On some conjectures on the arithmetic of elliptic curves

Abstract

The goal of the present thesis is twofold

we show the two conjectures concerning the arithmetic of elliptic curves: the Stein–Watkins conjecture (for 5-isogenies) and the Gross--Zagier conjecture.

Essentially, Stein--Watkins conjecture tells us about the relations of optimal curves in given rational isogeny class of elliptic curves. In this thesis we show the two optimal curves differ by a 5-isogeny if and only if the isogeny class is '11a'.

The Gross--Zagier conjecture provides a theoretical evidence to the strong form of Birch and Swinnerton-Dyer conjecture. We show when elliptic curves have particular types of rational torsion subgroups, the order of the torsion subgroup divides certain arithmetic invariants attached to the curve.