Number Theory Seminar

Speaker: Peter Dillery

Title talk 1 (at 13:00): The local Langlands conjectures

We introduce the local Langlands conjectures---results that allow one to pass from a representation of an algebraic group G over a local field F to representations of the Galois group of F, building up to a conjectural parametrization of L- packets (the fibers of the above map) for quasi-split reductive groups, taking care to introduce all the relevant machinery and provide examples. We also briefly discuss endoscopy—a version of “functoriality" in the context of the Langlands program.

Title talk 2 (at 14:15): Inner forms and the local Langlands

We talk about how inner forms of groups can be used to parametrize L-packets for arbitrary reductive groups, focusing in particular on when we should declare two inner forms equivalent, and the impact that this has on the study of endoscopy. We also connect inner forms to isocrystals and, time-permitting, discuss how isocrystals and, more generally, inner forms relate to the Fargues-Scholze construction of the correspondence, and how one can generalize the Fargues-Scholze construction beyond isocrystals.