Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Superelliptic curves with large Galois images

Pip Goodman

Abstract

Let $r>2$ and $\ell$ be primes. In this paper we study the mod $\ell$ Galois representations attached to curves of the form $y^r = f(x)$ where $f$ is monic and has coefficients belonging to the $r$-th cyclotomic field. We provide conditions on the coefficients (and degree) of $f$ which allow one to verify the mod $\ell$ image is large outside of a (typically small) finite explicit set of primes. We allow all values of $r$ for which the $r$-th cyclotomic field has odd class number. This appears to be the first explicit result for abelian varieties of dimension greater than two and not of ${\rm GL}_2$-type which allows the ground field to have unramified extensions. In proving the large image result we give a classification of the maximal subgroups containing transvections of certain classical groups and describe (in many cases) the images of inertia groups. The exact mod $\ell$ image is governed by the "endomorphism character", a certain algebraic Hecke character which generalises the CM character. When $r=3$, we depict the image in its entirety. To the author's knowledge, this is the first accurate description in the literature. Finally, we give several examples with genus ranging from 10 to 36. Applications to the Inverse Galois Problem are also included.

Superelliptic curves with large Galois images

Abstract

Let and be primes. In this paper we study the mod Galois representations attached to curves of the form where is monic and has coefficients belonging to the -th cyclotomic field. We provide conditions on the coefficients (and degree) of which allow one to verify the mod image is large outside of a (typically small) finite explicit set of primes. We allow all values of for which the -th cyclotomic field has odd class number. This appears to be the first explicit result for abelian varieties of dimension greater than two and not of -type which allows the ground field to have unramified extensions. In proving the large image result we give a classification of the maximal subgroups containing transvections of certain classical groups and describe (in many cases) the images of inertia groups. The exact mod image is governed by the "endomorphism character", a certain algebraic Hecke character which generalises the CM character. When , we depict the image in its entirety. To the author's knowledge, this is the first accurate description in the literature. Finally, we give several examples with genus ranging from 10 to 36. Applications to the Inverse Galois Problem are also included.

Paper Structure

This paper contains 25 sections, 54 theorems, 137 equations, 1 table.

Table of Contents

  1. Introduction
  2. $\lambda$-adic representations
  3. Group Theory
  4. Classical groups and their automorphisms
  5. Centralisers in the symplectic group
  6. Restrictions on the image of $\rho_\ell$
  7. Maximal subgroups of classical groups
  8. Generating products of classical groups
  9. Controlling inertia groups
  10. Image of inertia
  11. Irreducibility
  12. Primitivity
  13. Subfield subgroups and classical subgroups
  14. The endomorphism character
  15. Algebraic Hecke characters
  16. ...and 10 more sections

Key Result

Theorem 1.2

Let $d \geq 12$ be a natural number divisible by $2r$ which is also the sum of two distinct primes $q_1<q_2$. Suppose there exists a prime $q_2<q_3<d$. If $r>23$ assume the class number of ${\mathbb{Q}(\zeta_r)}$ is odd and $d=q_3+1$. Let $n = \frac{2g}{r-1}$. Then given a polynomial $f \in {\mathbb for all $\ell$ outside of a finite explicit set.

Theorems & Definitions (116)

  • Theorem 1.2: $\subseteq$ Theorem \ref{['thm_mod_l_image_is_huge']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • ...and 106 more