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College of Liberal Arts and Sciences | Mathematics

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October 23, 2018 | 2:00 p.m. - 3:00 p.m.
Category: Seminar
Location: Faculty/Administration #1140 | Map
656 W. Kirby
Detroit, MI 48202
Cost: Free

Speaker: Andrew Salch, Wayne State University

Title: The spectral Jacquet-Langlands correspondence

Abstract: Given a p-adic number field K, the Jacquet-Langlands correspondence establishes a bijection between irreducible supercuspidal representations of GL_n(O_K) and certain irreducible representations of the unit group of the maximal order O_D in the invariant 1/n central division algebra over K. In this talk I will explain a bit about why the JL correspondence matters, and how the approaches of Drinfeld, Carayol, and Faltings led to a proof of the existence of this correspondence by studying the limit, over m, of the etale cohomology of the deformation space of a height n formal group law with Drinfeld level p^m structures. In the case m=0, this deformation space is ordinary Lubin-Tate space, which (in the case K = Q_p) admits a spectral scheme refinement by the theorem of Goerss-Hopkins-Miller, but Schwanzl, Vogt, and Waldhausen proved that this spectral scheme structure cannot be extended to the m>0 stages in this tower of deformation spaces.

In this talk I will explain joint work with Matthias Strauch, from 2018, in which (in the case K = Q_p) we enlarge the moduli problem of formal groups with Drinfeld level structures by considering a more general notion of "degenerating level structures," so that the resulting moduli spaces do admit spectral scheme refinements at each stage in the tower, and such that the essential cohomological calculation of Faltings splits off, as a summand, from a certain K-theoretic calculation in these spectral schemes. Consequently we get a kind of "spectral Jacquet-Langlands correspondence," in which the group GL_n(O_K) and the unit group in O_D act on ell-adic spectra instead of ell-adic vector spaces. If time allows, I will give the p>3, height 2, level p case as an explicit example, which turns out to also be a special case of a construction being pursued in ongoing joint work with L. Candelori, on the construction of "topological Katz p-adic modular forms."

For more information about this event, please contact Department of Mathematics at 3135772479 or math@wayne.edu.