On quadratic distinction of automorphic sheaves

Details On quadratic distinction of automorphic sheaves On quadratic distinction of automorphic sheaves

2011-02-17

arxiv.org/abs/1102.3469v1

Mathematics

Algebraic Geometry

23 pages

Scientific paper

We prove a geometric version of a classical result on the characterization of an irreducible cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_E)$ being the base change of a stable cuspidal packet of the quasi-split unitary group associated to the quadratic extension $E/F$, via the nonvanishing of certain period integrals, called being distinguished. We show that certain cohomology of an automorphic sheaf of $\mathrm{GL}_{n,X'}$ is nonvanishing if and only if the corresponding local system $E$ on $X'$ is conjugate self-dual with respect to an \'{e}tale double cover $X'/X$ of curves, which directly relates to the base change from the associated unitary group. In particular, the geometric setting makes sense for any base field.

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