Conjectures for Moments Associated With Cubic Twists of Elliptic Curves

Abstract

We extend the heuristic introduced by Conrey, Farmer, Keating, Rubinstein, and Snaith in order to formulate conjectures for the $(k,\mathcal{l})$-moments of L-functions of elliptic curves twisted by cubic characters. We also apply the work of Keating and Snaith on the $(k,\mathcal{l})$-moments of characteristic polynomials of unitary matrices to extend our conjecture to $k,\mathcal{l}\in \mathbb{C}$ such that $\text{Re}(k),\text{Re}(\mathcal{l}),$ and $\text{Re}(k+\mathcal{l})>-1$. Our conjectures are then numerically tested for two families. A novelty of our conjectures is that cubic twists naturally lead us to consider the possibility $k\cancel{=}\mathcal{l}$.

Keywords:

• L-functions
• moments
• cubic twists
• random matrix theory

2010 Mathematics Subject Classification:

• Primary 11M41
• Secondary 11M50
• 14G10
• 11M26

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Acknowledgments

The authors thank to Hershy Kisilevsky for many useful discussions, Andrew Granville for suggesting the formula of Lemma 2.2 to show that $g_{k,\mathcal{l}}$ is an integer, Jon Keating and Michael Rubinstein for bringing our attention to the status of the computations of coefficients for zeta functions and quadratic twists, Pablo Bianucci for helpful advice in some of the numerical computations, and the anonymous referee for helpful corrections and suggestions. The authors are very grateful to the Centre de Recherches Mathématiques (CRM) for the access to their cluster and to Jack Fearnley for providing his L-value data.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Table 0 $\#{\mathcal{F}}_E(D)$ and $\#\mathcal{F}{\prime }_E(D)$ for 11a1 and 14a1 for $D=3\times {10}^6.$

Additional information

Funding

This work was supported by the Natural Sciences and Engineering Research Council of Canada—NSERC [Discovery Grant 155635-2019 to CD, 355412-2013 to ML] and the Fonds de recherche du Québec—Nature et technologies—FRQNT [Projet de recherche en équipe 256442 to CD and ML]