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Goldfeld conjecture for non-hyperelliptic direction

Keunyoung Jeong, Junyeong Park

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.

Goldfeld conjecture for non-hyperelliptic direction

Abstract

Since the curve 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 -functions. Also, we propose an analogue of the Goldfeld conjecture for the family following Katz--Sarnak philosophy.
Paper Structure (10 sections, 30 theorems, 200 equations)

This paper contains 10 sections, 30 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Review of Cardona--Lario
  4. Review of Fité--Sutherland
  5. Some arithmetic invariants
  6. Parametrization and some invariants
  7. Cluster picture and Conductor
  8. Main results
  9. Proof of the main theorem
  10. Type B

Key Result

Theorem 1.1

Suppose the generalized Riemann hypothesis for the $L$-function of $C_d$. Let $\phi$ be a positive test function whose Fourier inversion is supported in $[-\sigma, \sigma]$ for $\sigma > 0$. Then,

Theorems & Definitions (63)

  • Theorem 1.1
  • Conjecture 1
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Proposition 2.5
  • proof
  • ...and 53 more