Yoshida lifts and Selmer groups

TL;DR

This paper establishes a connection between congruences of Yoshida lifts and Selmer groups, providing new formulas and evidence supporting the Bloch-Kato conjecture for certain L-values.

Contribution

It proves that certain primes dividing algebraic parts of L-values correspond to congruences between Yoshida lifts and non-endoscopic cusp forms, with explicit formulas for pullback and norms.

Findings

Proves congruences between Yoshida lifts and cusp forms related to L-values.

Provides explicit pullback formula for genus-four Eisenstein series.

Constructs elements in Selmer groups consistent with Bloch-Kato conjecture.

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 $L(f\otimes g,\frac{k+k'-2}{2})$. 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.