Let
f and
g, of weights
k′ >
k ≥ 2, be normalised newforms for Γ
0(
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.
View full abstract