The distribution of $a$-numbers of hyperelliptic curves in characteristic three
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| Format: | Preprint |
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2024
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| author | Garton, Derek Thunder, Jeffrey Lin Weir, Colin |
| author_facet | Garton, Derek Thunder, Jeffrey Lin Weir, Colin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_00120 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The distribution of $a$-numbers of hyperelliptic curves in characteristic three Garton, Derek Thunder, Jeffrey Lin Weir, Colin Number Theory Algebraic Geometry 14G17, 11G20, 14K10 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. |
| title | The distribution of $a$-numbers of hyperelliptic curves in characteristic three |
| topic | Number Theory Algebraic Geometry 14G17, 11G20, 14K10 |
| url | https://arxiv.org/abs/2403.00120 |