相关论文: Exponential Sums over Points of Elliptic Curves

A uniform bound of intersection multiplicities of curves and divisors on abelian varieties is proved by algebraic geometric methods. It extends and improves a result obtained by A. Buium with a different method based on Kolchin's…

It is a well known result that the number of points over a finite field on the Legendre family of elliptic curves can be written in terms of a hypergeometric function modulo $p$. In this paper, we extend this result, due to Igusa, to a…

In this note, we presented a new decomposition of elements of finite fields of even order and illustrated that it is an effective tool in evaluation of some specific exponential sums over finite fields, the explicit value of some…

We prove several incidence theorems in vector spaces over finite fields using bounds for various classes of exponential sums and apply these to Erdos-Falconer type distance problems.

An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over…

We present new constructions of complex and p-adic Darmon points on elliptic curves over base fields of arbitrary signature. We conjecture that these points are global and present numerical evidence to support our conjecture.

We establish local $(L^p,L^q)$ mapping properties for averages on curves. The exponents are sharp except for endpoints.

We extend bounds on additive energies of modular square roots by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu and apply these results to obtain bounds on certain bilinear exponential sums with modular square roots. From here, we make…

This is an expository account of the proof of the theorem of Bourgain, Glibichuk and Konyagin which provides non-trivial bounds for exponential sums over very small multiplicative subgroups of prime finite fields.

We study a boundary-value quasilinear elliptic problem on a generic time scale. Making use of the fixed-point index theory, sufficient conditions are given to obtain existence, multiplicity, and infinite solvability of positive solutions.

We develop a theory of sesquilinear forms over finite fields, investigating their representations via polynomials and coefficient matrices, along with classification results for these forms. Through their connection to quadratic forms, we…

We derive upper and lower bounds on the determinant of an exponential matrix. They can be transformed into corresponding bounds for the determinant of a univariate Gaussian matrix.

We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice…

We construct pairs of elliptic curves over number fields with large intersection of projective torsion points.

Given an elliptic surface $\mathcal{E}\to\mathcal{C}$ over a field $k$ of characteristic zero equipped with zero section $O$ and another section $P$ of infinite order, we give a simple and explicit upper bound on the number of points where…

We study the distribution of algebraic points on curves in abelian varieties over finite fields.

In this article we study the inverse problem of determining a semilinear term appearing in an elliptic equation from boundary measurements. Our main objective is to develop flexible and general theoretical results that can be used for…

In this note, we investigate the measure of singular sets and critical sets of real-valued solutions of elliptic equations in two dimensions. These singular sets and critical sets are finitely many points in the plane. Adapting the Carleman…

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of…

We study residually transcendental extensions of a valuation $v$ on a field $E$ to function fields of hyperelliptic curves over $E$. We show that $v$ has at most finitely many extensions to the function field of a hyperelliptic curve over…