Yoshida lifts and Selmer groups

Abstract

Let f and g, of weights k′ > k ≥ 2, be normalised newforms for Γ₀(N), for square-free N > 1, such that, for each Atkin-Lehner involution, the eigenvalues of f and g are equal. Let λ | ℓ be a large prime divisor of the algebraic part of the near-central critical value L(f ⊗ g, (k + k′ − 2)/2). Under certain hypotheses, we prove that λ 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 λ-adic Galois representation attached to a genus-two cusp form, we produce, in an appropriate Selmer group, an element of order λ, as required by the Bloch-Kato conjecture on values of L-functions.