Related papers: \'Equir\'epartition de sommes exponentielles (trav…

By using a variant of Kowalski's large sieve for Frobenius in compatible systems, we obtain zero-density estimates for arguments of $\ell$-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic…

We show that under certain general conditions, short sums of $\ell$-adic trace functions over finite fields follow a normal distribution asymptotically when the origin varies, generalizing results of Erd\H{o}s-Davenport, Mak-Zaharescu and…

Corentin Perret-Gentil proved, under some very general conditions, that short sums of $\ell$-adic trace functions over finite fields of varying center converges in law to a Gaussian random variable or vector. The main inputs are…

These notes are devoted to the theory of exponential sums over finite fields. The first chapter recalls some of the number-theoretic interest of such sums. The second chapter discusses the $L$-functions attached to such sums, the "Weil…

This note concerns exponential sheaves and the "universal" Fourier transform on them. Fourier invertibility and the subsequent Fourier miracle is demonstrated. Further, t-structures and realizations are constructed and shown to have…

We study the arithmetic Fourier transforms of trace functions on general connected commutative algebraic groups. To do so, we first prove a generic vanishing theorem for twists of perverse sheaves by characters, and using this tool, we…

We survey some of the stratification theorems concerning exponential sums over finite fields, especially those due to Katz-Laumon and Fouvry-Katz, as well as some of their applications. Moreover, motivated partly by recent work of Bonolis,…

We investigate exponential sums over singular binary quartic forms, proving an explicit formula for the finite field Fourier transform of this set. Our formula shares much in common with analogous formulas proved previously for other vector…

We consider families of exponential sums indexed by a subgroup of invertible classes modulo some prime power $q$. For fixed $d$, we restrict to moduli $q$ so that there is a unique subgroup of invertible classes modulo $q$ of order $d$. We…

Given a prime $p$ and an integer $d>1$, we give a numerical criterion to decide whether the $\ell$-adic sheaf associated to the one-parameter exponential sums $t\mapsto \sum_x\psi(x^d+tx)$ over ${\mathbb F}_p$ has finite monodromy or not,…

The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on…

This is a sequel to math.AG/0003009. Here we study identities for the Fourier transform of "elementary functions" over finite field containing "exponents" of monomial rational functions. It turns out that these identities are governed by…

We prove that a perverse sheaf on a connected commutatitve algebraic group over a finite is generically unramified. This implies an equidistribution theorem for Tannakian monodromy groups in previously unavailable generality. We also prove…

We investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear in the study of the quantum cat map. While they are not…

We first show joint uniform distribution of values of Kloosterman sums or Birch sums among all extensions of a finite field $\mathbb{F}_q$, for almost all couples of arguments in $\mathbb{F}_q^\times$, as well as lower bounds on…

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums, as their parameter varies modulo a prime tending to infinity. Using independence of Kloosterman sheaves, we prove convergence in the…

We show that integral monodromy groups of Kloosterman $\ell$-adic sheaves of rank $n\ge 2$ on $\mathbb{G}_m/\mathbb{F}_q$ are as large as possible when the characteristic $\ell$ is large enough, depending only on the rank. This variant of…

We deduce Katz's theorems for $(A,B)$-exponential sums over finite fields using $\ell$-adic cohomology and a theorem of Denef-Loeser, removing the hypothesis that $A+B$ is relatively prime to the characteristic $p$. In some degenerate…

We prove the Sato--Tate distribution of Kloosterman sums over function fields with explicit error terms, when the places vary in arithmetic progressions or short intervals. A joint Sato--Tate distribution of two ``different" exponential…

In the framework of infinite ergodic theory, we derive equidistribution results for suitable weighted sequences of cusp points of Hecke triangle groups encoded by group elements of constant word length with respect to a set of natural…