Geometry and arithmetic of moduli spaces

Raphaël Beuzart-Plessis: On the Gan-Gross-Prasad conjecture for unitary groups
The Gan-Gross-Prasad conjecture stipulates the existence of relations between the central values of certain $L$-functions and some automorphic periods on classical groups. It can be seen as a higher rank generalization of a famous formula of Waldspurger for toric periods on $\text{GL}(2)$. In this talk, I plan to give a survey of some recent progress leading to a complete proof in the case of unitary groups. The basic strategy consists in a comparison of relative trace formulae, originally proposed by Jacquet and Rallis, but this will be explained and motivated during the talk. This is based on joint work with Pierre-Henri Chaudouard and Michal Zydor.

Carel Faber: Arithmetic aspects of the cohomology of moduli spaces of curves
The moduli spaces of stable $n$-pointed curves of genus $g$ are smooth and proper stacks over the integers. This imposes strong restrictions on the automorphic and Galois representations that can appear in the cohomology. E.g., the cusp forms associated to non-Tate cohomology of the moduli spaces in genus $1$ are of level one. I will present an overview of known and conjectured results about the cohomology of moduli spaces of (pointed) curves of low genus, focusing on the part of the cohomology associated to modular forms. I will also discuss some questions that seem relevant to me.

Frank Gounelas: Some new results on curves on K3s
The moduli theory of curves on K3 surfaces is now considered classical going back to Mukai, Mumford and Lazarsfeld in the 1970s. Questions abound about (non-)existence of curves which vary maximally in moduli, or, respectively, do not vary at all (the isotrivial case) - a classical open problem in the area, which goes back to Deligne/Schoen/Serre is whether a very general K3 surface can be dominated by the product of two smooth curves. I will summarise some recent results and in particular focus on recent work with Chen and Dutta regarding smooth curves.

Eugen Hellmann: Moduli spaces of equivariant vector bundles on the Fargues—Fontaine curve
The Fargues—Fontaine curve is the fundamental curve of $p$-adic Hodge theory and equivariant vector bundles on it naturally arise as a generalization of $p$-adic representations of Galois groups of $p$-adic local fields. Their moduli spaces (and coherent sheaves thereon) play an important role in the theory of $p$-adic automorphic forms and are expected to replace the moduli space of $L$-parameters in a categorical approach to the $p$-adic local Langlands correspondence. In this talk I will discuss similarities and differences of these moduli spaces with the classical theory of moduli spaces of vector bundles on algebraic curves.

Ben Heuer: The $p$-adic Simpson correspondence via moduli spaces
In analogy to the complex Corlette-Simpson correspondence, $p$-adic non-abelian Hodge theory studies $p$-adic representations of fundamental groups of smooth projective varieties with the methods of $p$-adic Hodge theory. In this talk, I will explain how this is achieved by the $p$-adic Simpson correspondence, which relates pro-étale vector bundles to Higgs bundles. I will then sketch how in joint work with Daxin Xu, we interpret the correspondence more geometrically as a twisted isomorphism between the moduli stacks of either side.

Bruno Klingler: Recent progress on Hodge loci
Given a quasi projective family $S$ of complex algebraic varieties, its Hodge locus is the locus of points of $S$ where the corresponding fiber admits exceptional Hodge classes (conjecturally: exceptional algebraic cycles). In this talk I will survey the many recent advances in our understanding of such loci, both geometrically and arithmetically, as well as the remaining open questions.

Laura Pertusi: Non-commutative abelian surfaces and generalized Kummer varieties
Examples of non-commutative K3 surfaces arise from semiorthogonal decompositions of the bounded derived category of certain Fano varieties. The most interesting cases are those of cubic fourfolds and Gushel-Mukai varieties of even dimension. Using the deep theory of families of stability conditions, locally complete families of hyperkahler manifolds deformation equivalent to Hilbert schemes of points on a K3 surface have been constructed from moduli spaces of stable objects in these non-commutative K3 surfaces. On the other hand, an explicit description of a locally complete family of hyperkahler manifolds deformation equivalent to a generalized Kummer variety is not yet available.
In this talk we will construct families of non-commutative abelian surfaces as equivariant categories of the derived category of K3 surfaces which specialize to Kummer K3 surfaces. Then we will explain how to induce stability conditions on them and produce examples of locally complete families of hyperkahler manifolds of generalized Kummer deformation type. This is a joint work in progress with Arend Bayer, Alex Perry and Xiaolei Zhao.

Michael Rapoport: An arithmetic fundamental lemma for the spherical Hecke algebra
We formulate an AFL conjecture for the whole spherical Hecke algebra, generalizing Wei Zhang's AFL which concerns the unit element of the spherical Hecke algebra. Even the formulation of the conjecture is non-trivial since a definition of "integral" Hecke operators is needed, a problem that also appears in other contexts. The conjecture can be proved in the first non-trivial case. This is joint work with Chao Li and Wei Zhang.

Ziquan Yang: Pointwise reduction criterion for $p$-adic local systems
Let $S$ be a connected smooth rigid analytic variety over a $p$-adic field $K$ and let $T$ be a $p$-adic local system over $S$. A celebrated theorem of Liu and Zhu says that if $V$ is de Rham at one classical point, then $V$ is globally de Rham. When $S$ has good reduction over $O_K$, one naturally asks about analogous statements when we replace "de Rham" by "crystalline" or "semistable". It is well known that the naive analogues are false. In a joint work with Haoyang Guo, we prove that Liu-Zhu's result can be remedied if one tests at "sufficiently many" points, when we choose a good integral model for $S$. In particular, if $V$ is crystalline or semistable at every classical point, then it is crystalline or semi-stable. I will also discuss the $l$-adic analogue of this result as well as its relation to purity statements. If time permits, I will discuss some speculations yet to be affirmed.

Wei Zhang: Faltings heights and the sub-leading terms of adjoint $L$-functions
The Kronecker limit formula may be interpreted as an equality relating the Faltings height of an CM elliptic curve to the sub-leading term (at $s=0$) of the Dirichlet $L$-function of an imaginary quadratic character. Colmez conjectured a generalization relating the Faltings height of any CM abelian variety to the subheading terms of certain Artin $L$-functions. In this talk we will formulate a “non-Artinian” generalization of (averaged) Colmez conjecture, relating the following two quantities: (1) the Faltings height of certain cycles on unitary Shimura varieties, and (2) the sub-leading terms of the adjoint $L$-functions of (cohomological) automorphic representations of $U(n)$.
The case $n=1$ amounts to the averaged Colmez conjecture. We are able to prove our conjecture when $n=2$. Work in progress with Ryan Chen and Weixiao Lu.