Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Serre's genus 50 example   This note presents explicit equations (up to birational equivalence over
$\mathbb{F}_2$) for a complete, smooth, absolutely irreducible curve $X$ over
$\mathbb{F}_2$ of genus $50$ satisfying $#X(\mathbb{F}_2)=40$. In his 1985
Harvard lecture notes on curves over finite fields, J-P.~Serre already showed
the existence of such a curve: he used class field theory to describe the
function field $\mathbb{F}_2(X)$ as a certain abelian extension of the function
field $\mathbb{F}_2(E)$ of some elliptic curve $E/\mathbb{F}_2$. Although
various more recent texts recall Serre's construction, explicit equations as
well as a description of intermediate curves $X\to Y\to E$ over $\mathbb{F}_2$
seem to be new. We also describe explicit equations for a curve over
$\mathbb{F}_2$ of genus $8$ with $11$ rational points, and for a curve over
$\mathbb{F}_2$ of genus $22$ with $21$ rational points.

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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph