\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \begin{thebibliography}{10} \bibitem{agashe} A.~Agash\'{e}, \emph{On invisible elements of the {T}ate-{S}hafarevich group}, C. R. Acad. Sci. Paris S\'er. I Math. \textbf{328} (1999), no.~5, 369--374. \bibitem{agashe:phd} A.~Agashe, \emph{The {B}irch and {S}winnerton-{D}yer formula for modular abelian varieties of analytic rank~$0$}, U.\thinspace{}C. Berkeley Ph.D. thesis (2000). \bibitem{agashe-stein:manin} A.~Agashe and W.\thinspace{}A. Stein, \emph{On the generalized manin constant for quotients of \protect{$J_0(N)$}}, in preparation. \bibitem{atkin-lehner} A.\thinspace{}O.\thinspace{}L. Atkin and J.~Lehner, \emph{Hecke operators on \protect{$\Gamma \sb{0}(m)$}}, Math. Ann. \textbf{185} (1970), 134--160. \bibitem{birch:atkin} B.~Birch, \emph{Atkin and the {A}tlas {L}ab}, Proceedings of the conference in honor of A.\thinspace{}O.\thinspace{}L. Atkin held at the University of Illinois, Chicago, IL, September 1995, Amer. Math. Soc., Providence, RI, 1998, pp.~13--20. \bibitem{antwerpiv} B.\thinspace{}J. Birch and W.~Kuyk (eds.), \emph{Modular functions of one variable. {I}{V}}, Springer-Verlag, Berlin, 1975, Lecture Notes in Mathematics, Vol. 476. \bibitem{bloch-kato} S.~Bloch and K.~Kato, \emph{\protect{${L}$}-functions and \protect{T}amagawa numbers of motives}, The Grothendieck Festschrift, Vol. \protect{I}, Birkh\"auser Boston, Boston, MA, 1990, pp.~333--400. \bibitem{neronmodels} S.~Bosch, W.~L{\"u}tkebohmert, and M.~Raynaud, \emph{N\'eron models}, Springer-Verlag, Berlin, 1990. \bibitem{magma} W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system {I}: {T}he user language}, J. Symb. Comp. \textbf{24} (1997), no.~3-4, 235--265, \\\protect{\sf http://www.maths.usyd.edu.au:8000/u/magma/}. \bibitem{breuil-conrad-diamond-taylor} C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor, \emph{On the modularity of elliptic curves over \protect{$\Q$}, or {W}ild $3$-adic exercises}, (2000), \\\protect{\sf http://www.math.harvard.edu/HTML/Individuals/Richard\_Taylor.html}. \bibitem{brumer:rank} A.~Brumer, \emph{The rank of \protect{${J}\sb 0({N})$}}, Ast\'erisque (1995), no.~228, 3, 41--68, Columbia University Number Theory Seminar (New York, 1992). \bibitem{buzzard-stein:artin} K.~Buzzard and W.\thinspace{}A. Stein, \emph{Modularity of some icosahedral {G}alois representations}, in preparation. \bibitem{cohen-oesterle:dimensions} H.~Cohen and J.~Oesterl{\'e}, \emph{Dimensions des espaces de formes modulaires}, (1977), 69--78. Lecture Notes in Math., Vol. 627. \bibitem{coleman:monodromy} R.~Coleman, \emph{The monodromy pairing}, Asian Math. Journal (1999). \bibitem{cremona:gammaone} J.\thinspace{}E. Cremona, \emph{Modular symbols for \protect{$\Gamma\sb 1({N})$} and elliptic curves with everywhere good reduction}, Math. Proc. Cambridge Philos. Soc. \textbf{111} (1992), no.~2, 199--218. \bibitem{cremona:algs} \bysame, \emph{Algorithms for modular elliptic curves}, second ed., Cambridge University Press, Cambridge, 1997. \bibitem{cremona:periods} \bysame, \emph{Computing periods of cusp forms and modular elliptic curves}, Experiment. Math. \textbf{6} (1997), no.