Logarithmic abelian varieties, III: Logarithmic elliptic curves and modular curves

Cycles on modular varieties and rational points on elliptic curves

This is a summary of a three-part lecture series given at the meeting on “Explicit methods in number theory” that was held in Oberwolfach from July 12 to 18, 2009. The theme of this lecture series was the explicit construction of algebraic points on elliptic curves from cycles on modular varieties. Given a fixed elliptic curve E over Q, the goal is to better understand the group E(Q̄) of algebra...

On the Logarithmic Connections over Curves

We study two different actions on the moduli spaces of logarithmic connections over smooth complex projective curves. Firstly, we establish a dictionary between logarithmic orbifold connections and parabolic logarithmic connections over the quotient curve. Secondly, we prove that fixed points on the moduli space of connections under the action of finite order line bundles are exactly the push-f...

Elliptic Curves on Abelian Surfaces

The purpose of this paper is to present two theorems which give an overview of the set of elliptic curves lying on an abelian surface and to discuss several applications. One of these applications is a classical theorem of Biermann (1883) and Humbert (1893) on the characterization of abelian surfaces containing elliptic curves in terms of the “singular relations” of Humbert. As a by–product one...

Distortion in Craniofacial Superimposition : Logarithmic Decay Curves

Elliptic Curves and Modular Forms

This is an exposition of some of the main features of the theory of elliptic curves and modular forms.