On the cuspidal divisor class group of a Drinfeld modular curve.
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 de Rham isomorphism for Drinfeld modules.
On the degree of a local zeta function
On the degree of the -function associated with an exponential sum
On the Difference of the Weil Height and the Néron-Tate Height.
On the Diophantine equation ... (X, Y) = ... (x, y).
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 distribution of rational points on certain Kummer surfaces
On the distribution of the Fₚ-points on an affine curve in r dimensions
On the divisibility properties of families of filtered F-crystals.
On the dual variety in characteristic 2
On the equation
On the equation a³ + b³ⁿ = c²
We study coprime integer solutions to the equation a³ + b³ⁿ = c² using Galois representations and modular forms. This case represents perhaps the last natural family of generalized Fermat equations descended from spherical cases which is amenable to resolution using the so-called modular method. Our techniques involve an elaborate combination of ingredients, ranging from ℚ-curves and a delicate multi-Frey approach, to appeal to intricate image of inertia arguments.
On the Euler-Poincaré characteristics of finite dimensional -adic Galois representations
On the existence of abelian varieties with good reduction.
On the existence of supersingular curves of given genus.
On the ...-factor attached to motives.
On the Finite Field Nullstellensatz for the intersection of two quadric hypersurfaces.
On the Fourier coefficients of modular forms. II.