~2, 97--107. \bibitem{cremona-mazur} J.\thinspace{}E. Cremona and B.~Mazur, \emph{Visualizing elements in the \protect{Shafarevich-Tate} group}, to appear in Experiment. Math. \bibitem{darmon-bsd} H.~Darmon, \emph{Wiles' theorem and the arithmetic of elliptic curves}, Modular forms and Fermat's last theorem (Boston, MA, 1995), Springer, New York, 1997, pp.~549--569. \bibitem{darmon-merel} H.~Darmon and L.~Merel, \emph{Winding quotients and some variants of {F}ermat's last theorem}, J. Reine Angew. Math. \textbf{490} (1997), 81--100. \bibitem{diamond-im} F.~Diamond and J.~Im, \emph{Modular forms and modular curves}, Seminar on {F}ermat's {L}ast {T}heorem, Providence, RI, 1995, pp.~39--133. \bibitem{dummigan:cp} N.~Dummigan, \emph{Period ratios of modular forms}, to appear in Math. Annalen. \bibitem{edixhoven:eisen} B.~Edixhoven, \emph{L'action de l'alg\`ebre de \protect{H}ecke sur les groupes de composantes des jacobiennes des courbes modulaires est ``\protect{E}isenstein''}, Ast\'erisque (1991), no.~196--197, 7--8, 159--170 (1992), Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). \bibitem{empirical} E.\thinspace{}V. Flynn, F.~\protect{Lepr\'{e}vost}, E.\thinspace{}F. Schaefer, W.\thinspace{}A. Stein, M.~Stoll, and J.\thinspace{}L. Wetherell, \emph{Empirical evidence for the \protect{B}irch and \protect{S}winnerton-\protect{D}yer conjectures for modular \protect{J}acobians of genus 2 curves}, Math. of Comp. (2000). \bibitem{goldfeld:complexity} D.~Goldfeld, \emph{On the computational complexity of modular symbols}, Math. Comp. \textbf{58} (1992), no.~198, 807--814. \bibitem{ganz-lario:manin} J.~Gonz\'{a}lez and J-C. Lario, \emph{$\mathbf{Q}$-curves and their {M}anin ideals}, preprint (2000). \bibitem{gross-zagier} B.~Gross and D.~Zagier, \emph{Heegner points and derivatives of \protect{${L}$}-series}, Invent. Math. \textbf{84} (1986), no.~2, 225--320. \bibitem{gross:central} B.\thinspace{}H. Gross, \emph{${L}$-functions at the central critical point}, Motives (Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp.~527--535. \bibitem{hatada:rationality} K.~Hatada, \emph{Multiplicity one theorem and modular symbols}, J. Math. Soc. Japan \textbf{33} (1981), no.~3, 445--470. \bibitem{hijikata:trace} H.~Hijikata, \emph{Explicit formula of the traces of \protect{H}ecke operators for \protect{$\Gamma_0(N)$}}, J. Math. Soc. Japan \textbf{26} (1974), no.~1, 56--82. \bibitem{katzmazur} N.\thinspace{}M. Katz and B.~Mazur, \emph{Arithmetic moduli of elliptic curves}, Princeton University Press, Princeton, N.J., 1985. \bibitem{kohel:hecke} D.\thinspace{}R. Kohel, \emph{Hecke module structure of quaternions}, preprint (1998). \bibitem{kolyvagin:mordellweil} V.\thinspace{}A. Kolyvagin, \emph{On the {M}ordell-{W}eil group and the {S}hafarevich-{T}ate group of modular elliptic curves}, Proceedings of the International Congress of Mathematicians, Vol.\ I, II (Kyoto, 1990) (Tokyo), Math. Soc. Japan, 1991, pp.