Triple Product L-values and Dihedral Congruences for Cusp Forms

Abstract

Let be a prime, and . In this paper, we prove that a certain trace of normalized, rightmost critical values of triple product L-functions, of cuspidal Hecke eigenforms of level one and weight k, is non-integral at p if and only if the class number . We use the Bloch-Kato conjecture to explain this, using “dihedral” congruences, modulo a divisor of p, for cuspidal Hecke eigenforms of level one and weight k (e.g., p = 23, k = 12, g = Δ). Exploiting the Galois interpretation of such congruences, we may produce global torsion elements which contribute to the denominator of the conjectural formula for some L-value contributing to the trace.