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Symmetric square L-values and dihedral congruences for cusp forms by Neil Dummigan; Bernhard Heim is a Mathematics article available to read on EtoBox.

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Let p ≡ 3 (mod 4) be a prime, and k = (p + 1)/2. In this paper we prove that two things happen if and only if the class number h( √ -p) > 1. One is the non-integrality at p of a certain trace of normalised critical values of symmetric square L-functions, of cuspidal Hecke eigenforms of level one and weight k. The other is the existence of such a form g whose Hecke eigenvalues satisfy "dihedral" congruences modulo a divisor of p (e.g. p = 23, k = 12, g = ). We use the Bloch-Kato conjecture to link these two phenomena, using the Galois interpretation of the congruences to produce global torsion elements which contribute to the denominator of the conjectural formula for an L-value. When h( √ -p) = 1, the trace turns out always to be a p-adic unit.

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Author

Neil Dummigan; Bernhard Heim

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0022-314X)

Published

2010

Language

EN

Field

Mathematics (Physical Sciences)