|Date: Wednesday, December 06, 2017
Location: 3866 East Hall (3:00 PM to 4:00 PM)
Title: A fun connection between class groups, conjugacy classes, and isogeny classes
Abstract: Let f be the characteristic polynomial of Frobenius of an abelian variety of odd prime dimension p over a finite field; we use f to relate three seemingly disjoint objects. First, we consider the factorizations of primes in Split(f), a degree 2p number field K. Second, we use a parametrization of Shinoda to describe certain conjugacy classes of the general symplectic group GSp(2p,Fq). Our main result is a product formula relating the class number of K to the relative densities of conjugacy classes of GSp(2p,Fq). Finally, we give a (conjectural) application of our formula to the size of isogeny classes of certain abelian varieties of odd prime dimension.
Speaker: Jonathan Gerhard