# Abstracts for Conference on "Motives and Automorphic Forms" in Honour of Günter Harder's 85th Birthday

Alternatively have a look at the program.

## Short course "Motivic Cohomology". Lecture 1: An introduction to higher Chow groups and triangulated categories of motives

Lecture 1/3.

## Short course "Shimura varieties and interior motives". Lecture 1: Weight structures

Short course 1/3.

Keywords: weight structures, weight filtrations, orthogonality, heart, weight filtrations avoiding certain weights.

We shall give a complete review of the abstract theory of (bounded) weight structures on triangulated categories, as developed by Bondarko. Using material from Levine's series "Three Lectures on Motivic cohomology", we shall see that Voevodsky's category of motives underlies a (unique) weight structure whose heart equals the additive category of Chow motives.

## Short talk: Motivic action conjectures

A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. A recent series of conjectures proposes an arithmetic explanation: a hidden degree-shifting action of a certain motivic cohomology group. We will give an overview of these conjectures, focusing on the examples of $GL_2$ over Q and over quadratic fields, and $GSp_4$ over Q.

## Short talk: Tamagawa motives in the GL(2+r)-automorphic cohomology

I will introduce the tantalizing problem of giving an automorphic interpretation of Bloch’s Tamagawa number reformulation of Birch-Swinnerton-Dyer, together with some results for the special case of Mordell-Weil rank r=1.

## Short talk: Special cycles on compactifications of Shimura varieties

I will discuss ongoing work with Jan Bruinier and Shaul Zemel on constructing special cycles of dimension 0 on toroidal compactifications of orthogonal Shimura varieties. We show the modularity of the generating series that have these special cycles as coefficients, generalizing the compact case.

## Short course "Eisenstein Cohomology and Special Values of L-functions". Lecture 1: Eisenstein Kohomologie: A short introduction

Short course 1/3, online talk,

In this introductory lecture, I will start by setting up the context to study the cohomology of a locally symmetric space. Then I will discuss the notions of inner, cuspidal, and Eisenstein cohomology, and their mutual relationships. Whereas to a large extent I will work with a general connected reductive group over a number field, I will often specialize to GL(n) over a totally real or a totally imaginary number field.

## Short course "Motivic Cohomology". Lecture 2: Motivic cohomology and triangulated categories of motives

Lecture 2/3.

We introduce motivic cohomology as maps in Voevodsky’s triangulated category of motives. We describe the comparison theorem identifying motivic cohomology with Bloch’s higher Chow groups and discuss the relation of mod n-motivic cohomology with étale cohomology. We use the comparison theorem to show how the category of Chow motives fully embeds in the category of geometric motives and briefly describe realization functors associated to de Rham cohomology and Betti (co)homology.

## Short course "Shimura varieties and interior motives". Lecture 2: Chow motives realising to interior cohomology of Shimura varieties

Short course 2/3.

Keywords: motivic weight structure, boundary motive, interior motive, semi-primality, minimal weight filtrations, weight conservativity.

After giving some motivation of the problem (applications to L-functions), we shall apply the theory presented in Lecture 1. This will culminate in a criterion concerning the existence of the interior motive. The criterion is formulated in "non-motivic" terms, i.e., it uses only (l-adic or Hodge theoretic) realisations.

## Short talk: Block graded multiple zeta values and period polynomials of modular forms

Multiple zeta values are a class of conjecturally transcendental numbers, arising as iterated integrals on the projective plane minus three points. They are known to satisfy many algebraic relations, and describing these relations remains a substantial challenge.

## Short talk: Motivic Mahler measures and Eisenstein series

Mahler measures of polynomials have been known to be periods of Deligne cohomology since the seminal work of Deninger. This explains some links between Mahler measures and special values of L-functions, which were observed by Boyd and Rodriguez-Villegas, and proven in some cases using the method of Rogers and Zudilin, which allows one to perform explicit regulator computations using Eisenstein series.

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