Harmonic Analysis on the Positive Rationals. Computation of Character Sums

It follows from the work of Artin and Hooley that, under assumption of the generalized Riemann hypothesis, the density of the set of primes $q$ for which a given non-zero rational number $r$ is a primitive root modulo $q$ can be written as an infinite product $\prod_p \delta_p$ of local factors $\delta_p$ reflecting the degree of the splitting field of $X^p-r$ at the primes $p$, multiplied by a somewhat complicated factor that corrects for the `entanglement' of these splittin...

In this paper, we consider modular forms for finite index subgroups of the modular group whose Fourier coefficients are algebraic. It is well-known that the Fourier coefficients of any holomorphic modular form for a congruence subgroup (with algebraic coefficients) have bounded denominators. It was observed by Atkin and Swinnerton-Dyer that this is no longer true for modular forms for noncongruence subgroups and they pointed out that unbounded denominator property is a clear ...

The purpose of this informal article is to introduce the reader to some of the objects and methods of the theory of p-adic representations. My hope is that students and mathematicians who are new to the subject will find it useful as a starting point. It consists mostly of an expanded version of the notes for my two lectures at the "Dwork trimester" in June 2001.

These informal notes concern some basic themes of harmonic analysis related to representations of groups.

We prove the rationality of some zeta functions associated tocharacters of pro-p groups of finite rank.

Let $m\ge 1$ be a rational integer. We give an explicit formula for the mean value $$\frac{2}{\phi(f)}\sum_{\chi (-1)=(-1)^m}\vert L(m,\chi )\vert^2,$$ where $\chi$ ranges over the $\phi (f)/2$ Dirichlet characters modulo $f>2$ with the same parity as $m$. We then adapt our proof to obtain explicit means values for products of the form $L(m_1,\chi_1)\cdots L(m_{n-1},\chi_{n-1})\overline{L(m_n,\chi_1\cdots\chi_{n-1})}$.

We study additive double character sums over two subsets of a finite field. We show that if there is a suitable rational self-map of small degree of a set $D$, then this set contains a large subset $U$ for which the standard bound on the absolute value of the character sum over $U$ and any subset $C$ (which satisfies some restrictions on its size $|C|$) can be improved. Examples of such suitable self-maps are inversion and squaring. Then we apply this new bound to trace produ...

The ring of finite ad\`eles $\Af$ of the rational numbers $\Q$ is obtained in this article as a completion of $\Q$ with respect to a certain non--Archimedean metric. This ultrametric allows to represent any finite ad\`ele as a series generalizing $m$--adic analysis. The description made provides a new perspective for the adelic Fourier analysis on $\Af$, which also permits to introduce new oscillatory integrals. By incorporating the infinite prime, the mentioned results are e...

This work introduces author's approach to harmonic analysis on algebraic groups over functional two-dimensional local fields. For a two-dimensional local field a Hecke algebra which is formed by operators which integrate pro-locally-constant complex functions over a non-compact domain is defined and its properties are described.

We survey the recent developments in the area that grew out of attempts to extend an analog of Hilbert's Tenth Problem to the field of rational numbers and rings of integers of number fields. The paper is based on a plenary talk the author gave at the North American Annual ASL meeting at the University of Notre Dame in May of 2009.