2010/12/28 by Siegfried Böcherer, Böcherer, Siegfried, Neil Dummigan +3
Mathematics · #11F33 #11F46 #11F67 #11F80 #11G40 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1012.5817
openalex publication_date 2010/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.