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...
On the coefficients of Drinfeld modular forms.
On the conductor formula of Bloch
In [6], S. Bloch conjectures a formula for the Artin conductor of the ℓ-adic etale cohomology of a regular model of a variety over a local field and proves it for a curve. The formula, which we call the conductor formula of Bloch, enables us to compute the conductor that measures the wild ramification by using the sheaf of differential 1-forms. In this paper, we prove the formula in arbitrary dimension under the assumption that the reduced closed fiber has normal crossings.
On the Conjecture of Birch and Swinnerton-Dyer
On the conjecture of Birch and Swinnerton-Dyer for abelian varieties over function fields in characteristic p > 0.
On the conjectures of Birch and Swinnerton-Dyer and a geometric analog
On the continued fractions of quadratic surds
On the critical values of Hecke L-functions for imaginary quadratic fields.
On the critical values of Hecke L-series
On the cuspidal divisor class group of a Drinfeld modular curve.
On the cycle map for products of elliptic curves over a p-adic field