~429--436. \bibitem{kolyvagin:structureofsha} \bysame, \emph{On the structure of {S}hafarevich-{T}ate groups}, Algebraic geometry (Chicago, IL, 1989), Springer, Berlin, 1991, pp.~94--121. \bibitem{kolyvagin-logachev:totallyreal} V.\thinspace{}A. Kolyvagin and D.\thinspace{}Y. Logachev, \emph{Finiteness of \protect{$\Sha$} over totally real fields}, Math. USSR Izvestiya \textbf{39} (1992), no.~1, 829--853. \bibitem{lang:algebra} S.~Lang, \emph{Algebra}, third ed., Addison-Wesley Publishing Co., Reading, Mass., 1993. \bibitem{winnie:newforms} W-C. Li, \emph{Newforms and functional equations}, Math. Ann. \textbf{212} (1975), 285--315. \bibitem{manin:parabolic} J.\thinspace{}I. Manin, \emph{Parabolic points and zeta functions of modular curves}, Izv. Akad. Nauk SSSR Ser. Mat. \textbf{36} (1972), 19--66. \bibitem{mazur:symboles} B.~Mazur, \emph{Courbes elliptiques et symboles modulaires}, S\'eminaire Bourbaki, 24\`eme ann\'ee (1971/1972), Exp. No. 414, Springer, Berlin, 1973, pp.~277--294. Lecture Notes in Math., Vol. 317. \bibitem{mazur:eisenstein} \bysame, \emph{Modular curves and the \protect{Eisenstein} ideal}, Inst. Hautes \'Etudes Sci. Publ. Math. (1977), no.~47, 33--186 (1978). \bibitem{mazur:rational} \bysame, \emph{Rational isogenies of prime degree (with an appendix by {D}. {G}oldfeld)}, Invent. Math. \textbf{44} (1978), no.~2, 129--162. \bibitem{mazur:arithmetic_values} \bysame, \emph{On the arithmetic of special values of ${L}$\ functions}, Invent. Math. \textbf{55} (1979), no.~3, 207--240. \bibitem{mazur:visthree} \bysame, \emph{Visualizing elements of order three in the {S}hafarevich-{T}ate group}, preprint (1999). \bibitem{mazur-sd} B.~Mazur and P.~Swinnerton-Dyer, \emph{Arithmetic of {W}eil curves}, Invent. Math. \textbf{25} (1974), 1--61. \bibitem{merel:1585} L.~Merel, \emph{Universal \protect{F}ourier expansions of modular forms}, On {A}rtin's conjecture for odd 2-dimensional representations (Berlin), Springer, 1994, pp.~59--94. \bibitem{merel:weil} \bysame, \emph{L'accouplement de {W}eil entre le sous-groupe de {S}himura et le sous-groupe cuspidal de ${J}\sb 0(p)$}, J. Reine Angew. Math. \textbf{477} (1996), 71--115. \bibitem{mestre:graphs} J.-F. Mestre, \emph{La m\'ethode des graphes. \protect{Exemples} et applications}, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata) (1986), 217--242. \bibitem{mestre-oesterle:crelle} J.-F. Mestre and J.~Oesterl{\'e}, \emph{Courbes de {W}eil semi-stables de discriminant une puissance \protect{$m$}-i\`eme}, J. Reine Angew. Math. \textbf{400} (1989), 173--184. \bibitem{milne:etale} J.\thinspace{}S. Milne, \emph{\'{E}tale cohomology}, Princeton University Press, Princeton, N.J., 1980. \bibitem{milne:abvars} \bysame, \emph{Abelian varieties}, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp.~103--150. \bibitem{milne:duality} \bysame, \emph{Arithmetic duality theorems}, Academic Press Inc., Boston, Mass., 1986. \bibitem{pizer:alg} A.~Pizer, \emph{An algorithm for computing modular forms on \protect{$\Gamma\sb{0}({N})$}}, J. Algebra \textbf{64} (1980), no.