From weston@math.lsa.umich.edu Sun Feb 25 13:49:50 2001 Date: Wed, 21 Feb 2001 16:55:09 -0500 (EST) From: Tom Weston To: Barry Mazur Cc: colwell@its.caltech.edu, eisentra@math.berkeley.edu, grigorov@math.harvard.edu, jjlee@math.jhu.edu, mohit@mast.queensu.ca, jpineiro@gc.cuny.edu, sinan@math.berkeley.edu, howard@math.stanford.edu, long@math.psu.edu, apacetti@mail.ma.utexas.edu, msadykov@math.columbia.edu, mschein@its.caltech.edu, yt156@columbia.edu, jvoight@math.berkeley.edu Subject: Re: various I thought that it might be worthwhile to send this to everyone. > I currently have two questions: > > 1) In Student Project, what are the good points of choosing > the elliptic curve E=X_0(11) to show its (p-primary part of) > the Shafarevich-Tate group is trivial? I mean, which property > of X_0(11) simplifies our computation? > > 2)What is motive and why it is called motive, and some > examples? 1) As I see it, there are two advantages to working with the case of X_0(11) rather than the general case: i) Heegner points and Kato's euler system initially live on modular curves X_0(N). For applications to elliptic curves, you need to transfer them to the elliptic curve E via a modular parameterization X_0(N) -> E for some appropriate N. Although it is now known that such a map always exists (at least for E defined over the rationals), it is not such a simple thing; it is basically given by an appropriate modular form of level N and weight 2. X_0(11), however, is itself an elliptic curve (unlike most modular curves), so that the Heegner points initially live on X_0(11) and we only have to study them via the identity map X_0(11) -> X_0(11). This immediately eliminates one level of abstraction and complexity from the argument. ii) X_0(11) is a fairly simple (or at least well-understood) elliptic curve. It's the curve 11A1 in Cremona's tables, it's given by the equation y^2+y=x^3-x^2-10x-20, it has exactly five rational points... More to the point, it is very well-behaved mod p except for p=5 and p=11. Most treatments of these Euler systems explicitly avoid these bad primes, but in this case we aren't losing all that much by ignoring them. (And it's probably not so hard to treat the p=5 case as well, at least for Heegner points.) 2) I doubt that we'll need to deal with motives in any serious way at the Winter school, but I hope that the somewhat long-winded discussion below may help to clarify some of the ideas we'll be dealing with. Let me begin with an example. Let E be an elliptic curve defined over the rational numbers. For any prime p we can consider the Tate module T_p(E) (defined to be the inverse limit of the p^n torsion points E[p^n](\Qbar) for all n); it is a free Z_p-module of rank two and it inherits an action of the absolute Galois group G=Gal(Qbar/Q) of Q via the action on the torsion points. Now, although literally speaking the Tate modules for different choices of the prime p have no relation (after all, they are defined via entirely different sets of torsion points), they are nevertheless quite similar. Before I explain what I mean by this, let me quickly recall the idea of Frobenius elements in G. Let l be a prime. For every Galois extension F/Q which is unramified at l, the conjugacy class of Frobenius elements at l is simply the set of elements of Gal(F/Q) which are the l-power map modulo some prime of F above l. It will generally cause us no harm to pick one particular Frobenius element Fr_l at l. If further l is such that E has good reduction at l, the criterion of Neron-Ogg-Shaferevich guarantees that the action of G on T_p(E) is unramified at l, (at least if p doesn't equal l), so that we inherit an action of Fr_l on T_p(E). (Again, this is really only defined up to conjugation, but it won't matter below.) Now fix the prime p and pick another prime l so that E has good reduction at l and l does not equal p. We consider the reduction of E mod l; it is an elliptic curve over the finite field F_l. Like any variety in characteristic l, E mod l has a Frobenius morphism fr_l. We can use fr_l to yield an endomorphism of T_p(E mod l) as in Silverman (Chapter 3, Section 7), which we also write as fr_l. We now have two remarkable (but not so