Chapter 6 - Modular forms and cohomology groups

Summary

In this chapter, we prove the Eichler-Shimura isomorphism between the space of modular forms and the cohomology group on each modular curve. This fact was first proven by Shimura in 1959 in [Sh1] (see also [Sh, VIII]). We shall give two proofs in §6.2 of this isomorphism. The first one is the original proof of Shimura based on the two dimension formulas. One is the formula for the space of cusp forms and the other is for the cohomology group. The other proof makes use of harmonic analysis on the modular curve. After studying the Hecke module structure of modular cohomology groups, in §6.5, we construct the p-adic standard L-function of GL(2)_/Q following the method (the so-called “p-adic Mellin transform”) of Mazur and Manin in [Mz1], [MTT] and [Mn1,2]. Throughout this chapter, we use without warning the cohomological notation and definition described in Appendix at the end of the book. If the reader is not familiar with cohomology theory, it is recommended to have a look at the appendix first.

Cohomology of modular groups

In this section, we shall prove the dimension formula of the cohomology group of congruence subgroups of SL₂(Z) following [Sh, VIII]. Let Γ be a congruence subgroup of SL₂(Z) and suppose for simplicity that Γ is torsion-free. (The general case without assuming the torsion-freeness of Γ is treated in [Sh, VIII].)