Asymptotics of analytic torsion for congruence quotients of $\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$
Tim Berland
TL;DR
This work sharpens the asymptotics for analytic torsion growth on congruence quotients of the noncompact symmetric space $SL(n,\mathbb{R})/SO(n)$ by expressing $\log T_{X(N)}(\tau)$ as $\log T^{(2)}_{X}(\tau)$ plus a controlled error term that scales with the volume and level. The authors develop strong heat-kernel bounds under a $\lambda$-strongly acyclic spectral gap, prove the existence of infinitely many such representations for $\delta(G(\mathbb{R}))\ge 1$, and adapt the Arthur trace formula to a noncocompact, level-$N$ tower to track error terms precisely. They obtain a bound of the form $\log T_{Y(N)}(\tau) = \log T^{(2)}_{Y(N)}(\tau) + O(\operatorname{vol}(Y(N))N^{-(n-1)}\log^a N)$, equivalently $O(\operatorname{vol}(Y(N))^{1-1/(n+1)}\log^a\operatorname{vol}(Y(N)))$, thereby advancing understanding of torsion growth and its arithmetic implications. The results connect to the Ash conjecture via torsion in cohomology and build on prior work by Bergeron–Venkatesh and Matz–Müller, potentially informing the construction of Galois representations in arithmetic settings.
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.