A study in sums of products

We give a general version of cancellation in exponential sums that arise as sums of products of trace functions satisfying a suitable independence condition related to the Goursat-Kolchin-Ribet criterion, in a form that is easily applicable in analytic number theory

Counting Sheaves Using Spherical Codes

Using the Riemann Hypothesis over finite fields and bounds for the size of spherical codes, we give explicit upper bounds, of polynomial size with respect to the size of the field, for the number of geometric isomorphism classes of geometrically irreducible l-adic middle-extension sheaves on a curve over a finite field, which are pointwise pure of weight 0 and have bounded ramification and rank. As an application, we show that "random" functions defined on a finite field cannot usually be approximated by short linear combinations of trace functions of sheaves with small complexity

Algebraic twists of modular forms and Hecke orbits

We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistribution property for twisted Hecke orbits. This is done by exploiting the amplification method and the Riemann Hypothesis over finite fields, relying in particular on the a""-adic Fourier transform introduced by Deligne and studied by Katz and Laumon

An inverse theorem for Gowers norms of trace functions over F-p

We study the Gowers uniformity norms of functions over Z/pZ which are trace functions of l-adic sheaves. On the one hand, we establish a strong inverse theorem for these functions, and on the other hand this gives many explicit examples of functions with Gowers norms of size comparable to that of "random" functions

Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions

We show that, in a restricted range, the divisor function of integers in residue classes modulo a prime follows a Gaussian distribution, and a similar result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore, we obtain the joint distribution of these arithmetic functions in two related residue classes. These results follow from asymptotic evaluations of the relevant moments, and depend crucially on results on the independence of monodromy groups related to products of Kloosterman sums

Новое применение дисперсионного метода Линника

Let $\alpha_m$ and $\beta_n$ be two sequences of real numbers supported on $[M, 2M]$ and $[N, 2N]$ with $M = X^{1/2 - \delta}$ and $N = X^{1/2 + \delta}$. We show that there exists a \delta_0 > 0 such that the multiplicative convolution of $\alpha_m$ and $\beta_n$ has exponent of distribution $\frac{1}{2} + \delta-\varepsilon$ (in a weak sense) as long as 0 \leq \delta < \delta_0,    the sequence  $\beta_n$ is Siegel-Walfisz and both sequences $\alpha_m$ and $\beta_n$ are bounded above by divisor functions. Our result is thus a general dispersion estimate for ``narrow'' type-II sums. The proof relies crucially on Linnik's dispersion method and recent bounds for trilinear forms in Kloosterman fractions due to Bettin-Chandee. We highlight an application related to the Titchmarsh divisor problem.Пусть $\alpha_{m}$ и $\beta_{n}$ --- две последовательности вещественных чисел с носителями наотрезках $[M,2M]$ и $[N,2N]$, где $M = X^{1/2-\delta}$ и $N = X^{1/2+\delta}$. Мы доказываемсуществование такой постоянной $\delta_{0}$, что мультипликативная свертка$\alpha_{m}$ и $\beta_{n}$ имеет уровень распределения $1/2+\delta-\varepsilon$ (в слабом смысле),если только 0\leqslant \delta<\delta_{0}, последовательность $\beta_{n}$ являетсяпоследовательностью Зигеля-Вальфиша, и обе последовательности $\alpha_{m}$ и $\beta_{n}$ограничены сверху функцией делителей.Наш результат, таким образом, представляет собой общую дисперсионную оценкудля "коротких"\, сумм II типа. Доказательство существенно использует дисперсионный метод Линникаи недавние оценки трилинейных сумм с дробями Клоостермана, принадлежащие Беттин и Чанди.Также мы остановимся на применении полученного результата к проблеме делителей Титчмарша