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October, 2012 Yoshida lifts and Selmer groups
Siegfried BÖCHERER, Neil DUMMIGAN, Rainer SCHULZE-PILLOT
J. Math. Soc. Japan 64(4): 1353-1405 (October, 2012). DOI: 10.2969/jmsj/06441353

Abstract

Let $f$ and $g$, of weights $k' > k \geq 2$, be normalised newforms for $\Gamma_0(N)$, for square-free $N > 1$, such that, for each Atkin-Lehner involution, the eigenvalues of $f$ and $g$ are equal. Let $\lambda\mid\ell$ be a large prime divisor of the algebraic part of the near-central critical value $L(f\otimes g,(k+k'-2)/2)$. Under certain hypotheses, we prove that $\lambda$ is the modulus of a congruence between the Hecke eigenvalues of a genus-two Yoshida lift of (Jacquet-Langlands correspondents of) $f$ and $g$ (vector-valued in general), and a non-endoscopic genus-two cusp form. In pursuit of this we also give a precise pullback formula for a genus-four Eisenstein series, and a general formula for the Petersson norm of a Yoshida lift.

Given such a congruence, using the 4-dimensional $\lambda$-adic Galois representation attached to a genus-two cusp form, we produce, in an appropriate Selmer group, an element of order $\lambda$, as required by the Bloch-Kato conjecture on values of $L$-functions.

Citation

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Siegfried BÖCHERER. Neil DUMMIGAN. Rainer SCHULZE-PILLOT. "Yoshida lifts and Selmer groups." J. Math. Soc. Japan 64 (4) 1353 - 1405, October, 2012. https://doi.org/10.2969/jmsj/06441353

Information

Published: October, 2012
First available in Project Euclid: 29 October 2012

zbMATH: 1276.11069
MathSciNet: MR2998926
Digital Object Identifier: 10.2969/jmsj/06441353

Subjects:
Primary: 11F46
Secondary: 11F33 , 11F67 , 11F80 , 11G40

Keywords: Bloch-Kato conjecture , doubling method , pullback formula , Yoshida lift

Rights: Copyright © 2012 Mathematical Society of Japan

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Vol.64 • No. 4 • October, 2012
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