Misc

The aim of this workshop was to discuss recent developments on the arithmetic of Shimura varieties and on related topics within the Langlands program, and to initiate and support further research in this direction. The talks presented new methods and results covering topics ranging from geometric questions on the reduction of Shimura varieties to representations in their cohomology, automorphic forms, and questions on the geometry and arithmetic of moduli spaces of bundles on the Fargues-Fontaine curve.

Arithmetic geometry is at the interface between algebraic geometry and number theory, and studies schemes over the ring of integers of number fields, or their p -adic completions. The talks covered a wide range of topics including the categorical Langlands program, Shimura varieties, complex and p -adic Hodge theory, homotopy theory, and Diophantine geometry.

In this note, we discuss several forms of the Eichler-Shimura relation for the compactly supported cohomology of local Shimura varieties, using the work of Fargues-Scholze.

In 1994, van Geemen and Top constructed a non-selfdual motive of rank three over $\mathbb{Q}$ conjecturally associated with a cuspidal non-selfdual automorphic representation of $\mathrm{GL}_3(\mathbb{A}_{\mathbb{Q } })$ of level $\Gamma_0(128)$. They experimentally confirmed the coincidence of the local $L$-factors at finitely many primes using computer. In this paper, we shall prove the coincidence of the local $L$-factors at every prime. To show this, we use the recent results of Harris-Lan-Taylor-Thorne and Scholze on the construction of Galois representations, and Greni\'e's results to compare three-dimensional $2$-adic Galois representations. We also prove the local-global compatibility at $p = 2$, including the case $p = \ell$.

In this paper we prove a new characterization of the distinguished unipotent orbits of a connected reductive group over an algebraically closed field of characteristic 0. For classical groups we prove the characterization by a combinatorial computation, and for exceptional groups we check it with a computer. This characterization is needed in the theory of cuspidal sheaves on the stack of L-parameters in forthcoming work of the first two named authors.

We introduce a logarithmic variant of the notion of $\delta$-rings, which we call $\delta_{\log}$-rings, and use it to define a logarithmic version of the prismatic site introduced by Bhatt and Scholze. In particular, this enables us to construct the Breuil-Kisin cohomology in the semistable case.

We study heights of motives with integral coefficients over number fields introduced by Kato. It is a generalization of the Faltings height of an abelian variety and we establish generalizations of some properties of the Faltings height in our context as conjectured by Kato. This sheds some light on integral structures of motives.

Using the work of Fargues-Scholze, we prove a vanishing theorem for the generic unramified part of the cohomology of local Shimura varieties of general linear groups. This gives an alternative approach to vanishing results of Caraiani-Scholze for the cohomology of unitary Shimura varieties.