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PATCHING AND THE COMPLETED HOMOLOGY OF LOCALLY SYMMETRIC SPACES

Published online by Cambridge University Press:  27 May 2020

Toby Gee
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK (toby.gee@imperial.ac.uk)
James Newton
Affiliation:
Department of Mathematics, King’s College London, LondonWC2R 2LS, UK (j.newton@kcl.ac.uk)

Abstract

Under an assumption on the existence of $p$-adic Galois representations, we carry out Taylor–Wiles patching (in the derived category) for the completed homology of the locally symmetric spaces associated with $\operatorname{GL}_{n}$ over a number field. We use our construction, and some new results in non-commutative algebra, to show that standard conjectures on completed homology imply ‘big $R=\text{big}~\mathbb{T}$’ theorems in situations where one cannot hope to appeal to the Zariski density of classical points (in contrast to all previous results of this kind). In the case where $n=2$ and $p$ splits completely in the number field, we relate our construction to the $p$-adic local Langlands correspondence for $\operatorname{GL}_{2}(\mathbb{Q}_{p})$.

Type
Research Article
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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Footnotes

The first author was supported in part by a Leverhulme Prize, EPSRC grant EP/L025485/1, Marie Curie Career Integration Grant 303605, and by ERC Starting Grant 306326. The second author was supported by ERC Starting Grant 306326.

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