Geodesic
Supersingular elliptic curves, theta series and weight two modular forms
Emerton, Matthew1
1 Department of Mathematics, Northwestern University, 2033 Sheridan Rd., Evanston, Illinois 60208-2730
Journal of the American Mathematical Society, Tome 15 (2002) no. 3, pp. 671-714

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-02-00390-9

Emerton, Matthew 1

1 Department of Mathematics, Northwestern University, 2033 Sheridan Rd., Evanston, Illinois 60208-2730 
@article{10_1090_S0894_0347_02_00390_9,
     author = {Emerton, Matthew},
     title = {Supersingular elliptic curves, theta series and weight two modular forms},
     journal = {Journal of the American Mathematical Society},
     pages = {671--714},
     publisher = {mathdoc},
     volume = {15},
     number = {3},
     year = {2002},
     doi = {10.1090/S0894-0347-02-00390-9},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-02-00390-9/}
}
TY  - JOUR
AU  - Emerton, Matthew
TI  - Supersingular elliptic curves, theta series and weight two modular forms
JO  - Journal of the American Mathematical Society
PY  - 2002
SP  - 671
EP  - 714
VL  - 15
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-02-00390-9/
DO  - 10.1090/S0894-0347-02-00390-9
ID  - 10_1090_S0894_0347_02_00390_9
ER  - 
%0 Journal Article
%A Emerton, Matthew
%T Supersingular elliptic curves, theta series and weight two modular forms
%J Journal of the American Mathematical Society
%D 2002
%P 671-714
%V 15
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-02-00390-9/
%R 10.1090/S0894-0347-02-00390-9
%F 10_1090_S0894_0347_02_00390_9
Emerton, Matthew. Supersingular elliptic curves, theta series and weight two modular forms. Journal of the American Mathematical Society, Tome 15 (2002) no. 3, pp. 671-714. doi: 10.1090/S0894-0347-02-00390-9

[1] Atkin, A. O. L., Lehner, J. Hecke operators on Γ₀(𝑚) Math. Ann. 1970 134 160

[2] Coleman, Robert F. A 𝑝-adic Shimura isomorphism and 𝑝-adic periods of modular forms 1994 21 51

[3] Coleman, Robert F. A 𝑝-adic inner product on elliptic modular forms 1994 125 151

[4] Arf, Cahit Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper J. Reine Angew. Math. 1939 1 44

[5] Gross, Benedict H. Heights and the special values of 𝐿-series 1987 115 187

[6] Gross, Benedict H. A tameness criterion for Galois representations associated to modular forms (mod 𝑝) Duke Math. J. 1990 445 517

[7] Groupes de monodromie en géométrie algébrique. I 1972

[8] Hartshorne, Robin Residues and duality 1966

[9] Hecke, Erich Lectures on the theory of algebraic numbers 1981

[10] Hijikata, Hiroaki, Saito, Hiroshi On the representability of modular forms by theta series 1973 13 21

[11] Igusa, Jun-Ichi Class number of a definite quaternion with prime discriminant Proc. Nat. Acad. Sci. U.S.A. 1958 312 314

[12] Mazur, B. Modular curves and the Eisenstein ideal Inst. Hautes Études Sci. Publ. Math. 1977

[13] Modular forms and Fermat’s last theorem 1997

[14] Mazur, B., Ribet, K. A. Two-dimensional representations in the arithmetic of modular curves Astérisque 1991

[15] Ohta, Masami On theta series mod 𝑝 J. Fac. Sci. Univ. Tokyo Sect. IA Math. 1981

[16] Ribet, Kenneth A. Mod 𝑝 Hecke operators and congruences between modular forms Invent. Math. 1983 193 205

[17] Ribet, K. A. On modular representations of 𝐺𝑎𝑙(\overline{𝑄}/𝑄) arising from modular forms Invent. Math. 1990 431 476

[18] Ribet, Kenneth A. Multiplicities of Galois representations in Jacobians of Shimura curves 1990 221 236

[19] Ribet, Kenneth A. Multiplicities of 𝑝-finite mod 𝑝 Galois representations in 𝐽₀(𝑁𝑝) Bol. Soc. Brasil. Mat. (N.S.) 1991 177 188

[20] Ribet, Kenneth A. Torsion points on 𝐽₀(𝑁) and Galois representations 1999 145 166

[21] Serre, Jean-Pierre Sur les représentations modulaires de degré 2 de 𝐺𝑎𝑙(\overline{𝐐}/𝐐) Duke Math. J. 1987 179 230

Cité par Sources :