PhD Defence: Classification of Isogeny classes of Supersingular Abelian Varieties over Finite Fields
Speaker: Vijaykumar Singh
Date: Thu 9th June 2011
Location: Mathematical Sciences Seminar Room
We study the intersection of two particular Fermat hyper-surfaces in P^3 over a finite field. Using the Kani-Rosen decomposition we study arithmetic properties of this curve in terms of its quotients. Explicit computation of the quotients is done using a Groebner basis algorithm. We also study the p-rank, zeta function, and number of rational points, of the modulo p reduction of the curve. We show that the Jacobian of the genus 2 quotient is (4, 4)-split. We also give a complete classification of the isogeny classes of supersingular abelian varieties for all dimension by explicitly finding all possible characteristic polynomial of Frobenius endomorphism up to dimension 7 and giving an algorithm to find for all dimension using Honda-Tate Theory.
(This talk is part of the Algebra/Claude Shannon Institute series.)