# Abstracts for Conference on Arithmetic and Automorphic Forms on the occasion of Günter Harder's 80th birthday, March 12 - 14, 2018

Alternatively have a look at the program.

## Modular forms for genus $2$ and $3$

For genus $2$ and $3$ modular forms are intimately connected with the moduli of curves of genus $2$ and $3$. We describe an explicit way to construct such modular forms for genus $2$ and $3$ using invariant theory and give some applications. This is based on joint work with Fabien Clery and Carel Faber.

## Hodge structure and motivic gamma function associated to the Apéry family

(Joint work with M. Vlasenko) Inspired by recent work of V. Golyshev and D. Zagier, we associate to the Apéry family (a rank 3 variation of Hodge structure on an open set in the projective line) a "motivic Gamma function" which is a Mellin transform of a suitable Picard Fuchs solution. The Taylor series at $s=0$ of this Mellin transform has coefficients which numerical calculations suggest are (multiple) zeta values. We show how this data can be interpreted as a variation of mixed Hodge structure.

## A higher weight generalization of the Hermite-Minkowski theorem

Let $E$ be a number field, $N$ an ideal of its ring of integers, and $w \geq 0$ an integer. Consider the set of cuspidal algebraic automorphic representations of $GL_n$ over $E$ whose conductor is $N$, and whose ''weights'' are in the interval $\{0,\dots,w\}$ (with $n$ varying). If the root-discriminant of $E$ is less than a certain explicit function $f$ of $w$, then I show that this set is finite. For instance, we have $f(w)>1$ if, and only if, $w<24$.

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## Ordinary points mod $p$ of hyperbolic $3$-manifolds

For each $d<0$ there is an anti-holomorphic involution of the $(Sp(4, R))$ Siegel modular variety whose fixed point set is a finite union of hyperbolic $3$-manifolds with fundamental group $SL(2,\mathcal{O}_d)$.

## Eisenstein cohomology and automorphic L-functions

Günter Harder has pioneered the theory of Eisenstein cohomology over the last few decades. This involves my own work with Harder on rank one Eisenstein cohomology for $GL(N)$ over a totally real field and the arithmetic of Rankin-Selberg $L$-functions for $GL(n) \times GL(m)$. Since then I have been involved in several projects which have the common theme of Eisenstein cohomology of some ambient reductive group and the special values of certain automorphic $L$-functions.

## Towards Harder-Narasimhan filtrations for Fukaya-Seidel type categories with coefficients

We report on progress in our joint work with F. Haiden, L. Katzarkov, and P. Pandit on a program of extending the Bridgeland-Smith construction of stability conditions to the case of $SL(3)$ spectral curves. We consider Fukaya-Seidel categories of graph Lagrangians with coefficients in a constant category, in our case of type $A2-CY2$, on a contractible flat Riemann surface. As in the recent theory of "perverse schobers", objects involve putting triangles at the threefold vertices of the underlying graph.

## Some conjectures on Weil cohomology theories over $\overline{F}_p$

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## On the automorphic spectrum of non-quasi-split groups

Langlands' conjectures provide a description of the discrete automorphic representations of connected reductive groups defined over global fields, as well as of the irreducible admissible representations of such groups defined over local fields. When the group in question is quasi-split, a precise form of these conjectures has been known for a long time and important special cases have recently been proved.

## Algebraic groups with good reduction

Let $G$ be an absolutely almost simple algebraic group over a field $K$. Assume that $K$ is equipped with a "natural" set $V$ of discrete valuations. We are interested in the $K$-forms of $G$ that have good reduction at all $v$ in $V$. In the case $K$ is the fraction field of a Dedekind domain, a similar question was considered by G.~Harder; the case where $K = \mathbb{Q}$ and $V$ is the set of all $p$-adic places was analyzed in detail by B.H.~Gross and B.~Conrad.