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Supersingular elliptic curves, theta series and weight two modular forms

Author: Matthew Emerton
Journal: J. Amer. Math. Soc. 15 (2002), 671-714
MSC (2000): Primary 11F11, 11F27, 11F37
Published electronically: February 27, 2002
MathSciNet review: 1896237
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Abstract: Let $p$ be a prime, and let $\mathcal {M}$ denote the space of weight two modular forms on $\Gamma _{0}(p)$ all of whose Fourier coefficients are integral, except possibly for the constant term, which should be either integral or half-integral. We prove that $\mathcal {M}$ is spanned as a $\mathbb {Z}$-module by theta series attached to the unique quaternion algebra that is ramified at $p$, at infinity, and at no other primes.

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  • A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma _{0}(m)$, Math. Ann. 185 (1970), 134–160. MR 268123, DOI
  • Robert F. Coleman, A $p$-adic Shimura isomorphism and $p$-adic periods of modular forms, $p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) Contemp. Math., vol. 165, Amer. Math. Soc., Providence, RI, 1994, pp. 21–51. MR 1279600, DOI
  • Robert F. Coleman, A $p$-adic inner product on elliptic modular forms, Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991) Perspect. Math., vol. 15, Academic Press, San Diego, CA, 1994, pp. 125–151. MR 1307394
  • EIF Eichler, M., Über die Idealklassenzahl total definiter Quaternionenalgebren, Math. Z. 43 (1938), 102–109. EIS Eichler, M., Über die Darstellbarkeit von Modulformen durch Thetareihen, J. Reine Angew. Math. 195 (1955), 156–171.
  • Benedict H. Gross, Heights and the special values of $L$-series, Number theory (Montreal, Que., 1985) CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 115–187. MR 894322
  • Benedict H. Gross, A tameness criterion for Galois representations associated to modular forms (mod $p$), Duke Math. J. 61 (1990), no. 2, 445–517. MR 1074305, DOI
  • GRL Gross, B.H., Course at Harvard University, Spring, 1996.
  • Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, Vol. 288, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I); Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim. MR 0354656
  • Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093
  • Erich Hecke, Lectures on the theory of algebraic numbers, Graduate Texts in Mathematics, vol. 77, Springer-Verlag, New York-Berlin, 1981. Translated from the German by George U. Brauer, Jay R. Goldman and R. Kotzen. MR 638719
  • Hiroaki Hijikata and Hiroshi Saito, On the representability of modular forms by theta series, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 13–21. MR 0357332
  • Jun-ichi Igusa, Class number of a definite quaternion with prime discriminant, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 312–314. MR 98728, DOI
  • KIL Kilford, L., Some examples of non-Gorenstein Hecke algebras associated to modular forms, preprint, available at
  • B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287
  • Gary Cornell, Joseph H. Silverman, and Glenn Stevens (eds.), Modular forms and Fermat’s last theorem, Springer-Verlag, New York, 1997. Papers from the Instructional Conference on Number Theory and Arithmetic Geometry held at Boston University, Boston, MA, August 9–18, 1995. MR 1638473
  • B. Mazur and K. A. Ribet, Two-dimensional representations in the arithmetic of modular curves, Astérisque 196-197 (1991), 6, 215–255 (1992) (English, with French summary). Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). MR 1141460
  • Masami Ohta, On theta series mod $p$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 679–686 (1982). MR 656043
  • Kenneth A. Ribet, Mod $p$ Hecke operators and congruences between modular forms, Invent. Math. 71 (1983), no. 1, 193–205. MR 688264, DOI
  • K. A. Ribet, On modular representations of ${\rm Gal}(\overline {\bf Q}/{\bf Q})$ arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476. MR 1047143, DOI
  • Kenneth A. Ribet, Multiplicities of Galois representations in Jacobians of Shimura curves, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 3, Weizmann, Jerusalem, 1990, pp. 221–236. MR 1159117
  • Kenneth A. Ribet, Multiplicities of $p$-finite mod $p$ Galois representations in $J_0(Np)$, Bol. Soc. Brasil. Mat. (N.S.) 21 (1991), no. 2, 177–188. MR 1139564, DOI
  • Kenneth A. Ribet, Torsion points on $J_0(N)$ and Galois representations, Arithmetic theory of elliptic curves (Cetraro, 1997) Lecture Notes in Math., vol. 1716, Springer, Berlin, 1999, pp. 145–166. MR 1754687, DOI
  • Jean-Pierre Serre, Sur les représentations modulaires de degré $2$ de ${\rm Gal}(\overline {\bf Q}/{\bf Q})$, Duke Math. J. 54 (1987), no. 1, 179–230 (French). MR 885783, DOI
  • STE Stevens, G., Coleman’s $\mathcal {L}$-invariant and families of modular forms, preprint (1996).

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Additional Information

Matthew Emerton
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Rd., Evanston, Illinois 60208-2730

Received by editor(s): November 1, 2000
Received by editor(s) in revised form: September 19, 2001
Published electronically: February 27, 2002
Article copyright: © Copyright 2002 American Mathematical Society