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Oort's conjecture and automorphisms of supersingular curves of genus four

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arxiv 2405.01282 v1 pith:QUWJR353 submitted 2024-05-02 math.AG math.NT

classification math.AGmath.NT
keywords curvessupersingularautomorphismconjectureoortgenussmoothabelian
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abstract

We show that every component of the locus of smooth supersingular curves of genus $4$ in characteristic $p>2$ has a trivial generic automorphism group. As a result, we prove Oort's conjecture about automorphism groups of supersingular abelian fourfolds for $p>2$. Our main idea consists of estimating dimensions of the loci of smooth supersingular curves that admit an automorphism of prime order by considering possible choices of the corresponding quotient curves. This reasoning also results in a new proof of Oort's conjecture for $g = 3$ and $p>2$, previously proved by Karemaker, Yuboko, and Yu.

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  1. The Torelli locus and Newton polygons

    math.AG 2025-08 conditional novelty 3.0 of 10

    A survey of the Torelli locus and Newton polygons that corrects a previously erroneous genus computation and thereby proves new infinite families of supersingular curves of genus δp(p−1)/2.

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