2-Adic Arithmetic-Geometric Mean and Elliptic Curves

Elliptic Curves over Q and 2-adic Images of Galois

We give a classification of all possible 2-adic images of Galois representations associated to elliptic curves over Q. To this end, we compute the ‘arithmetically maximal’ tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.

Arithmetic of Elliptic Curves

Here are the notes I am taking for Wei Zhang’s ongoing course on the arithmetic of elliptic curves offered at Columbia University in Fall 2014 (MATH G6761: Topics in Arithmetic Geometry). As the course progresses, these notes will be revised. I recommend that you visit my website from time to time for the most updated version. Due to my own lack of understanding of the materials, I have inevita...

Arithmetic of Elliptic Curves

Our research focuses on 9 specific elliptic curves E over Q, each with complex multiplication by the maximal order in an imaginary quadratic field. Viewed over C, each E gives rise to tori, defined by the generators ω1, ω2 ∈ C of the period lattice. Using SAGE, information and characteristics about the curves and their tori were calculated and compiled. Additionally, the tori were virtually con...

On the Arithmetic-geometric Mean for Curves of Genus 2 I. Dolgachev and D. Lehavi

We study the relationship between two genus 2 curves whose jacobians are isogenous with kernel equal to a maximal isotropic subspace of p-torsion points with respect to the Weil pairing. When p = 2 this relationship is a generalization of Gauss’s arithmetic-geometric mean for elliptic curves studied by Richelot (1837) and Humbert (1901), and in modern terms by Bost-Mestre (1988) and Donagi-Livn...

Abelian L-adic representation and elliptic curves