Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication.
The arithmetic of curves defined by iteration
We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...
The Arithmetic of Elliptic Curves.
The automorphism group of the modular curve
The canonical height and integral points on elliptic curves.
The Chow ring of the stack of cyclic covers of the projective line
In this paper we compute the integral Chow ring of the stack of smooth uniform cyclic covers of the projective line and we give explicit generators.
The Clifford dimension of a projective curve
The Congruences of Clausen- von Staudt and Kummer for Bernoulli-Hurwitz Numbers.
The curve , is not set-theoretic complete intersection of two quartic surfaces
The degree of the secant variety and the join of monomial curves.
A monomial curve is a curve parametrized by monomials. The degree of the secant variety of a monomial curve is given in terms of the sequence of exponents of the monomials defining the curve. Likewise, the degree of the join of two monomial curves is given in terms of the two sequences of exponents.
The Diophantine equation f(x) = g(y)
The discriminants of curves of genus 2
The enumeration of simultaneous higher-order contacts between plane curves
The enumerative geometry of plane cubics. II. Nodal and cuspidal cubics.
The Fixed Points of the Symmetrie Hilbert Modular Group of a Real Quadratic Field with Arbitrary Discriminant.
The Genus of Curves in P4 and P5.
The Genus of Curves in P4 verifying certain Flag conditions.
The genus of curves over finite fields with many rational points.
The Genus of Maximal Function Fields over Finite Fields.
The Genus of Space Curves.