arXiv subjects

Thibaud van den Hove

Publications and source records attributed to Thibaud van den Hove.

8 recordsLinked to original sources

arXiv2026

The integral motivic Satake equivalence for ramified groups

We construct the geometric Satake equivalence for quasi-split reductive groups over nonarchimedean local fields, using étale Artin-Tate motives with $\mathbb{Z}[\frac{1}{p}]$-coefficients. We consider local fields of both equal and mixed characteristic. Along the way, we extend the work of Gaussent--Littelmann on the connection between LS galleries and MV cycles to the case of residually split reductive groups. As an application, we generalize Zhu's integral Satake isomorphism for spherical Hecke algebras to ramified groups. Moreover, for residually split groups, we define generic spherical Hecke algebras, and construct generic Satake and Bernstein isomorphisms.

math.RT↗

arXiv2026

Exponential motives on the affine Grassmannian

We develop a notion of exponential motives on general prestacks equipped with a $\mathbf{G}_a$-action, and compare them with Whittaker motives via Gaitsgory's Kirillov model. We then establish foundational results for exponential motives on affine flag varieties concerning Tate motives and t-structures. We use this to prove a motivic Casselman-Shalika equivalence, relating exponential Tate motives on the affine Grassmannian to ind-coherent sheaves on the classifying stack of the Langlands dual group. The decategorification of this equivalence provides a new construction of the Whittaker module for the spherical Hecke algebra which works for arbitrary coefficients, including a generic version.

math.AG↗

arXiv2026

Cycles on splitting models of Shimura varieties

We construct exotic Hecke correspondences between the special fibers of different PEL type Shimura varieties, when the local groups are restrictions of scalars of unramified groups. In particular, the local groups themselves are not necessarily unramified, and the Shimura varieties can have bad reduction. By adapting the methods of Xiao-Zhu in the case of good reduction, we use this to construct new instances of the geometric Jacquet-Langlands correspondence, including a motivic refinement, and verify generic instances of the Tate conjecture for the special fibers of these Shimura varieties at very special level. Our main tool is to resolve the integral models by the splitting models of Pappas-Rapoport. We also define splitting versions of the moduli stacks of local shtukas and affine Deligne-Lusztig varieties, and study their geometry.

math.AG↗

arXiv2025

Central motives on parahoric flag varieties

We construct a refinement of Gaitsgory's central functor for integral motivic sheaves, and show it preserves stratified Tate motives. Towards this end, we develop a reformulation of unipotent motivic nearby cycles, which also works over higher-dimensional bases. We moreover introduce Wakimoto motives and use them to show that our motivic central functor is t-exact. A decategorification of these functors yields a new approach to generic Hecke algebras for general parahorics.

math.AG↗

arXiv2025

The geometric Satake equivalence for integral motives

We prove the geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum. This refines previous versions of the geometric Satake equivalence for split reductive groups. Our new geometric results include Whitney--Tate stratifications of Beilinson--Drinfeld Grassmannians and cellular decompositions of semi-infinite orbits. With future global applications in mind, we also achieve an equivalence relative to a power of the affine line. Finally, we use our equivalence to give Tannakian constructions of Deligne's modification of the dual group and a modified form of Vinberg's monoid over the integers.

math.AG↗

arXiv2025

The stack of spherical Langlands parameters

For a reductive group over a nonarchimedean local field, we define the stack of spherical Langlands parameters, using the inertia-invariants of the Langlands dual group. This generalizes the stack of unramified Langlands parameters in case the group is unramified. We then use this stack to deduce the Eichler--Shimura congruence relations for Hodge type Shimura varieties, without restrictions on the ramification.

math.NT↗

arXiv2023

Quasi-isogeny groups of supersingular abelian surfaces via pro-étale fundamental groups

We consider a $J_b(\mathbb{Q}_p)$-torsor on the supersingular locus of the Siegel threefold constructed by Caraiani-Scholze, and show that it induces an isomorphism between a free group on a finite number of generators, and the group of self-quasi-isogenies of a supersingular abelian surface, respecting a principal polarization and a prime-to-$p$ level structure. Along the way, we classify certain pro-étale torsors in terms of the pro-étale fundamental group, describe the category of geometric covers of non-normal schemes, and use this to compute pro-étale fundamental groups of curves.

math.AG↗

arXiv2021

A classification of ideals in Steinberg and Leavitt path algebras over arbitrary rings

We give a one-to-one correspondence between ideals in the Steinberg algebra of a Hausdorff ample groupoid $G$, and certain families of ideals in the group algebras of isotropy groups in $G$. This generalises a known ideal correspondence theorem for Steinberg algebras of strongly effective groupoids. We use this to give a complete graph-theoretic description of the ideal lattice of Leavitt path algebras over arbitrary commutative rings, generalising the classification of ideals in Leavitt path algebras over fields.

math.RA↗