Series on Number Theory and Its Applications | 2021
Cohomological modular forms and p-adic L-functions
Abstract
Contents 1. Introduction 2 1.1. Cohomology groups 2 1.2. Cohomology of G m (C) and Dirichlet L-values 5 1.3. Relative cohomology 7 1.4. Hecke operators 8 1.5. p-Adic measure 9 1.6. p-Adic measure and Hecke operators 11 2. Modular p-adic L-functions 15 2.1. Elliptic modular forms 15 2.2. Modular cohomology group 16 2.3. Hecke operators 17 2.4. Duality 19 2.5. Modular Hecke L-functions 22 2.6. Rationality of Hecke L-values 23 2.7. p-Old and p-new forms 24 2.8. Elliptic modular p-adic measure 25 References 28 In this course, assuming basic knowledge of algebraic number theory, elliptic modular forms, commutative algebra and topology, we will make p-adic study of cohomological modular forms on GL(1) and GL(2). We plan to discuss the following four topics: (1) Isomorphism of Eichler-Shimura type connecting modular forms and cohomology groups, (2) Rationality and integrality of L-values, (3) p-adic measure theory, (4) Construction of analytic p-adic L-functions. Along with these main topics, we will give a brief description of different cohomology theory we will use. In this note, all rings are supposed to have the identity.