Date of Degree

9-2019

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Gautam Chinta

Committee Members

Alexander Gamburd

Abhijit Champanerkar

Subject Categories

Algebra | Number Theory

Keywords

subgroup growth zeta functions, cotype zeta functions, formal Hecke series of Hecke algebras, nilpotent groups, classical algebraic groups, counting finite p-groups

Abstract

This thesis is concerned with zeta functions and generating series associated with two families of groups that are intimately connected with each other: classical groups and class two nilpotent groups. Indeed, the zeta functions of classical groups count some special subgroups in class two nilpotent groups.

In the first chapter, we provide new expressions for the zeta functions of symplectic groups and even orthogonal groups in terms of the cotype zeta function of the integer lattice. In his paper on universal $p$-adic zeta functions, J. Igusa computed explicit formulae for the zeta functions of classical algebraic groups. These zeta functions were later extended to more general algebraic groups and their representations by M. du Sautoy and A. Lubotzky. We show computations of local Hecke series by A. Andrianov and later by T. Hina and T. Sugano lead to new expressions for the zeta functions of classical groups.

In the second chapter, we attempt to generalize the notion of a cotype zeta function to subgroup growth zeta functions of class two nilpotent groups. G. Chinta, N. Kaplan and S. Koplewitz recently used the cotype zeta function of the integer lattice to prove interesting facts about the proportion of sublattices of given corank and to match the distribution of sublattices with a particular Cohen-Lenstra distribution. Motivated by the method of subgroup counting in nilpotent groups in the seminal paper of F. J. Grunewald, D. Segal and G. C. Smith, we give a general definition of the cotype of a subgroup of finite index and use it to compute new multivariable zeta functions. Such zeta functions help in determining the distribution of subgroups of finite index.

In the last chapter, we make use of a generating series in several variables to count finite class two nilpotent groups. In 2009, C. Voll computed the number $g(n,2,2)$ of nilpotent groups of order $n$, of class at most $2$ generated by at most $2$ generators, by giving an explicit formula for the Dirichlet generating function of $g(n,2,2)$. Later in 2012, A. Ahmad, A. Magidin and R. F. Morse gave a direct enumeration of such groups building on the works of M. Bacon, L. Kappe, et al. We use their enumeration to provide a natural multivariable extension of the generating function counting such groups and as a result rederive Voll's explicit formula. Similar formulas or enumerations for finite groups of nilpotency class 2 on more than 2 generators or of at least class 3 on 2 or more generators is currently unknown.

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