Counting bundles with extra structures over curves II

BIMSA-YMSC Number Theory Lunch Seminar

Organizers：Yong Suk Moon, Koji Shimizu

Speaker：Tao Su

Time：

Thur., 12:00-13:00, May 14, 2026

Venue：

BIMSA A4-1

Title:

Counting bundles with extra structures over curves II

Abstract:

In this talk I will discuss a counting problem for vector bundles over a smooth projective curve defined over a finite field. We consider bundles equipped with parabolic structures at finitely many points and a nilpotent endomorphism, and study the corresponding generating functions. A theorem of Anton Mellit shows that this counting problem admits a striking factorization: each marked point contributes independently, and the contribution is given by a modified Macdonald polynomial. As a consequence, these polynomials admit a geometric interpretation as weighted point counts on affine Springer fibers associated to constant nilpotent matrices. If time permits, I will briefly explain how this perspective leads to a proof of the conjectural formula of Hausel--Letellier--Rodriguez-Villegas for the Poincaré polynomials of character varieties of punctured Riemann surfaces.