Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves

TL;DR

This paper demonstrates that a positive proportion of quartic Hecke L-functions do not vanish at the central point, implying many related elliptic curves have Mordell-Weil rank zero over .

Contribution

It extends non-vanishing results of quartic Hecke L-functions to elliptic curves with quartic twists, establishing rank zero for a positive proportion of such curves.

Findings

A positive proportion of quartic Hecke L-functions do not vanish at the central point.

Elliptic curves with quartic twists have Mordell-Weil rank zero over  for many squarefree q.

The method applies to Hecke characters associated with quartic residue symbols.

Abstract

We show that a positive proportion of Hecke $L$-functions attached to the quartic residue symbols $\big( \frac{\cdot}{q} \big)_4$ for squarefree $q \in \mathbb{Z}[i]$ do not vanish at the central point. Our method also extends to the Hecke characters associated to quartic twists of the congruent number curve $E : y^2 = x^3 - x$. In particular, we prove that the elliptic curve $E^{(q)} : y^2 = x^3 - qx$ has Mordell-Weil rank $0$ over $\mathbb{Q}(i)$ for a positive proportion of squarefree $q \in \mathbb{Z}[i]$ ordered by norm.