Modular forms in characteristic \(\ell\) and special values of their \(L\)-functions. (English) Zbl 0618.10026

Ash, Avner; Stevens, Glenn

Duke Math. J. 53, No. 3, 849-868 (1986).

The authors use group cohomological methods to study mod \(\ell\) modular forms of weight \(k>2\) and level \(N\), prime to \(\ell\). In the course of their study they show that the systems of mod \(\ell\) Hecke eigenvalues occurring in the space of modular forms of level \(N\) and weight \(>2\) coincide with those of twisted forms occurring in the space of weight two forms on \(\Gamma_ 1(N\ell)\). As a consequence they obtain the fact that there are only finitely many systems of Hecke eigenvalues mod \(\ell\) occurring in the space of forms of level \(N\) and weight \(>2.\)
The authors also study modular symbols of weight \(>2\), and show how one can use these to prove mod \(\lambda\) congruences between the algebraic parts of special values of \(L\)-functions of higher weight cusp forms on \(\text{SL}_ 2({\mathbb Z})\) and special values of \(L\)-functions of weight 2 cusp forms on \(\Gamma_ 1(\ell)\) for a prime \(\lambda\) dividing \(\ell\).

Reviewer: S.Kamienny

MSC:

11F75	Cohomology of arithmetic groups
11F33	Congruences for modular and \(p\)-adic modular forms
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols

Keywords:

group cohomology; modular forms; Hecke eigenvalues; modular symbols; congruences; special values of L-functions

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