Let @math be a number field with ring of integers @math . We compute explicitly the local factors of the normal zeta functions of the Heisenberg groups @math that are indexed by rational primes which are unramified in @math . We show that these local zeta functions satisfy functional equations upon the inversion of the prime.
More generally, it is of interest to compute the (normal) subgroup zeta functions of other finitely generated nilpotent groups, and their behavior under base extension. We refer the reader to @cite_6 for a comprehensive list of examples. In his MSc thesis @cite_4 , Bauer has generalized many of our results to the normal zeta functions of the higher Heisenberg groups @math for all @math , where @math is a centrally amalgamated product of @math Heisenberg groups. In other words, if @math is a ring and we view elements of @math as row vectors, and if @math denotes the @math identity matrix, then ) a , b R^m, , c R . The paper @cite_8 arose from the (uncompleted) project to compute the subgroup zeta functions @math .
@cite_4 @cite_6 @cite_8
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We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are groups of rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and @math -adic integration, we prove rationality and functional equations for almost all local factors of the Euler products of these zeta functions. We further give explicit formulae, in terms of Dedekind zeta functions, for the zeta functions of class- @math -nilpotent groups obtained from three infinite families of group schemes, generalizing the integral Heisenberg group. As an immediate corollary, we obtain precise asymptotics for the representation growth of these groups, and key analytic properties of their zeta functions, such as meromorphic continuation. We express the local factors of these zeta functions in terms of generating functions for finite Weyl groups of type @math . This allows us to establish a formula for the joint distribution of three functions, or statistics'', on such Weyl groups. Finally, we compare our explicit formulae to @math -adic integrals associated to relative invariants of three infinite families of prehomogeneous vector spaces. We compute explicitly the normal zeta functions of the Heisenberg groups H(R), where R is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form H(O K ), where O K is the ring of integers of an arbitrary number field K, at the rational primes which are non-split in K. We show that these local zeta functions satisfy functional equations upon inversion of the prime. Let F2,d denote the free class-2-nilpotent group on d generators. We compute the normal zeta functions ζ � F2,d (s), prove that they satisfy local functional equations and determine their abscissae of convergence and pole orders. Abstract We calculate the representation zeta function of the Heisenberg group over the integers of a quadratic number field. In general, the representation zeta function of a finitely generated torsion-free nilpotent group enumerates equivalence classes of representations, called twist-isoclasses. This calculation is based on an explicit description of a representative from each twist-isoclass. Our method of construction involves studying the eigenspace structure of the elements of the image of the representation and then picking a suitable basis for the underlying vector space. We introduce a new method to calculate local normal zeta functions of finitely generated, torsion-free nilpotent groups. It is based on an enumeration of vertices in the Bruhat-Tits building for Sln(Qp). It enables us to give explicit computations for groups of class 2 with small centres and to derive local functional equations. Examples include formulae for non-uniform normal zeta functions. We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or T -)groups, and the normal zeta functions of T -groups of class 2. In particular we solve the two problems posed in [9, Section 5]. We deduce our theorems from a ‘blueprint result’ on certain p-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to ‘linearise’ the problems of counting subgroups and representations in T -groups, respectively. Let @math be a number field with ring of integers @math . We compute explicitly the local factors of the normal zeta functions of the Heisenberg groups @math that are indexed by rational primes which are unramified in @math . We show that these local zeta functions satisfy functional equations upon the inversion of the prime. 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 SL_3(O), where O is a compact discrete valuation ring of characteristic 0, and of the corresponding unitary groups. 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 A_2. 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 A_2 defined over number fields. The pro-isomorphic zeta function ( ^ _ (s) ) of a finitely generated nilpotent group ( ) is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of ( ). Such zeta functions can be expressed as Euler products of p-adic integrals over the ( Q _p )-points of an algebraic automorphism group associated to ( ). In this way they are closely related to classical zeta functions of algebraic groups over local fields. This is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. These zeta functions encode, respectively, the numbers of isomorphism classes of irreducible complex representations of finite dimensions and the numbers of conjugacy classes of congruence quotients of the associated groups. In this paper, we show that such bivariate zeta functions specialise to (univariate) class number zeta functions. In case of nilpotency class 2, bivariate representation zeta functions also specialise to (univariate) twist representation zeta functions. Moreover, we show that these zeta functions satisfy Euler factorisations and almost all of their Euler factors satisfy rationality and functional equations. In the second part of this work, we compute the above mentioned zeta functions of three infinite families of nilpotent groups