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
On the de Rham and -adic realizations of the elliptic polylogarithm for CM elliptic curves
In this paper, we give an explicit description of the de Rham and -adic polylogarithms for elliptic curves using the Kronecker theta function. In particular, consider an elliptic curve defined over an imaginary quadratic field with complex multiplication by the full ring of integers of . Note that our condition implies that has class number one. Assume in addition that has good reduction above a prime unramified in . In this case, we prove that the specializations of the -adic elliptic...
On the diophantine equation f(x)f(y) = f(z)²
Let f ∈ ℚ [X] and deg f ≤ 3. We prove that if deg f = 2, then the diophantine equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in ℚ (t). In the case when deg f = 3 and f(X) = X(X²+aX+b) we show that for all but finitely many a,b ∈ ℤ satisfying ab ≠ 0 and additionally, if p|a, then p²∤b, the equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in rationals.
On the Diophantine equation f(y) - x = 0
On the Diophantine equation
On the discrete logarithm problem for plane curves
In this article the discrete logarithm problem in degree 0 class groups of curves over finite fields given by plane models is studied. It is proven that the discrete logarithm problem for non-hyperelliptic curves of genus 3 (given by plane models of degree 4) can be solved in an expected time of , where is the cardinality of the ground field. Moreover, it is proven that for every fixed natural number the following holds: We consider the discrete logarithm problem for curves given by plane models...
On the dispersions of the polynomial maps over finite fields.
On the distribution of analytic values on quadratic twists of elliptic curves.
On the distribution of distances between the points of affine curves over finite fields.
On the distribution of rational points on certain Kummer surfaces
On the distribution of the elliptic subset sum generator of pseudorandom numbers.