Testing rationality of coherent cohomology of Shimura varieties
Michael Harris

TL;DR
The paper proves a new rationality criterion for coherent cohomological automorphic forms on Shimura varieties by analyzing pairings between automorphic representations of related groups using ergodic and positivity methods.
Contribution
It introduces a novel approach combining ergodic theory, positivity, and classification of discrete series to establish rationality criteria for automorphic forms on Shimura varieties.
Findings
Existence of non-trivial pairings between automorphic representations of G and G'
Explicit rationality criteria for coherent cohomological forms
Application to inner forms of unitary groups
Abstract
Let be an inclusion of reductive groups whose real points have a non-trivial discrete series. Combining ergodic methods of Burger-Sarnak and the author with a positivity argument due to Li and the classification of minimal -types of discrete series, due to Salamanca-Riba, we show that, if is a cuspidal automorphic representation of whose archimedean component is a sufficiently general discrete series, then there is a cuspidal automorphic representation of , of (explicitly determined) discrete series type at infinity, that pairs non-trivially with . When and are inner forms of U(n) and , respectively, this result is used to define rationality criteria for sufficiently general coherent cohomological forms on .
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic Geometry and Number Theory · Advanced Mathematical Identities
