)$ then $b_\rho c_\tau\mod\frak m=b_{\tau,\frak m}c_{\tau,\frak m}$ where $\rho_{\frak m}(\tau)=\big({a_{\tau,\frak m}\atop c_{\tau,\frak m}}\ {b_{\tau,\frak m}\atop d_{\tau,\frak m}}\big).$ The existence of a $\tau$ such that $b_\tau c_\tau\not\equiv0(\frak m)$ comes from the irreducibility of $\rho_{\frak m}$.) With this normalization one checks that $\rho'$ actually takes values in the (closed) subring of $\bold T_{\frak m}$ generated over $\bold Z_p$ by the traces. One can even construct the representation directly from the representations in Theorem 0.1 using this ring which is reduced. This is the method of Carayol which requires also the characterization of $\rho$ by the traces and determinants (Theorem 1 of [Ca3]). One can also often interpret the $U_q$ operators in terms of $\rho$ for $q|N$ using the $\pi_q\simeq\pi(\sigma_q)$ theorem of Langlands (cf. [Ca1]) and the\linebreak $U_q$ operator in case (ii) using Theorem 2.1.4 of [Wi1]. \vskip6pt {\it Proof} ({\it of theorem}). The important technique for proving such multiplicity-one results is due to Mazur and is based on the $q$-expansion principle in characteristic $p$. Since the kernel of $J_H(N)(\overline{\bold Q})\rightarrow J_1(N)(\overline{\bold Q})$ is an abelian group on which Gal$(\overline{\bold Q}/\bold Q)$ acts through an abelian extension of $\bold Q$, the intersection with $\ker\frak m$ is trivial when $\rho_{\frak m}$ is irreducible. So it is enough to verify the theorem for $J_1(N)$ in part (i) (resp. $J_1(Np)$ in part (ii)). The method for part (i) was developed by Mazur in [Ma2, Ch. II, Prop. 14.2]. It was extended to the case of $\Gamma_0(N)$ in [Ri1, Th. 5.2] which summarizes Mazur's argument. The case of $\Gamma_1(N)$ is similar (cf. [E1, Th. 9.2]). Now consider case (ii). Let $\Delta_{(p)}=\{\langle a\rangle:a\equiv1(N)\}\subseteq\Delta$. Let us first\linebreak assume that $\Delta_{(p)}$ is nontrivial mod $\frak m$, i.e., that $\delta-1\!\notin\!\frak m$ for some $\delta\!\in\!\Delta_{(p)}$. This\linebreak case is essentially covered in [Ti1] (and also in [Gro]). We briefly review the argument for use later. Let $K=\bold Q_p(\zeta_p),\zeta_p$ being a primitive $p^\roman{th}$ root of unity, and let $\Cal O$ be the ring of integers of the completion of the maximal unramified extension of $K$. Using the