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

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.

Theorems & Definitions (46)

• Theorem 1.1: Positive proportion of rank $0$
• Theorem 1.2: Positive proportion of non-vanishing
• Remark 1.3
• Proposition 1.4
• Proposition 1.5
• Proposition 1.6
• Proposition 1.7
• Corollary 1.8
• proof
• Lemma 4.1