arXiv subjects

Jack Sempliner

Publications and source records attributed to Jack Sempliner.

4 recordsLinked to original sources

arXiv2026

On the normality of the commuting scheme

If $G$ is a connected reductive algebraic group over a field $k$ of characteristic $\ell$, we show that both the Lie-theoretic and group-theoretic commuting schemes associated to $G$ are normal and Cohen--Macaulay when $\ell = 0$, or $\ell$ is larger than the Coxeter numbers of all simple factors of $G$, and that $\ell \neq 19$ (resp. $\ell \neq 31$) when $G$ has a simple factor of type $E_7$ (resp. $E_8$). The main new ingredients are provided by studying the unipotent categorical Langlands functor of Zhu. Along the way we prove a $t$-exactness result for a finite unipotent categorical Langlands functor which may be of independent interest.

math.RT↗

arXiv2026

A geometric Jacquet--Langlands correspondence for Shimura varieties

We formulate a conjecture predicting the existence of exotic isomorphisms between Igusa stacks associated to different Shimura data whose underlying groups are pure inner forms of each other, and prove it in many cases of interest. In combination with the spectral action of Fargues--Scholze, this allows us to relate the cohomology groups of the two Shimura varieties, giving a geometric incarnation of the Jacquet--Langlands correspondence. An important ingredient in our proofs is the work of Xiao--Zhu on exotic Hecke correspondences between the perfect special fibers of different Shimura varieties, which we reinterpret in terms of exotic isomorphisms of perfect Igusa stacks.

math.NT↗

arXiv2025

On the Piatetski-Shapiro construction for integral models of Shimura varieties

We study the Piatetski-Shapiro construction, which takes a totally real field F and a Shimura datum (G,X) and produces a new Shimura datum (H,Y). If F is Galois, then the Galois group Gamma of F acts on (H,Y), and we show that the Gamma-fixed points of the Shimura varieties for (H,Y) recover the Shimura varieties for (G,X) under some hypotheses. For Shimura varieties of Hodge type with parahoric level, we show that the same is true for the p-adic integral models constructed by Pappas--Rapoport. We also study the Gamma-fixed points of the Igusa stacks of Zhang for (H,Y) and prove optimal results.

math.NT↗

arXiv2024

Cocycles for Kottwitz cohomology

In his work on defining the pointed set B(G) for all local and global fields, Kottwitz introduced certain Galois gerbes and considered their 'algebraic' cohomology with values in algebraic groups. However, the gerbes so constructed are only canonical up to conjugation by their bands. This is enough in order to ensure that the cohomology set B(G) is canonical, but not enough data to pin down the set of algebraic cocycles used to compute the set B(G). For arithmetic applications it is often desirable to have the finer control provided by these cocycles readily at hand. To this end we propose in this paper a framework with which to work with such spaces of algebraic Kottwitz cocycles. This will play a crucial role in our forthcoming work on the formalism of Shimura varieties.

math.NT↗