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Oort's conjecture and automorphisms of supersingular curves of genus four
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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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The Torelli locus and Newton polygons
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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