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We study the value distribution of the Epstein zeta function for and a random lattice of large dimension . For any fixed and , we prove that the random variable has a limit distribution, which we give explicitly (here is the volume of the -dimensional unit ball). More generally, for any fixed , we determine the limit distribution of the random function , . After compensating for the pole at , we even obtain a limit result on the whole interval , and as a special case we deduce the following strengthening of a result by Sarnak and Strömbergsson concerning the height function of the flat torus : the random variable has a limit distribution as , which we give explicitly. Finally, we discuss a question posed by Sarnak and Strömbergsson as to whether there exists a lattice for which has no zeros in .
We construct holomorphic families of proper holomorphic embeddings of into (), so that for any two different parameters in the family, no holomorphic automorphism of can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of , we derive the existence of families of holomorphic -actions on () so that different actions in the family are not conjugate. This result is surprising in view of the long-standing holomorphic linearization problem, which, in particular, asked whether there would be more than one conjugacy class of -actions on (with prescribed linear part at a fixed point).
We introduce new methods from -adic integration into the study of representation zeta functions associated to compact -adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of -adic analytic pro- groups obtained from a global, “perfect” Lie lattice satisfy functional equations. In the case of “semisimple” compact -adic analytic groups, we exhibit a link between the relevant -adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by the centralizer dimension.
Based on this algebro-geometric description, we compute explicit formulas for the representation zeta functions of principal congruence subgroups of the groups , where is a compact discrete valuation ring of characteristic , and of the groups , where is an unramified quadratic extension of . These formulas, 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 . 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 defined over number fields.