# Schur indices and the p-adic langlands program

- Responsibility
- David Alfred Sherman.
- Publication
- [Stanford, California] : [Stanford University], 2018.
- Copyright notice
- ©2018
- Physical description
- 1 online resource.

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Call number | Status |
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3781 2018 S | In-library use |

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## Description

### Creators/Contributors

- Author/Creator
- Sherman, David Alfred, author.
- Contributor
- Conrad, Brian, 1970- degree supervisor.
- Bump, Daniel, 1952- degree committee member.
- Howe, Sean, degree committee member.
- Stanford University. Department of Mathematics.

### Contents/Summary

- Summary
- This dissertation uses the lens of the p-adic Langlands program to understand arithmetic questions about representations of some finite groups of Lie type. Any irreducible complex representation V of finite group G is realizable as a representation on a K-vector space for some number field K. Whether this is possible for a particular field K (necessarily containing all of the values of the character of V) is essentially controlled by a cohomological obstruction (belonging to the Brauer group of K), which is encoded in "local obstructions" at each place of K. To be specific, we consider cuspidal representations of the degree-two general and special linear groups GL_2(F_p) and SL_2(F_p) over the field with p elements, p an odd prime, and focus on the obstructions at p-adic places of K. These obstructions have previously been shown (via group-theoretic means) to vanish. In this dissertation, we present a new proof along the following lines: relate the original representation V to an (infinite-dimensional) p-adic Banach space representation of the corresponding p-adic group GL_2(Q_p) or SL_2(Q_p), use the p-adic Langlands correspondence to further relate that to a p-adic Galois representation W (or a close cousin), and compute the obstruction using W. The p-adic Langlands correspondence was already known for the degree-two general linear group over the p-adic numbers, but here we prove that it is suitably "natural" to transfer the Brauer obstruction from V to W (making our strategy possible). For the special linear group, on the other hand, there is no existing p-adic correspondence. Therefore, in this dissertation we construct a functor D_S, which we expect to realize the correspondence. This functor is a relative of the "Montreal functor" D that realizes the GL_2(Q_p) correspondence. Using the GL_2(Q_p) case as a guide, we then prove enough (though not all) of the expected properties of the SL_2(Q_p) correspondence, including its "naturality, " to carry out our above strategy.

### Bibliographic information

- Publication date
- 2018
- Copyright date
- 2018
- Note
- Submitted to the Department of Mathematics.
- Note
- Thesis Ph.D. Stanford University 2018.