Titles and abstracts

I will introduce Harder's conjecture, on congruences between the Hecke eigenvalues of genus 2 vector-valued Siegel modular forms and genus 1 forms, modulo divisors of algebraic parts of critical L-values of the latter. His belief is that such congruences are realised in the cohomology of local systems on Siegel modular varieties, between elements of inner cohomology and Eisenstein classes in cohomology on the boundary of a Borel-Serre compactification.

I will place the conjecture in the context of Ramanujan's mod 691 congruence between a cusp form and an Eisenstein series, the Bloch-Kato conjecture on L-values, and various congruences involving Klingen-Eisenstein series and Saito-Kurokawa lifts, introducing the necessary definitions or properties, and will try to impart some appreciation of why it is difficult. This is in preparation for surveying various approaches that have succeeded in proving instances of the conjecture. The first, which will be mentioned only briefly, is that of Chenevier and Lannes using algebraic modular forms for orthogonal groups of rank-24 lattices. I will also outline the main ideas in recent work of Atobe, Chida, Ibukiyama, Katsurada and Yamauchi, using Klingen-Eisenstein series and Ikeda lifts of higher genus, and of myself with Pacetti, Rama and Tornaria, using orthogonal groups of quinary forms.

The analytic class number formula relates the residue at s=1 of the Dedekind zeta function of a number field to a transcendental quantity, the regulator, which is a determinant of logarithms of units of the number field. At the end of the 80s, Zagier conjectured a generalization of this classical result to all special values of Dedekind zeta functions, where polylogarithms replace the logarithm function. The existence of higher regulators linked to these special values results from Borel’s computation of the stable cohomology of the general linear group, and Zagier’s conjecture can be viewed as a quest for explicit cocycles for these cohomology groups. A more conceptual interpretation, connected to the theory of (mixed Tate) motives, was given by Beilinson and Deligne, and developed by Goncharov. In this mini-course we will review the links between all these objects and introduce the tools involved in recent progress on mixed Tate motives, including Goncharov and Rudenko’s proof of Zagier’s conjecture in weight 4.

Plan:

1) Dedekind zeta values, polylogarithms, and Zagier’s conjecture

We will review classical facts about Dedekind zeta functions and their special values and introduce Zagier’s conjecture, which is a conjectural generalization of the classical class number formula. The special functions which replace the logarithm are the polylogarithms functions, that we will study. The relationship to K-theory is explained by Borel’s celebrated results on higher regulators.

2) Cohomology and (mixed Tate) motives

We will introduce (mixed Tate) motives with a special emphasis on concrete examples: motives associated to logarithms, polylogarithms, and volumes of hyperbolic manifolds.

3) The Tannakian formalism and motivic complexes

We will review the classical Tannakian formalism and apply it to mixed Tate motives and the construction of motivic complexes. In low weight we will recover classical results in K-theory.

4) Recent progress

We will review recent progress on mixed Tate motives, including Goncharov and Rudenko’s proof of Zagier’s conjecture in weight 4.

I will first review Shimura varieties, their compactifications and their canonical models, concentrating on the example of Siegel modular varieties. I will then explain how to relate the cohomology of compact Shimura varieties and automorphic representations using Matsushima's formula, and the consequences for the Langlands programme. For noncompact Shimura varieties, we can hope to salvage some of this thanks to Zucker's conjecture relating $L^2$ cohomology and intersection cohomology. The introduction of weighted cohomology by Goresky, Harder and MacPherson was motivated by this conjecture. I will explain the original definition of the weighted cohomology complexes and some later applications of these ideas by Saper. Then I will talk about the reinterpretation of the weighted cohomology complexes in the $\ell$-adic

étale context based on the theory of weights (in the sense of Deligne) and a formula of Pink. This last definition is very motivic in nature, so I will finally talk about the motivic versions of weighted cohomology.

This mini-course aims for the participants to become familiar with some of the techniques commonly used for computations in motivic homotopy theory. We will focus on calculations of algebraic and hermitian K-groups for rings of integers in number fields. Contributing ingredients include the effective and very effective slice filtrations of the stable motivic homotopy category and the action of Steenrod operations on motivic cohomology groups. Several mathematicians have made significant contributions to this aspect of the subject, including Bachmann, Kylling, Levine, Morel, Rondigs, Spitzweck, and Voevodsky.

Lecture 1. Background and motivation.

Lecture 2. Motivic homotopy theory.

Lecture 3. Slice filtrations.

Lecture 4. Computations and open problems.

The lectures are aimed at graduate students.

In this mini-course, we will focus on some relations between the theory of algebraic cycles and the cohomology of locally symmetric spaces. Specifically, we will focus on the phenomenon of the same system of Hecke eigenvalues occurring in different cohomological degrees. In the case of Shimura varieties, it is well known that non-tempered representations can contribute to different cohomological degrees, and this is related to the Lefschetz operator. However, in the non-Shimura variety case, even tempered representations can contribute to different cohomological degrees. A possible conjectural explanation for this, using a purported action of motivic cohomology on the cohomology of locally symmetric spaces, is suggested in a recent joint paper with Venkatesh (Asterisque 428), and much of the course will be devoted to motivating this conjecture and explaining what we know about it (very little!)

Rough outline (to be covered in four talks):

(1) We will start by discussing in detail the case of GL_2 over number fields, and the connection with Beilinson's conjecture on special values of motivic L-functions, which helps motivate the main conjecture.

(2) Next we will formulate the main conjecture precisely in the general case, and explain some evidence for it, including compatibility with the Gan-Gross-Prasad conjectures on periods of automorphic forms.

(3) Finally, we will discuss the case of coherent cohomology of Shimura varieties, where this phenomenon also occurs. This case has an added element of mystery since one does not yet know how to (conjecturally) construct the motivic action, even though we believe it exists.