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 discriminant locus of an ample and spanned line bundle.
On the discriminant of the Artin component
On the discriminant variety of a projective manifold.
On the distribution of ramification points in trigonal curves.
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 Donaldson polynomials of elliptic surfaces.
On the double ...-functions in connexion with a 16-nodal quartic surface.
On the dual variety in characteristic 2
On the embedding dimension of an affine variety.
On the embedding of 1-convex manifolds with 1-dimensional exceptional sets.
On the Enriques classification of algebraic surfaces
On the enumerative geometry of real algebraic curves having many real branches.
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 Equations Defining Abelian Varieties. I.
On the Equations Defining Abelian Varieties. II.