of class two and deduce some combinatorial corollaries. We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class- @math -nilpotent Lie rings, introduced in [8], in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced, and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question. Publisher Summary This chapter focuses on zeta functions of finite graphs and representations of p-adic groups. It discusses two different subjects: first is a combinatorial problem in algebraic graph theory, and the other is arithmetic of discrete subgroups of p-adic groups and their representations. The chapter presents the notation and basic definitions in graph theory. It also presents a generalization of the definition of zeta function. Spectrum of a finite multigraph is analyzed in the chapter. Moreover, the chapter also describes harmonic functions and the Hodge decomposition. The chapter also presents the computation of zeta functions Zx(u) for some well known families of graphs. These computations give many examples of graphs that are not Ramanujan graphs. We develop a practical method for computing local zeta functions of groups, algebras, and modules in fortunate cases. Using our method, we obtain a complete classification of generic local representation zeta functions associated with unipotent algebraic groups of dimension at most six. We also determine the generic local subalgebra zeta functions associated with gl2(Q). Finally, we introduce and compute examples of graded subobject zeta functions. We enumerate traceless square matrices over finite quotients of compact discrete valuation rings by their image sizes. We express the associated rational generating functions in terms of statistics on symmetric and hyperoctahedral groups, viz. Coxeter groups of types A and B, respectively. These rational functions may also be interpreted as local representation zeta functions associated to the members of an infinite family of finitely generated class-2-nilpotent groups. We conjecture that every finite groupGhas a short presentation (in terms of generators and relations) in the sense that the totallengthof the relations is (log|G|)O(1). We show that it suffices to prove this conjecture for simple groups. Motivated by applications in computational complexity theory, we conjecture that for finite simple groups, such a short presentation is computable in polynomial time from the standard name ofG, assuming in the case of Lie type simple groups overGF(pm) that an irreducible polynomialfof degreemoverGF(p) and a primitive root ofGF(pm) are given. We verify this (stronger) conjecture for all finite simple groups except for the three families of rank 1 twisted groups: we do not handle the unitary groupsPSU(3, q) = 2A2(q), the Suzuki groupsSz(q) = 2B2(q), and the Ree groupsR(q) = 2G2(q). In particular,all finite groups G without composition factors of these types have presentations of length O((log|G|)3). For groups of Lie type (normal or twisted) of rank ≥ 2, we use a reduced version of the Curtis–Steinberg–Tits presentation. Comparison between formulae for the counting functions of the heights tn of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the tn are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian Hcl. Many features of Hcl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the tn have a similar structure to those of the semiclassical En; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian Hcl=XP. In this paper, we deduce the vanishing of Selmer groups for the Rankin-Selberg convolution of a cusp form with a theta series of higher weight from the nonvanishing of the associated @math -value, thus establishing the rank 0 case of the Bloch-Kato conjecture in these cases. Our methods are based on the connection between Heegner cycles and @math -adic @math -functions, building upon recent work of Bertolini, Darmon and Prasanna, and on an extension of Kolyvagin's method of Euler systems to the anticyclotomic setting. In the course of the proof, we also obtain a higher weight analogue of Mazur's conjecture (as proven in weight 2 by Cornut-Vatsal), and as a consequence of our results, we deduce from Nekovar's work a proof of the parity conjecture in this setting. This paper studies the Lerch zeta-function L(λ αs) which was introduced by M. Lerch in 1887. The case of non-integer parameter λ is considered, and then L(λ αs) is an entire function. First, the universality of this function with transcendental parameter a is obtained. From this the functional independence is derived. The third part of the paper is devoted to the zero-distribution of the Lerch zeta-function. Zero-free regions are indicated on the right as well on the left of the imaginary axis when λ ≠ 1 2. If λ = 1 2, then zero-free regions lie in the left half-plane. Finally, estimates for the number of zeros in the critical strip are given. We prove a two-part theorem on local ideal zeta functions of Lie rings of upper-triangular matrices. First, we prove that these local zeta functions display a strong uniformity. Secondly, we prove that these zeta functions satisfy a local functional equation. Some explicit examples of these zeta functions are also presented. Finally, we consider certain quotients of these Lie rings, showing that the strong uniformity continues to hold, and that under certain circumstances the functional equation does too. In this paper, new probability estimates are derived for ideal lattice codes from totally real number fields using ideal class Dedekind zeta functions. In contrast to previous work on the subject, it is not assumed that the ideal in question is principal. In particular, it is shown that the corresponding inverse norm sum depends not only on the regulator and discriminant of the number field, but also on the values of the ideal class Dedekind zeta functions. Along the way, we derive an estimate of the number of elements in a given ideal with a certain algebraic norm within a finite hypercube. We provide several examples which measure the accuracy and predictive ability of our theorems.