Let G be a semisimple algebraic group defined over Q p , and let Γ be a compact open subgroup of G(Q p ). We relate the asymptotic representation theory of Γ and the singularities of the moduli space of G-local systems on a smooth projective curve, proving new theorems about both:1. We prove that there is a constant C, independent of G, such that the number of n-dimensional representations of Γ grows slower than n C , confirming a conjecture of Larsen and Lubotzky. In fact, we can take C = 3 · dim(E 8 ) + 1 = 745. We also prove the same bounds for groups over local fields of large enough characteristic.2. We prove that the coarse moduli space of G-local systems on a smooth projective curve of genus at least C/2 + 1 = 374 has rational singularities.For the proof, we study the analytic properties of push forwards of smooth measures under algebraic maps. More precisely, we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities.
Abstract. We introduce new methods from p-adic integration into the study of representation zeta functions associated to compact p-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of p-adic analytic pro-p groups obtained from a global, 'perfect' Lie lattice satisfy functional equations. In the case of 'semisimple' compact p-adic analytic groups, we exhibit a link between the relevant p-adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by centraliser dimension.Based on this algebro-geometric description, we compute explicit formulae for the representation zeta functions of principal congruence subgroups of the groups SL3(o), where o is a compact discrete valuation ring of characteristic 0, and of the groups SU3(O, o), where O is an unramified quadratic extension of o. These formulae, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type A2. Assuming a conjecture of Serre on the Congruence Subgroup Problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type A2 defined over number fields.
Let Γ be an arithmetic lattice in a semisimple algebraic group over a number field. We show that if Γ has the congruence subgroup property, then the number of n-dimensional irreducible representations of Γ grows like n α , where α is a rational number.
We begin the systematic study of the spectral theory of periodic Jacobi matrices on trees including a formal definition. The most significant result that appears here for the first time is that these operators have no singular continuous spectrum. We review important previous results of Sunada and Aomoto and present several illuminating examples. We present many open problems and conjectures that we hope will stimulate further work.
Abstract. In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.
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