Linked Grassmannians and Local Models of Shimura Varieties: Unramified Type A

with B. Li

Linked Grassmannians are objects that arise in the study of algebraic curves, namely in limit linear series and Brill-Noether theory. It turns out that local models of Shimura varieties of PEL-type admit natural interpretations as Linked Grassmannians, and that this allows for richer class of conditions that we can impose on the local model moduli problem, of a much different flavor than ones in standard use (e.g. "wedge" or "spin" conditions). This allows us to give a moduli-theoretic definition of local models even in cases where the definition must be modified due to the naive moduli space not being flat. In this paper, we work out the case where the PEL Shimura variety is of type A, that is, for Shimura varieties associated with unitary groups.

Last Updated: 12/12/15 (Ramified case under revision)

Globalizing Self-Dual Automorphic Representations on GL(2n+1).

Many naturally arising Galois representations are self-dual, for example, the ones attached to elliptic curves over the rationals or to modular forms. Under the global Langlands correspondence, such Galois representations correspond to self-dual automorphic representations. However, self-duality is a very delicate condition and so it is subtle to construct such automorphic representations without assuming some strong technical hypotheses. However, in this paper, we construct self-dual automorphic representations on GL(2n+1) over totally real fields without assuming such hypotheses, by exploiting some special behavior that holds under such a restriction on the number field. Guided by the techniques of twisted endoscopy in the Langlands program and applying the Arthur-Selberg trace formula, we use local harmonic analysis methods to show such self-dual representations can be constructed, even if we fix finitely many local components. This extends similar results obtained by Chenevier and Clozel in the GL(2n) case.