Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0894-0347-02-00390-9
Published electronically: February 27, 2002
MathSciNet review: 1896237
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. Atkin, A.O.L., Lehner, J., Hecke operators on $\Gamma _{0}(m)$, Math. Ann. 185 (1970), 134-160. MR 42:3022
  • 2. Coleman, R.F., A $p$-adic Shimura isomorphism and $p$-adic periods of modular forms, Contemp. Math. 165 (1994), 21-51. MR 96a:11050
  • 3. Coleman, R.F., A $p$-adic inner product on elliptic modular forms, Barsotti symposium in algebraic geometry, Academic Press, 1994, pp. 125-151. MR 95k:11078
  • 4. Eichler, M., Über die Idealklassenzahl total definiter Quaternionenalgebren, Math. Z. 43 (1938), 102-109.
  • 5. Eichler, M., Über die Darstellbarkeit von Modulformen durch Thetareihen, J. Reine Angew. Math. 195 (1955), 156-171. MR 18:297d
  • 6. Gross, B.H., Heights and the special values of $L$-series, Number theory (Montreal, Que., 1985), CMS Conf. Proc., vol. 7, Amer. Math. Soc. , 1987, pp. 115-187. MR 89c:11082
  • 7. Gross, B.H., On a tameness criterion for Galois representations associated to modular forms $\pmod p$, Duke Math. J. 61 (1990), 445-517. MR 91i:11060
  • 8. Gross, B.H., Course at Harvard University, Spring, 1996.
  • 9. Grothendieck, A., SGA VII, Exposé IX, SLN, vol. 288, 1972, pp. 313-523. MR 50:7134
  • 10. Hartshorne, R., Residues and Duality, SLN 20 (1966). MR 36:5145
  • 11. Hecke, E., Lectures on the theory of algebraic numbers, Springer-Verlag, 1981. MR 83m:12001
  • 12. Hijikata, H., Saito, H., 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 50:9800
  • 13. Igusa, J.-I., Class number of a definite quaternion with prime discriminant, Proc. Nat. Acad. Sci. USA 44 (1958), 312-314. MR 20:5183
  • 14. Kilford, L., Some examples of non-Gorenstein Hecke algebras associated to modular forms, preprint, available at http://www.ma.ic.ac.uk/~ljpk/maths/maths.html.
  • 15. Mazur, B., Modular curves and the Eisenstein ideal, Publ. Math., Inst. Hautes Etud. Sci. 47 (1977), 33-186. MR 80c:14015
  • 16. Mazur, B., letter to K. Ribet and J. Tilouine, in ``Hecke algebras and the Gorenstein property'', by J. Tilouine, Modular forms and Fermat's last theorem (G. Cornell, J.H. Silverman, G. Stevens, eds.), Springer-Verlag, 1997. MR 99k:11004
  • 17. Mazur, B., Ribet, K.A., Two-dimensional representations in the arithmetic of modular curves, Astérisque 196-197 (1991), 215-255. MR 93d:11056
  • 18. Ohta, M., Theta series mod $p$, J. Fac. Sci. Tokyo 28 (1981), 679-686. MR 83h:10058
  • 19. Ribet, K.A., Mod $p$ Hecke operators and congruences between modular forms, Invent. Math. 71 (1983), 193-205. MR 84j:10040
  • 20. Ribet, K.A., On modular representations of $\mathop{\rm {Gal}} ({\overline {\mathbb Q }} / \mathbb Q )$ arising from modular forms, Invent. Math. 100 (1990), 431-476. MR 91g:11066
  • 21. Ribet, K.A., Multiplicities of Galois representations in Jacobians of Shimura curves, Israel Math. Conf. Proc. 3 (1990), 221-236. MR 93c:11043
  • 22. Ribet, K.A., Multiplicities of $p$-finite mod $p$ Galois representations in $J_{0}(N p)$, Bol. Soc. Brasil. Mat. (N. S.) 21 (1991), 177-188. MR 93b:11078
  • 23. Ribet, K.A., Torsion points on $J_{0}(N)$ and Galois representations, Arithmetic Theory of Elliptic Curves (J. Coates, ed.), SLN, vol. 1716, 1999. MR 2001b:11054
  • 24. Serre, J.-P., Sur les représentations modulaires de degré 2 de $\mathop{\rm {Gal}} (\overline{\mathbb Q} /\mathbb Q )$, Duke Math. J. 54 (1987), 179-230. MR 88g:11022
  • 25. Stevens, G., Coleman's $\mathcal{L}$-invariant and families of modular forms, preprint (1996).

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 11F11, 11F27, 11F37

Retrieve articles in all journals with MSC (2000): 11F11, 11F27, 11F37


Additional Information

Matthew Emerton
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Rd., Evanston, Illinois 60208-2730
Email: emerton@math.northwestern.edu

DOI: https://doi.org/10.1090/S0894-0347-02-00390-9
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

American Mathematical Society