Goldfeld conjecture for non-hyperelliptic direction
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.
Fields of Study & Subjects
Analytic Number Theory Research
Algebraic Geometry and Number Theory
Advanced Algebra and Geometry
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 25 February 2026
- Language
- EN
- Document type
- Preprint