The even parity Goldfeld conjecture: congruent number elliptic curves

TL;DR

This paper discusses recent progress on Goldfeld's conjecture, focusing on the case of congruent number elliptic curves, which predicts that half of their quadratic twists have analytic rank zero.

Contribution

It reviews recent developments towards proving Goldfeld's conjecture for congruent number elliptic curves, highlighting new insights and partial results.

Findings

Progress towards Goldfeld's conjecture for congruent number elliptic curves

Evidence supporting the 50% rank zero prediction

Advances in understanding quadratic twists of elliptic curves

Abstract

In 1979 Goldfeld conjectured: 50\% of the quadratic twists of an elliptic curve defined over the rationals have analytic rank zero. In this expository article we present a few recent developments towards the conjecture, especially its first instance - the congruent number elliptic curves.