Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.17060

Modular functoriality for finite groups

Sean Cotner
View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop an extension of Deligne--Lusztig theory to certain (possibly infinite type) disconnected reductive groups arising from the special fibers of point stabilizers in the Bruhat--Tits building, which we call \emph{paraductive}. We then compute explicit lower bounds for the Tate cohomology of representations of paraductive groups, relating these to Shintani descent, Lusztig restriction, and the Glauberman correspondence. As an application, using Feng's modular functoriality and Scholze's independence of $\ell$, we compute the Fargues--Scholze L-parameters of non-singular depth $0$ cuspidal representations of a (possibly wildly ramified) reductive group over a nonarchimedean local field.

Explore related subjects

math.NTmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sean Cotner. 2026-09-15. Modular functoriality for finite groups. https://arxiv.org/abs/2609.17060

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

KEEP EXPLORING

Related papers

arXiv2026

Asymptotic density of k-almost primes

Landau's well known asymptotic formula $$N_k(x):=\ \mid\{n\leq x : Ω(n)=k\}\mid \ \sim \left( \frac{x}{\log x} \right) \frac{(\log\log x)^{k-1}}{(k - 1)!}\ \ (x \rightarrow \infty),$$ which also holds for $$π_k(x):=\ \mid\{n\leq x : ω(n)=k\}\mid,$$ is known to be fairly poor for $k > 1$, and when $k$ is allowed to tend to infinity with $x$, the study of $N_k(x)$ and $π_k(x)$ becomes very technical [1, Chapter II.6, $§$ 6.1, p.200]. I hope to show that the method described below provides not only a more accurate approach, but rather increases in its asymptotic accuracy as $k$ tends to infinity.

math.NT↗
arXiv2026

Real quadratic base changes for $\mathrm{GL}_3$ and integral periods relations

We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation of $\mathrm{GL}_3(\mathbb{Q})$ and the periods of its Arthur-Clozel's base change to some real quadratic field $E$. This generalizes earlier works of Tilouine-Urban and of Hida in the case of classical modular forms. The divisibility we prove involves a new kind of automorphic periods, defined using the middle degree of the cuspidal cohomology of $\mathrm{GL}_3(E)$, instead of the top or bottom degrees. We also investigate the Rogawski's stable base change from the quasi-split unitary group $U_E$ associated with $E$ to $\mathrm{GL}_3(E)$. In this situation, we also obtain some results toward a $p$-adic divisibility of automorphic periods.

math.NT↗