Goldfeld conjecture for non-hyperelliptic direction

TL;DR

This paper investigates the average analytic rank of twist families from non-hyperelliptic directions of a specific curve, proposing an analogue of the Goldfeld conjecture under GRH.

Contribution

It provides an explicit upper bound on the average analytic rank for these families assuming GRH and formulates an analogue of Goldfeld's conjecture in this context.

Findings

Established an explicit upper bound on average analytic rank

Proposed an analogue of Goldfeld conjecture for non-hyperelliptic twist families

Connected the results to Katz--Sarnak philosophy

Abstract

Since the curve $y^2 = x^6+1$ has a large automorphism group, there exist twist families arising from non-hyperelliptic directions. In this paper, we give an explicit upper bound on the average analytic rank of such a family, assuming the generalized Riemann hypothesis for the $L$-functions. Also, we propose an analogue of the Goldfeld conjecture for the family following Katz--Sarnak philosophy.