Multiplicative Order of Gauss Periods

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.Comment: 9 page

Globally nilpotent differential operators and the square Ising model

We recall various multiple integrals related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility, to the (lattice) form factors, to the two-point correlation functions and to their lambda-extensions. These integrals are holonomic and even G-functions: they satisfy Fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties. We recall the explicit forms, found in previous work, of these Fuchsian equations. These differential operators are very selected Fuchsian linear differential operators, and their remarkable properties have a deep geometrical origin: they are all globally nilpotent, or, sometimes, even have zero p-curvature. Focusing on the factorised parts of all these operators, we find out that the global nilpotence of the factors corresponds to a set of selected structures of algebraic geometry: elliptic curves, modular curves, and even a remarkable weight-1 modular form emerging in the three-particle contribution $\chi^{(3)}$ of the magnetic susceptibility of the square Ising model. In the case where we do not have G-functions, but Hamburger functions (one irregular singularity at 0 or $\infty$) that correspond to the confluence of singularities in the scaling limit, the p-curvature is also found to verify new structures associated with simple deformations of the nilpotent property.Comment: 55 page

On matrix elements for the quantized cat map modulo prime powers

The quantum cat map is a model for a quantum system with underlying chaotic dynamics. In this paper we study the matrix elements of smooth observables in this model, when taking arithmetic symmetries into account. We give explicit formulas for the matrix elements as certain exponential sums. With these formulas we can show that there are sequences of eigenfunctions for which the matrix elements decay significantly slower then was previously conjectured. We also prove a limiting distribution for the fluctuation of the normalized matrix elements around their average.Comment: 26 pages, final version, to appear in AH

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

On the denominators of the Taylor coefficients of G-functions

Let $\sum\_{n=0}^\infty a\_n z^n\in \overline{\mathbb Q}[[z]]$ be a $G$-function, and, for any $n\ge0$, let $\delta\_n\ge 1$ denote the least integer such that $\delta\_n a\_0, \delta\_n a\_1, ..., \delta\_n a\_n$ are all algebraic integers. By definition of a $G$-function, there exists some constant $c\ge 1$ such that $\delta\_n\le c^{n+1}$ for all $n\ge 0$. In practice, it is observed that $\delta\_n$ always divides $D\_{bn}^{s} C^{n+1}$ where $D\_n=lcm\{1,2, ..., n\}$, $b, C$ are positive integers and $s\ge 0$ is an integer. We prove that this observation holds for any $G$-function provided the following conjecture is assumed: {\em Let $\mathbb{K}$ be a number field, and $L\in \mathbb{K}[z,\frac{d }{d z}]$ be a $G$-operator; then the generic radius of solvability $R\_v(L)$ is equal to 1, for all finite places $v$ of $\mathbb{K}$ except a finite number.} The proof makes use of very precise estimates in the theory of $p$-adic differential equations, in particular the Christol-Dwork Theorem. Our result becomes unconditional when $L$ is a geometric differential operator, a special type of $G$-operators for which the conjecture is known to be true. The famous Bombieri-Dwork Conjecture asserts that any $G$-operator is of geometric type, hence it implies the above conjecture