## A ribbon obstruction and derivatives of knots

We define an obstruction for a knot to be Z[Z]-homology ribbon, and use this to provide restrictions on the integers that can occur as the triple linking numbers of derivative links of knots that are either homotopy ribbon or doubly slice. Our main application finds new non-doubly slice knots. In particular this gives new information on the doubly solvable filtration of Taehee Kim: doubly algebraically slice ribbon knots need not be doubly (1)-solvable, and doubly algebraically slice knots need not be (0.5,1)-solvable. We also discuss potential connections to unsolved conjectures in knot concordance, such as generalised versions of Kauffman's conjecture. Moreover it is possible that our obstruction could fail to vanish on a slice knot.

## Anticyclotomic p-adic spinor L-functions for PGSp(4)

The Boecherer conjecture is a generalization of the Waldspurger formula and relates squares of Bessel periods

of genus two Siegel cusp forms to the central L-values.

This conjecture was currently proved by Furusawa and Morimoto for the special Bessel period, and the general

case is a work-in-progress. In this talk I will construct a square root of an anticyclotomic p-adic L-function with

explicit interpolation formulas for Siegel cusp forms of genus 2 and scalar weight greater than 1 with respect to

paramodular groups of square-free level, assuming the Boecherer conjecture for the L-values with anticyclotomic twist.

This is a joint work with Ming-Lun Hsieh.

## Theta integrals and generalized error functions

Recently Alexandrov, Banerjee, Manschot and Pioline constructed generalizations of Zwegers theta functions for lattices of signature $(n-2,2)$. They also suggested a generalization to the case of arbitrary signature $(n-q,q)$, and this case was subsequently proved by Nazaroglu. Their functions, which depend on certain collections $\mathcal{C}$ of negative vectors, are obtained by 'completing' a non-modular holomorphic generating series

by means of a non-holomorphic theta type series involving generalized error functions.

In joint work with Jens Funke, we show that their completed modular series arises as integrals of the $q$-form valued theta functions, defined in old joint work of the author and John Millson, over a certain singular $q$-cube determined by the data $\mathcal{C}$. This gives an alternative construction of such series and a conceptual basis for their modularity. I will discuss the simplicial case and a curious 'convexity' problem for Grassmannians that arises in this context.

## Tamagawa numbers in the function field case

The Tamagawa number of a semi-simple simply connected group over a function field is equal to 1, and the space of connections on a principal bundle is contractible. In the 1980s, Professor Harder pointed out to me the intriguing but mysterious relationship between these two statements. A recent proof of Weil's Tamagawa number conjecture in the function field case by Gaitsgory and Lurie using derived geometry sheds new light on this old conundrum. I will report on this and related work.

## 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. I will discuss the emerging results in the higher-dimensional situation where $K$ is the function field $k(C)$ of a smooth geometrically irreducible curve $C$ over a number field $k$, or even an arbitrary finitely generated field. I will also indicate connections with other questions involving the genus of $G$ (i.e., the set of isomorphism classes of $K$-forms of $G$ having the same isomorphism classes of maximal $K$-tori as $G$), the Hasse principle, weakly commensurable Zariski-dense subgroups, etc. (Joint work with V. Chernousov and I. Rapinchuk.)

## 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. For non-quasi-split groups (such as special linear, symplectic, and special orthogonal groups over division algebras), the conjectures have been vague and their proof out of reach.

In this talk we will present a precise formulation of the local and global conjectures for arbitrary connected reductive groups in characteristic zero. It is based on the construction of certain Galois gerbes defined over local and global fields and the study of their cohomology. These cohomological results place the conjectures for classical groups within reach of the currently available methods.

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

tba.

## 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. The main question for defining a stability condition is to try to construct a deformation of an object to bring out the first step of its Harder-Narasimhan filtration. We sketch our current view of this process and discuss some of the remaining unsolved problems.

## 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. Interesting new cases involve (i) Rankin-Selberg L-functions over a CM field, (ii) the degree $2n$ $L$-functions for $SO(n,n)$, in joint work with Bhagwat, and (iii) the degree $n^2$ Asai $L$-functions for $GL(n)$ for a quadratic extension of totally real fields, in joint work with Krishnamurthy. In this talk, which is a celebration of Harder's ideas on cohomology and $L$-functions, I will review the general principles of Eisenstein cohomology that apply to all these different contexts and how they give arithmetic information about $L$-values.

## 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 $SL(2,\mathcal{O}_d)$. Its points correspond to (principally polarized) Abelian surfaces with anti-holomorphic multiplication by $\mathcal{O}_d$, meaning a real action of $\mathcal{O}_d$ such that $\sqrt{d}$ acts anti-holomorphically. Using Deligne`s description of the category of ordinary Abelian varieties, we are able to make sense of anti-holomorphic multiplication for ordinary Abelian varieties over a finite field, thus providing a candidate for the ordinary points, mod $p$ of these hyperbolic $3$-manifolds. This is ongoing joint work with Yung sheng Tai.

## tba.

tba.

## 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, $w<24$. Under a suitable form of GRH, we may replace $f(w)$ by $8*\Pi*e^{-\psi(1+w)}$, where $\psi$ is the classical digamma function.

## 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.

## 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.

## Collapse with bounded curvature

## New guests at the MPIM

## tba

## tba

## A heuristic for a Manin-type conjecture for K3 surfaces

In the late 1980's Manin conjectured the asymptotic growth of the number of rational points up to bounded height in a suitable open subvariety of any Fano variety. Since his original conjecture, much work has been done in (dis)proving several cases and many modifications and refinements have been proposed. My upcoming PhD thesis partly concerns the 'next case' in dimension 2: that of K3 surfaces. Using the circle method, I was able to find heuristics for diagonal quartic surfaces that agree with numerical data produced by my supervisor Ronald van Luijk. In my talk I will explain the method, sketch the proof, and discuss its limitations, along with explaining some problems that need to be overcome before a reasonably strong conjecture can be stated.