Two-torsion statistics for curves in characteristic two
Vortrag von Dr. Wouter Castryck
Datum: 13.10.14 Zeit: 11.00 - 12.00 Raum: UNINE, B217
We study the statistical behaviour of the 2-torsion subgroup of the Jacobian Jac(C) of a randomly chosen curve C over a finite field k of characteristic 2, for various notions of randomness. More precisely, we analyze the distribution of the rank of Jac(C)[2], of Jac(C)[2](k) and of the Hasse-Witt matrix of C. There are some surprising phenomena, such as: almost all smooth plane curves of odd degree have a k-rational 2-torsion point on their Jacobian, and similarly for hyperelliptic curves of odd genus. Our discussion will mention old work of Stöhr and Voloch, more recent results by Cais, Ellenberg and Zureick-Brown, and some joint observations with Marco Streng, Damiano Testa and Fréderik Vercauteren.