Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 64, Number 4 (2012), 1353-1405.
Yoshida lifts and Selmer groups
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.
J. Math. Soc. Japan Volume 64, Number 4 (2012), 1353-1405.
First available in Project Euclid: 29 October 2012
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F80: Galois representations 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50] 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
BÖCHERER, Siegfried; DUMMIGAN, Neil; SCHULZE-PILLOT, Rainer. Yoshida lifts and Selmer groups. J. Math. Soc. Japan 64 (2012), no. 4, 1353--1405. doi:10.2969/jmsj/06441353. https://projecteuclid.org/euclid.jmsj/1351516778.