Potential Automorphy for $GL_n$

DOI10.1007/S00222-022-01161-6arXiv2104.09761MaRDI QIDQ6365699

Publication date: 20 April 2021

Abstract: We prove potential automorphy results for a single Galois representation

G_{F} i g h t a r r o w G L_{n} (o v e r l i n e {m a t h b b Q}_{l})

where

F

is a CM number field. The strategy is to use the

p, q

switch trick and modify the Dwork motives employed in cite{HSBT} to break self-duality of the motives, but not the Hodge-Tate weights. Another key result to prove is the ordinarity of certain

p

-adic representations, which follows from log geometry techniques. One input is the automorphy lifting theorem in cite{tap}.

Mathematics Subject Classification ID

Elliptic curves over global fields (11G05) Holomorphic modular forms of integral weight (11F11) Galois representations (11F80) Representation-theoretic methods; automorphic representations over local and global fields (11F70)

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