arXiv subjects
Publications and source records attributed to Jasmin Matz.
We prove that for almost all symmetric spaces $X$ and for any sequence of compact locally symmetric spaces $Y_n$ which is uniformly discrete, has a uniform spectral gap, and converges in the sense of Benjamini--Schramm to $X$, the joint eigenfunctions of all invariant differential operators on $Y_n$ delocalize on average when their spectral parameters are taken to lie in a fixed spectral window.
In this paper we define a regularized version of the analytic torsion for quotients of a symmetric space of non-positive curvature by arithmetic lattices. The definition is based on the study of the renormalized trace of the corresponding heat operators, which is defined as the geometric side of the Arthur trace formula applied to the heat kernel. Then we study the limiting behavior of the analytic torsion as the lattices run through a sequence of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for sequences of principal congruence subgroups, which converge to 1 at a fixed finite set of places and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the $L^2$-analytic torsion.
We study the limiting behavior of Maass forms on Benjamini-Schramm convergent sequences of compact quotients of ${\rm SL}_d(\mathbb{R})/{\rm SO}(d)$, $d\ge 3$, whose spectral parameter stays in a fixed window. We prove a form of Quantum Ergodicity in this level aspect which extends results of Le Masson and Sahlsten to the higher rank case.
We establish the Sato-Tate equidistribution of Hecke eigenvalues on average for families of Hecke--Maass cusp forms on SL(n,R)/SO(n). For each of the principal, symmetric square and exterior square L-functions we verify that the families are essentially cuspidal and deduce the level distribution with restricted support of the low-lying zeros. We also deduce average estimates toward Ramanujan.
In this paper, we study the asymptotic behavior of the traces of Hecke operators for spherical discrete automorphic representations of fixed level on general split reductive groups over $\mathbb{Q}$. Under a condition on the analytic behavior of intertwining operators, which is known for the classical groups and the exceptional group $G_2$, we obtain the expected asymptotics in terms of the spherical Plancherel measure and an explicit estimate for the remainder.
We prove that the family of lattices ${\rm SL}_2(\mathcal{O}_F)$, $F$ running over number fields with fixed archimedean signature $(r_1, r_2)$, in ${\rm SL}_2(\mathbb{R}^{r_1}\oplus\mathbb{C}^{r_2})$ has the limit multiplicity property.
In [MzM] we defined a regularized analytic torsion for quotients of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ by arithmetic lattices. In this paper we study the limiting behaviour of the analytic torsion as the lattices run through sequences of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for principal congruence subgroups and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the $L^2$-analytic torsion.
In this paper we define a regularized version of the analytic torsion for arithmetic quotients of the symmetric space SL(n,R)/\SO(n). The definition is based on the study of the renormalized trace of the corresponding heat operators, which is defined as the geometric side of the Arthur trace formula applied to the heat operator
We prove a mean value theorem for the traces of Hecke operators acting on cusp forms of GL(n) over imaginary quadratic number fields together with an upper bound for the error term depending explicitly on the Hecke operator.
We give upper bounds for the absolute value of the global coefficients appearing in the fine geometric expansion of Arthur's trace formula for GL(n).
We define zeta functions for the adjoint action of GL(n) on its Lie algebra and study their analytic properties. For n<4 we are able to fully analyse these functions, and recover the Shintani zeta function for the prehomogeneous vector space of binary quadratic forms for n=2. Our construction naturally yields a regularisation, which is necessary for the improvement of the properties of these zeta function, in particular for the analytic continuation if n>2. We further obtain upper and lower bounds on the mean value X^{-5/2}\sum_{E} \res_{s=1}ζ_E(s) as X->\infty, where E runs over totally real cubic number fields whose second successive minimum of the trace form on its ring of integers is bounded by X. To prove the upper bound we use our new zeta function for GL(3). These asymptotic bounds are a first step towards a generalisation of density results obtained by Datskovsky in case of quadratic field extensions.