Upcoming Talks

Ista white

On the Ramanujan conjecture over function fields

Date: Thursday, January 24, 2019 13:30 - 15:30
Speaker: Will Sawin (Columbia University)
Location: Big Seminar room Ground floor / Office Bldg West (I21.EG.101)
Series: Mathematics and CS Seminar
Host: Timothy Browning
Contact: MARR Lena
Lab building west seminar room

Deligne proved the Ramanujan conjecture bounding the Hecke eigenvalues of modular forms by constructing two-dimensional Galois representations associated to them. The same strategy was used by Laurent Lafforgue to prove the Ramanujan conjecture for automorphic forms on GL_n over function fields as a corollary of his proof of the Langlands correspondence, building on ideas of Drinfeld who handled the GL_2 case. With Nicolas Templier, we have a different approach to proving the Ramanujan conjecture over function fields, based on estimating the trace of the Hecke operator on a whole family of automorphic forms at once. Our main tools are from geometry, but a different sort of geometry than the proofs of Drinfeld and Lafforgue - we use the moduli space of G-bundles, rather than the moduli space of shtukas. We can prove the conjecture under two conditions (one local condition and one assumption about cyclic base change).
Qr image
Download ICS Download invitation
Back to eventlist