Every BT₁ group scheme appears in a Jacobian

2021/01/20 by Pries, Rachel, Ulmer, Douglas
#11G20 #14F40 #14G17 #14H10 #14H40 #14K15 #14L15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary 11D41 #Secondary 11G10

paper · doi:10.48550/arxiv.2101.07946

Abstract

Let p be a prime number and let k be an algebraically closed field of characteristic p. A BT₁ group scheme over k is a finite commutative group scheme which arises as the kernel of p on a p-divisible (Barsotti--Tate) group. Our main result is that every BT₁ scheme group over k occurs as a direct factor of the p-torsion group scheme of the Jacobian of an explicit curve defined over \mathbbF_p. We also treat a variant with polarizations. Our main tools are the Kraft classification of BT₁ group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves.

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