The computation of values of Riemann zeta and Dirichlet L-functions at non-positive integers is a classical theme which goes back to Euler. By using the excellent idea of the cone decomposition, T. Shintani extended this to the case of general totally real number fields. Recently, in joint works with K. Bannai, K. Hagihara and K. Yamada, I obtained a new interpretation of Shintani's result in the framework of the cohomology of equivariant sheaves on a family of algebraic tori, which gives a more natural understanding. I will lecture about this work, together with a formula on the values of p-adic L-functions at positive integers given in terms of the p-adic polylogarithm (which is an extension of Coleman's theorem in the rational case).
The main theme of the lecture is the construction of a certain cohomology class, called the Shintani generating class. In the classical rational case, the Shintani generating class is just the rational function t/(1-t). To consider cohomology classes as a natural generalization of functions in such a way may be useful to generalize classical results about functions. I hope that the participants will learn this viewpoint through our method to treat such a cohomology class.
To understand the following things:
- the way in which the values of Lerch zeta functions of the rational number field are described by one generating function
- the outline of Shintani's work on zeta functions of totally real number fields
- the definition of the family of algebraic tori associated with a totally real number field, and the description of its equivariant cohomology via the Cech complex
- the construction of the Shintani generating class and the computation of its specializations at torsion points
- the construction of the p-adic polylogarithm of a totally real number field, and its relationship with values of the p-adic L-function
totally real number field, Lerch zeta function, Hecke L-function, Shintani generating class, p-adic L-function, p-adic polylogarithm
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
This is a standard lecture course. There will be some assignments.
|Course schedule||Required learning|
|Class 1||The following topics will be covered: - computation of special values of the Lerch zeta function of the rational number field - Lerch zeta function of a totally real field and Shintani's computation of its special values - the family of algebraic tori, and a description of its equivariant cohomology - construction of the Shintani generating class - computation of specializations of the Shintani generating class at torsion points - p-adic L-function, p-adic polylogarithm, and relationship between them||Details will be provided during each class session|
Bannai et al.: Canonical equivariant cohomology classes generating zeta values of totally real fields, arXiv:1911.02650
Bannai et al.: p-adic polylogarithms and p-adic Hecke L-functions for totally real fields, arXiv:2003.08157