Every $BT_1$ group scheme appears in a Jacobian

Abstract

PRIES AND DOUGLAS ULMER A. Let p be a prime number and let k be an algebraically closed field of characteristic p. A BT 1 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 BT 1 scheme group over k occurs as a direct factor of the p-torsion group scheme of the Jacobian of an explicit curve defined over F p . We also treat a variant with polarizations. Our main tools are the Kraft classification of BT 1 group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves.