Non-abelian congruences between L-values of elliptic curves

Abelian L-adic representation and elliptic curves

Non-Abelian Zeta Functions for Elliptic Curves

In this paper, new local and global non-abelian zeta functions for elliptic curves are defined using moduli spaces of semi-stable bundles. To understand them, we also introduce and study certain refined Brill-Noether locus in the moduli spaces. Examples of these new zeta functions and a justification of using only semi-stable bundles are given too. We end this paper with an appendix on the so-c...

Congruences between Heegner Points and Quadratic Twists of Elliptic Curves

We establish a congruence formula between p-adic logarithms of Heegner points for two elliptic curves with the same mod p Galois representation. As a first application, we use the congruence formula when p = 2 to explicitly construct many quadratic twists of analytic rank zero (resp. one) for a wide class of elliptic curves E. We show that the number of twists of E up to twisting discriminant X...

Review of Abelian l-adic Representations and Elliptic Curves

Addison-Wesley has just reissued Serre’s 1968 treatise on l-adic representations in their Advanced Book Classics series. This circumstance presents a welcome excuse for writing about the subject, and for placing Serre’s book in a historical perspective. The theory of l-adic representations is an outgrowth of the study of abelian varieties in positive characteristic, which was initiated by Hasse...

Congruences of models of elliptic curves

Let OK be a discrete valuation ring with field of fractions K and perfect residue field. Let E be an elliptic curve over K, let L/K be a finite Galois extension and let OL be the integral closure of OK in L. Denote by X ′ the minimal regular model of EL over OL. We show that the special fibers of the minimal Weierstrass model and the minimal regular model of E over OK are determined by the infi...