~2, 340--390. \bibitem{ribet:modreps} K.\thinspace{}A. Ribet, \emph{On modular representations of \protect{${\rm {G}al}(\overline{\bf{Q}}/{\bf {Q}})$} arising from modular forms}, Invent. Math. \textbf{100} (1990), no.~2, 431--476. \bibitem{ribet:raising} \bysame, \emph{Raising the levels of modular representations}, S\'eminaire de Th\'eorie des Nombres, Paris 1987--88, Birkh\"auser Boston, Boston, MA, 1990, pp.~259--271. \bibitem{rubin:main-conjectures} K.~Rubin, \emph{The ``main conjectures'' of {I}wasawa theory for imaginary quadratic fields}, Invent. Math. \textbf{103} (1991), no.~1, 25--68. \bibitem{rubin:book} \bysame, \emph{{E}uler {S}ystems}, Princeton University Press, Spring 2000, {A}nnals of {M}athematics {S}tudies {\bf 147}, \protect{\sf http://math.Stanford.EDU/\~{ }rubin/weyl.html}. \bibitem{scholl:motivesinvent} A.\thinspace{}J. Scholl, \emph{Motives for modular forms}, Invent. Math. \textbf{100} (1990), no.~2, 419--430. \bibitem{scholl:kato} \bysame, \emph{An introduction to {K}ato's {E}uler systems}, Galois Representations in Arithmetic Algebraic Geometry, Cambridge University Press, 1998, pp.~379--460. \bibitem{shafarevich:exp} I.\thinspace{}R. Shafarevich, \emph{Exponents of elliptic curves}, Dokl. Akad. Nauk SSSR (N.S.) \textbf{114} (1957), 714--716. \bibitem{shimura:surles} G.~Shimura, \emph{Sur les \protect{int\'egrales} \protect{attach\'ees} aux formes automorphes}, J. Math. Soc. Japan \textbf{11} (1959), 291--311. \bibitem{shimura:factors} \bysame, \emph{On the factors of the jacobian variety of a modular function field}, J. Math. Soc. Japan \textbf{25} (1973), no.~3, 523--544. \bibitem{shimura:intro} \bysame, \emph{Introduction to the arithmetic theory of automorphic functions}, Princeton University Press, Princeton, NJ, 1994, Reprint of the 1971 original, Kan Memorial Lectures, 1. \bibitem{sokurov:modsym} V.~V. {\v{S}}okurov, \emph{Modular symbols of arbitrary weight}, Funkcional. Anal. i Prilo\v zen. \textbf{10} (1976), no.~1, 95--96. \bibitem{stein:hecke} W.\thinspace{}A. Stein, \emph{\protect{{\tt HECKE}: The} modular symbols calculator}, Software (available online) (1999). \bibitem{stevens:thesis} G.~Stevens, \emph{Arithmetic on modular curves}, Birkh\"auser Boston Inc., Boston, Mass., 1982. \bibitem{sturm:cong} J.~Sturm, \emph{On the congruence of modular forms}, Number theory (New York, 1984--1985), Springer, Berlin, 1987, pp.~275--280. \bibitem{tate:bsd} J.~Tate, \emph{On the conjectures of {B}irch and {S}winnerton-{D}yer and a geometric analog}, S\'eminaire Bourbaki, Vol.\ 9, Soc. Math. France, Paris, 1995, pp.~Exp.\ No.\ 306, 415--440. \bibitem{cime-1997} C.~Viola, \emph{{Arithmetic theory of elliptic curves. Lectures given at the 3rd session of the {C}entro {I}nternazionale {M}atematico {E}stivo \protect{(CIME)}.}}, Springer-Verlag, Berlin, 1997 (English). \bibitem{zagier:parametrizations} D.~Zagier, \emph{Modular parametrizations of elliptic curves}, Canad. Math. Bull. \textbf{28} (1985), no.~3, 372--384. \end{thebibliography}