Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2503.00696

On Almost Strong Approximation in Reductive Algebraic Groups

Andrei S. Rapinchuk·Wojciech Tralle
View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate a slight weakening of the classical property of strong approximation, which we call almost strong approximation, for connected reductive algebraic groups over global fields with respect to special sets of valuations. While nonsimply connected groups (in particular, all algebraic tori) always fail to have strong approximation -- and even almost strong approximation -- with respect to any finite set of valuations, we show that under appropriate assumptions they do have almost strong approximation with respect to certain infinite sets of valuations that can be characterized in terms of Dirichlet density and include tractable sets of valuations, i.e. those sets that contain all archimedean valuations and a generalized arithmetic progression minus a set of Dirichlet density zero. Almost strong approximation is likely to have a variety of applications, and as an example we use almost strong approximation in tori with respect to tractable sets to extend the essential part of the result of Radhika and Raghunathan on the congruence subgroup problem for inner forms of type $\textsf{A}_n$ to all absolutely almost simple simply connected groups. This version of the paper has been updated to reflect recent results of Y. Cao and Y. Wang (arXiv:2511.00824).

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Andrei S. Rapinchuk, Wojciech Tralle. 2026-09-07. On Almost Strong Approximation in Reductive Algebraic Groups. https://arxiv.org/abs/2503.00696

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Shrinking dynamic on multidimensional tropical series

Let $Ω\subset\mathbb R^n$ be a compact convex domain. An $Ω$-tropical series is a nonnegative, concave, integral-slope, piecewise-affine function on $Ω$ that vanishes on $\partialΩ$. For a finite set $P\subsetΩ^\circ$, we study the least such function above prescribed initial data whose corner locus contains $P$. It is obtained by repeatedly applying one-point shrinking operators $G_p$. We prove that every fair order of these operators stabilizes after finitely many nontrivial steps. We also describe an event-driven implementation that records the lowest monomials at each point and updates only affected watcher lists. Finally, we show that, on every compact subset of $Ω^\circ$, the resulting dynamics can be approximated by a finite path whose intermediate tropical hypersurfaces have only mild singularities on that compact set; equivalently, the corresponding local cells of the dual regular subdivision contain no lattice points other than their vertices.

math.AG↗
arXiv2026

A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus

Let $C$ be an algebraically closed perfectoid field over $\mathbb{Q}_p$ with the ring of integer $\mathcal{O}_C$ and the infinitesimal thickening $\Ainf$. Let $\mathfrak X$ be a semi-stable formal scheme over $\mathcal{O}_C$ with a fixed flat lifting $\widetilde{\mathfrak X}$ over $\Ainf$. Let $X$ be the generic fiber of $\mathfrak{X}$ and $\widetilde X$ be its lifting over $\BdRp$ induced by $\widetilde{\mathfrak X}$. Let $\MIC_r(\widetilde X)^{{\rm H}\text{-small}}$ and $\rL\rS_r(X,\BBdRp)^{{\rm H}\text{-small}}$ be the $v$-stacks of rank-$r$ Hitchin-small integrable connections on $X_{\et}$ and $\BBdRp$-local systems on $X_{v}$, respectively. In this paper, we establish an equivalence between these two stacks by introducing a new period sheaf with connection $(\calO\bB_{\dR,\pd}^+,\rd)$ on $X_{v}$.

math.AG↗
arXiv2026

A refinement of the coherence conjecture of Pappas and Rapoport

The coherence conjecture of Pappas and Rapoport, proved by Zhu, asserts the equality of dimensions for the global sections of a line bundle over a spherical Schubert variety in the affine Grassmannian and those of another line bundle over a certain union of Schubert varieties in a partial affine flag variety. We refine this equality of dimensions to an isomorphism of representations. The comparison is established by introducing a parahoric Bruhat-Tits group scheme $\mathcal{G}$ over the affine line, ramified at 0. We further strengthen this comparison by equipping any line bundle on the global Schubert variety of $\mathcal{G}$ with a unique equivariant structure under the global jet group scheme. As an application, we obtain new relations among affine Demazure modules.

math.AG↗