Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2507.10339

Asymptotics of analytic torsion for congruence quotients of $\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$

Tim Berland
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we prove a sharpened asymptotic for the growth of analytic torsion of congruence quotients of $\SL(n,\R)/\SO(n)$ in terms of the volume. The result is based on bounds on the trace of the heat kernel, allowing control of the large time behaviour of certain orbital integrals, as well as a careful analysis of error terms. The result requires the existence of $λ$-strongly acyclic representations, which we define and show exists in plenitude for any $λ>0$. The motivation is possible applications to torsion in the cohomology of arithmetic groups.

Explore related subjects

math.NTmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tim Berland. 2025-10-06. Asymptotics of analytic torsion for congruence quotients of $\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$. https://arxiv.org/abs/2507.10339

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

KEEP EXPLORING

Related papers

arXiv2026

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.

math.NT↗
arXiv2026

Exceptional poles of archimedean Rankin-Selberg L-functions for irreducible generic representations of GL(n,R)

For irreducible generic representations $π_1$ and $π_2$ of $\operatorname{GL}_n(\mathbb R)$, we prove that the notions of exceptional pole of type $1$ and type $2$ coincide at every level. When both representations are in general position, we use this identification to express the Rankin--Selberg $L$-function $L(s,π_1\timesπ_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of their derivatives.

math.NT↗