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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Congruences between Modular Forms,
Cyclic Isogenies of Modular Elliptic Curves,
and Integrality of $p$-adic $L$-Functions


Author: Shu-Leung Tang
Journal: Trans. Amer. Math. Soc. 349 (1997), 837-856
MSC (1991): Primary 11G05, 11G18; Secondary 11S40
MathSciNet review: 1376558
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Abstract: Let $\Gamma $ be a congruence subgroup of type $(N_1,N_2)$ and of level $N$. We study congruences between weight 2 normalized newforms $f$ and Eisenstein series $E$ on $\Gamma $ modulo a prime $\gp $ above a rational prime $p$. Assume that $p\nmid 6N$, $E$ is a common eigenfunction for all Hecke operators and $f$ is ordinary at $\gp $. We show that the abelian variety associated to $f$ and the cuspidal subgroup associated to $E$ intersect non-trivially in their $p$-torsion points. Let $A$ be a modular elliptic curve over $\Q $ with good ordinary reduction at $p$. We apply the above result to show that an isogeny of degree divisible by $p$ from the optimal curve $A_1$ in the $\Q $-isogeny class of elliptic curves containing $A$ to $A$ extends to an étale morphism of Néron models over $\Z _p$ if $p>7$. We use this to show that $p$-adic distributions associated to the $p$-adic $L$-functions of $A$ are $\Z _p$-valued.


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Additional Information

Shu-Leung Tang
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Address at time of publication: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01748-0
PII: S 0002-9947(97)01748-0
Keywords: Modular forms, elliptic curves, $p$-adic $L$-functions
Received by editor(s): May 10, 1995
Received by editor(s) in revised form: September 21, 1995
Article copyright: © Copyright 1997 American Mathematical Society