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Conductor exponents for families of hyperelliptic curves

Martin Azon, Mar Curcó-Iranzo, Maleeha Khawaja, Céline Maistret, Diana Mocanu

Abstract

We compute the conductor exponents at odd places using the machinery of cluster pictures of curves for three infinite families of hyperelliptic curves. These are families of Frey hyperelliptic curves constructed by Kraus and Darmon in the study of the generalised Fermat equations of signatures $(r,r,p)$ and $(p,p,r)$, respectively. Here, $r$ is a fixed prime number and $p$ is a prime that is allowed to vary. In the context of the modular method, Billerey-Chen-Dieulefait-Freitas computed all conductor exponents for the signature $(r,r,p)$. We recover their computations at odd places, providing an alternative approach. In a similar setup, Chen-Koutsianas computed all conductor exponents for the signature $(p,p,5)$. We extend their work to the general case of signature $(p,p,r)$ at odd places. Our work can also be used to compute local arithmetic data for the curves in these families.

Conductor exponents for families of hyperelliptic curves

Abstract

We compute the conductor exponents at odd places using the machinery of cluster pictures of curves for three infinite families of hyperelliptic curves. These are families of Frey hyperelliptic curves constructed by Kraus and Darmon in the study of the generalised Fermat equations of signatures and , respectively. Here, is a fixed prime number and is a prime that is allowed to vary. In the context of the modular method, Billerey-Chen-Dieulefait-Freitas computed all conductor exponents for the signature . We recover their computations at odd places, providing an alternative approach. In a similar setup, Chen-Koutsianas computed all conductor exponents for the signature . We extend their work to the general case of signature at odd places. Our work can also be used to compute local arithmetic data for the curves in these families.

Paper Structure

This paper contains 13 sections, 16 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Main results
  4. Related and future work
  5. Overview
  6. Notation and Background
  7. Notation
  8. Curves, Jacobians and conductors
  9. Cluster pictures
  10. On the factorization of $x^r + y^r$
  11. $C_{r}$ family associated to the signature $(r, r, p)$
  12. Roots of the defining polynomial $f_{r}$
  13. Cluster pictures for $C_{r}$

Key Result

Theorem 1.8

Consider $C_{r}/{\mathbb Q}$, $C_{r}^{-}/{\mathbb Q}$ and $C_{r}^{+}/{\mathbb Q}$ as in Definitions def:cr and def: Cminus & Cplus.

Theorems & Definitions (41)

  • Remark 1.1
  • Definition 1.3: $C_r$ family
  • Example 1.4
  • Definition 1.5: $C_{r}^{-}, C_{r}^{+}$ families
  • Example 1.6
  • Remark 1.7
  • Theorem 1.8: Theorems \ref{['cor:conductorQ']}, \ref{['thm: conductorCminusQ']}, \ref{['thm: conductorCplusQ']}
  • Theorem 1.9: Theorems \ref{['thm:conductorQplus']}, \ref{['thm: conductorCminusQpl']}, \ref{['thm: conductorCplusQpl']}
  • Remark 1.10
  • Corollary 1.11: Corollaries \ref{['cor:twistCr']}, \ref{['cor:twistCmius']} and \ref{['cor:twistCplus']}
  • ...and 31 more