On some equalities for the Weierstrass modular units of level p
On some equations over finite fields
In this paper, following L. Carlitz we consider some special equations of variables over the finite field of elements. We obtain explicit formulas for the number of solutions of these equations, under a certain restriction on and .
On special values of theta functions of genus two
We study a certain finitely generated multiplicative subgroup of the Hilbert class field of a quartic CM field. It consists of special values of certain theta functions of genus 2 and is analogous to the group of Siegel units. Questions of integrality of these specials values are related to the arithmetic of the Siegel moduli space.
On square values of quadratics
On squares of squares
On squares of squares II
On symmetric square values of quadratic polynomials
On Taylor's conjecture for Kummer orders
On the analogue of the division polynomials for hyperelliptic curves.
On the arithmetic of an elliptic curve over a -extension
On the arithmetic of certain modular curves
On the arithmetic of genus two fibrations
On the arithmetic of some division algebras.
On the arithmetic of the hyperelliptic curve
We study the arithmetic properties of hyperelliptic curves given by the affine equation by exploiting the structure of the automorphism groups. We show that these curves satisfy Lang’s conjecture about the covering radius (for some special covering maps).
On the arithmetic of twists of superelliptic curves
On the average value of the canonical height in higher dimensional families of elliptic curves
Given an elliptic curve E over a function field K = ℚ(T₁,...,Tₙ), we study the behavior of the canonical height of the specialized elliptic curve with respect to the height of ω ∈ ℚⁿ. We prove that there exists a uniform nonzero lower bound for the average of the quotient over all nontorsion P ∈ E(K).
On the Birch and Swinnerton-Dyer conjecture
On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields
Let be a modular elliptic curve defined over a totally real number field and let be its associated eigenform. This paper presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of over suitable quadratic imaginary extensions . In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachëv, that is, when is even and not new at any prime.
On the Brauer-Manin obstruction for zero-cycles on curves
On the classification of 3-dimensional non-associative division algebras over -adic fields
Let be a prime and a -adic field (a finite extension of the field of -adic numbers ). We employ the main results in [12] and the arithmetic of elliptic curves over to reduce the problem of classifying 3-dimensional non-associative division algebras (up to isotopy) over to the classification of ternary cubic forms over (up to equivalence) with no non-trivial zeros over . We give an explicit solution to the latter problem, which we then relate to the reduction type of the jacobian...