CONGRUENCES OF MODULAR FORMS AND TENSOR PRODUCT $L$-FUNCTIONS

Abstract

Two normalised Hecke eigenforms $f$ and $g$ are considered in this paper, of level one and weights $k'\,{>}\,k$, lying in a $p$-adic family, such that $f\,{\equiv}\,g\pmod{p^r}$ as $q$-expansions. Interpreting the congruence in terms of the associated Galois representations leads to the existence of non-trivial global $p^r$-torsion for the motive associated with the tensor product $L$-function at $s\,{=}\,k'\,{-}\,1$. (It must be assumed that the (mod $p$) Galois representation attached to $g$ is irreducible.) This contributes a factor of $p^r$ to the denominator of the Bloch–Kato conjectural formula for the $L$-value. The $p$-part of the numerator is considered, using recent work of Diamond, Flach and Guo. Using Shimura's Rankin–Selberg integral formula and the Clausen–von Staudt theorem, the ratio of $L$-values is examined at $s\,{=}\,k'-1$, for the tensor product and a quadratic twist; confirmation is given (under certain conditions) that, at $p$, this is as predicted by Bloch and Kato.