difficult) facts: a) there is an identification of T_p(E) with T_p(E mod l) so that Fr_l identifies with the inverse of fr_l. (Don't worry so much about the inverse; also, the particular identification depends on the choice of Fr_l, so it's ok that fr_l is completely canonical while Fr_l is not.) In particular, if we think of Fr_l^{-1} and fr_l as acting on free Z_p-modules of rank 2, they have precisely the same characteristic polynomial. (This is now independent of the choice of Fr_l in its conjugacy class.) b) It is shown in Silverman (Chapter 5, Section 2) that the characteristic polynomial of our fr_l acting on T_p(E mod l) is: P_l(T,E) = T^2-(a_l)T+l where a_l = l + 1 - #E(F_l). In particular, it has integral (rather than p-adic) coefficients and it is totally independent of the prime p (not equal to l) used to define it! If we combine these two facts, we get the following picture: the characteristic polynomial of Fr_l (or Fr_l^{-1}) acting on T_p(E) doesn't depend on the prime p. So even though the T_p(E) for different p appear quite different, the Galois actions are actually quite closely related. We can also now happily define the L-function L(E,s) = prod_{all l} P_l(E,p^{-s})^{-1} where we define P_l(E,T) as above for any p not equal to l. There is actually a way to see at least a little bit of a connection between these Tate modules as p varies. For this we need to consider the complex points E(C). Recall that by the standard uniformization theory we can define a holomorphic isomorphism E(C) = C/L for some lattice L; that is, L is a free Z-module of rank 2. Note also that we can use this to describe the torsion points: for any integer m, E(C)[m] = (m^{-1}L)/L inside of C/L. In particular, if we stare at this for a moment we realize that the p-adic Tate module T_p(E(C)) is naturally isomorphic to L tensored with Z_p (just think about the definition as an inverse limit). Thus the holomorphic isomorphism E(C) = C/L yields an isomorphism T_p(E) = L tensor Z_p for all p. We are thus led to following picture: we begin with our free rank 2 Z-module L. L itself has no action of the Galois group G, but if we tensor L with Z_p (for any p) we obtain a Galois action via the isomorphism with T_p(E). Furthermore, these Galois actions are all "compatible" in the sense (involving characteristic polynomials) discussed above. Roughly speaking, that's what a motive is. We should begin with some free Z-module of finite rank, call it M. M itself need have no Galois action, but for every prime p, M tensor Z_p should have a Galois action, and the characteristic polynomials of a Frobenius element at l should have integer coefficients and should be independent of the choice of p (not equal to l). In particular, we should be able to define a natural L-function L(M,s) as above. (I'm suppressing some additional structure related to Hodge theory here.) There is one last requirement for M to be a motive: it should "come from algebraic geometry" in some appropriate sense. This is the trickiest part of the theory, and really remains unresolved to this point. (There are many proposed definitions, none of which can be proven to behave well.) Roughly speaking, though, the basic example is the following: let X be a smooth, projective variety over Q. Let M = H^{n}(X(C),Z) for some fixed n; this is the usual singular cohomology of the topological space X(C). To get the Galois action on M tensor Z_p we need to use the comparison theorem of etale cohomology H^{n}(X(C),Z) tensor Z_p = H^{n}(X_Qbar,Z_p) where the right-hand side is etale cohomology, which has the desired Galois actions. The compatibility of these actions (for varying p) is known "almost always" thanks to Deligne's proof of the Weil conjectures, but beyond that not a lot is known in general. Our elliptic curve example is the case X=E, n=1 above (although we also need a Tate twist to get thinks just right). If you know a little about these things, the following exercise may be illuminating: Let F be a number field, and take X = Spec F, regarded as a zero dimensional variety over Q. Consider the motive as above with n=0, and show that the L-function is just the usual zeta function of the number field F. -Tom Weston