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The distribution of $a$-numbers of hyperelliptic curves in characteristic three

Derek Garton, Jeffrey Lin Thunder, Colin Weir

Abstract

In this paper we present a new approach to counting the proportion of hyperelliptic curves of genus $g$ defined over a finite field $\mathbb{F}_q$ with a given $a$-number. In characteristic three this method gives exact probabilities for curves of the form $Y^2=f(X)$ with $f(X)\in\mathbb{F}_q[X]$ monic and cubefree, probabilities that match the data presented by Cais et al. in previous work. These results are sufficient to derive precise estimates (in terms of $q$) for these probabilities when restricting to squarefree $f$. As a consequence, for positive integers $a$ and $g$ we show that the nonempty strata of the moduli space of hyperelliptic curves of genus $g$ consisting of those curves with $a$-number $a$ are of codimension $2a-1$. This contrasts with the analogous result for the moduli space of abelian varieties in which the codimensions of the strata are $a(a+1)/2$. Finally, our results allow for an alternative heuristic conjecture to that of Cais et al.; one that matches the available data.

The distribution of $a$-numbers of hyperelliptic curves in characteristic three

Abstract

In this paper we present a new approach to counting the proportion of hyperelliptic curves of genus defined over a finite field with a given -number. In characteristic three this method gives exact probabilities for curves of the form with monic and cubefree, probabilities that match the data presented by Cais et al. in previous work. These results are sufficient to derive precise estimates (in terms of ) for these probabilities when restricting to squarefree . As a consequence, for positive integers and we show that the nonempty strata of the moduli space of hyperelliptic curves of genus consisting of those curves with -number are of codimension . This contrasts with the analogous result for the moduli space of abelian varieties in which the codimensions of the strata are . Finally, our results allow for an alternative heuristic conjecture to that of Cais et al.; one that matches the available data.
Paper Structure (15 sections, 23 theorems, 162 equations)

This paper contains 15 sections, 23 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Previous work
  4. Organization
  5. Competing interests declaration
  6. Preliminary results
  7. The Cartier operator and a-numbers
  8. Heights and a-numbers
  9. Heights of subspaces
  10. Intermediate counting results
  11. Main counting results
  12. The a-numbers of hyperelliptic curves in characteristic three
  13. The proportion of curves with a given a-number
  14. A heuristic model and conjecture
  15. Results on the moduli space of hyperelliptic curves

Key Result

Theorem 1.1.1

Suppose $p=3$ and $g,a$ are nonnegative integers. If $g=0$, then so suppose $g\geq1$. If $g\equiv 0\pmod 3$ and $\epsilon = 1,2$ or if $g\equiv 2\pmod 3$ and $\epsilon = 2$, then On the other hand, if $g\equiv 1\pmod 3$ and $\epsilon = 1,2$ or if $g\equiv 2\pmod 3$ and $\epsilon =1$, then

Theorems & Definitions (50)

  • Theorem 1.1.1
  • Corollary 1.1.2
  • Corollary 1.1.3
  • Corollary 1.1.4
  • Definition 2.1.1
  • Lemma 2.1.2
  • Example 2.1.3
  • Example 2.1.4
  • proof
  • Lemma 2.1.5
  • ...and 40 more