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Level raising mod 2 and arbitrary 2-Selmer ranks

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 June 2016

Bao V. Le Hung  and
Chao Li
Bao V. Le Hung
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA email lhvietbao@googlemail.com
Chao Li
Affiliation:
Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA email chaoli@math.columbia.edu
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Abstract

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We prove a level raising mod $\ell =2$ theorem for elliptic curves over $\mathbb{Q}$. It generalizes theorems of Ribet and Diamond–Taylor and also explains different sign phenomena compared to odd $\ell$. We use it to study the 2-Selmer groups of modular abelian varieties with common mod 2 Galois representation. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families.

Keywords

modular formsSelmer groups

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms
Secondary: 11G05: Elliptic curves over global fields 11G10: Abelian varieties of dimension $> 1$
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 8 , August 2016 , pp. 1576 - 1608
Copyright
© The Authors 2016 

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