We extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring ${{\mathbb{F}}_{q}}\left[ x \right]\,/\,M\left( x \right)$ for collections of polynomials either of the form ${{f}^{-1}}{{g}^{-1}}$ or of the form ${{f}^{-1}}{{g}^{-1}}\,+\,afg$, where $f$ and $g$ are polynomials coprime to $M$ and of very small degree relative to $M$, and $a$ is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields.

This note signals an error in the above paper by giving a counter-example.

Given a smooth projective curve $C$ of positive genus $g$, Torelli's theorem asserts that the pair $\left( J\left( C \right),\,{{W}^{g-1}} \right)$ determines $C$. We show that the theorem is true with ${{W}^{g-1}}$ replaced by ${{W}^{d}}$ for each $d$ in the range $1\,\le \,d\,\le \,g\,-\,1$.

The method of moving frames, introduced by Elie Cartan, is a powerful tool for the solution of various equivalence problems. The practical implementation of Cartan's method, however, remains challenging, despite its later significant development and generalization. This paper presents two new variations on the Fels and Olver algorithm, which under some conditions on the group action, simplify a moving frame construction. In addition, the first algorithm leads to a better understanding of invariant differential forms on the jet bundles, while the second expresses the differential invariants for the entire group in terms of the differential invariants of its subgroup.

The infinitely divisible distributions on ${{\mathbb{R}}^{+}}$ of random variables ${{C}_{t}},\,{{S}_{t}}\,\text{and}\,{{T}_{t}}$ with Laplace transforms

$${{\left( \frac{1}{\cosh \sqrt{2\lambda }} \right)}^{t}},{{\left( \frac{\sqrt{2\lambda }}{\sinh \sqrt{2\lambda }} \right)}^{t}},\,\text{and}\,{{\left( \frac{\tanh \sqrt{2\lambda }}{\sqrt{2\lambda }} \right)}^{t}}$$

respectively are characterized for various $t\,>\,0$ in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Lévy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for $t\,=\,1\,\,\text{or}\,2$ in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of ${{C}_{1}}\,\text{and}\,{{S}_{2}}$ are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet $L$-function associated with the quadratic character modulo 4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from ${{S}_{t}}\,\text{and}\,{{C}_{t}}$ by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways.

We prove that the maximum number of rational points on a smooth, geometrically irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible by combining techniques from algebraic geometry with a computer verification. The appendix shows that 26 points is not possible by examining the zeta functions.

Let $F$ be a $p$-adic local field and let $A_{i}^{\times }$ be the unit group of a central simple $F$-algebra ${{A}_{i}}$ of reduced degree $n\,>\,1\,(i\,=\,1,\,2)$. Let ${{\mathcal{R}}^{2}}\left( A_{i}^{\times } \right)$ denote the set of irreducible discrete series representations of $A_{i}^{\times }$. The “Abstract Matching Theorem” asserts the existence of a bijection, the “Jacquet-Langlands” map, $J{{L}_{{{A}_{2}},{{A}_{1}}}}\,:\,{{R}^{2}}\left( A_{1}^{\times } \right)\,\to \,{{R}^{2}}\left( A_{2}^{\times } \right)$ which, up to known sign, preserves character values for regular elliptic elements. This paper addresses the question of explicitly describing the map 𝒥ℒ, but only for “level zero” representations. We prove that the restriction $J{{L}_{{{A}_{2}},{{A}_{1}}}}\,:\,R_{0}^{2}\left( A_{1}^{\times } \right)\,\to \,R_{0}^{2}\left( A_{2}^{\times } \right)$ is a bijection of level zero discrete series (Proposition 3.2) and we give a parameterization of the set of unramified twist classes of level zero discrete series which does not depend upon the algebra ${{A}_{i}}$ and is invariant under $J{{L}_{{{A}_{2}},{{A}_{1}}}}$ (Theorem 4.1).

Every classical inner function $\varphi$ in the unit disk gives rise to a certain factorization of functions in Hardy spaces. This factorization, which we call the generalized Riesz factorization, coincides with the classical Riesz factorization when $\varphi (z)\,=\,z$. In this paper we prove several results about the generalized Riesz factorization, and we apply this factorization theory to obtain a new description of the commutant of analytic Toeplitz operators with inner symbols on a Hardy space. We also discuss several related issues in the context of the Bergman space.

We give upper and lower Gaussian estimates for the diffusion kernel of a divergence and nondivergence form elliptic operator in a Lipschitz domain.

Let $p$ be an odd prime number, $K$ a $p$-adic field of degree $r$ over ${{\mathbf{Q}}_{p}}$, $O$ the ring of integers in $K,\,B\,=\,\{{{\beta }_{1}},\ldots .{{\beta }_{r}}\}$ an integral basis of $K$ over ${{\mathbf{Q}}_{p}}$, $u$ a unit in $O$ and consider sets of the form $N\,=\,\{{{n}_{1}}{{\beta }_{1}}\,+\ldots +\,{{n}_{r}}{{\beta }_{r}}\,:\,1\,\le \,{{n}_{j}}\,\le \,{{N}_{j}},\,1\,\le \,j\,\le \,r\}$. We show under certain growth conditions that the pair correlation of $\{u{{z}^{2}}\,:\,z\,\in N\}$ becomes Poissonian.