4:00 pm Tuesday, September 12, 2017
AGNT: **Counting Lattices by Cotype**
by **Nathan Kaplan** (UC Irvine) in HBH 227
The zeta function of Z^d is a generating function that encodes the number of its sublattices of each index. This function can be expressed as a product of Riemann zeta functions and analytic properties of the Riemann zeta function then lead to an asymptotic formula for the number of sublattices of Z^d of index at most X. Nguyen and Shparlinski have investigated more refined counting questions, giving an asymptotic formula for the number of cocyclic sublattices L of Z^d, those for which Z^d/L is cyclic. Building on work of Petrogradsky, we generalize this result, counting sublattices for which Z^d/L has at most m invariant factors. We will see connections to cokernels of random integer matrices and the Cohen-Lenstra heuristics. Joint work with Gautam Chinta (CCNY) and Shaked Koplewitz (Yale). Host Department: Rice University-Mathematics Submitted by jb93@rice.edu |