# Schur indices and the p-adic langlands program

## Abstract/Contents

- Abstract
- 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.

## Description

Type of resource | text |
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Form | electronic resource; remote; computer; online resource |

Extent | 1 online resource. |

Place | California |

Place | [Stanford, California] |

Publisher | [Stanford University] |

Copyright date | 2018; ©2018 |

Publication date | 2018; 2018 |

Issuance | monographic |

Language | English |

## Creators/Contributors

Author | Sherman, David Alfred |
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Degree supervisor | Conrad, Brian, 1970- |

Thesis advisor | Conrad, Brian, 1970- |

Thesis advisor | Bump, Daniel, 1952- |

Thesis advisor | Howe, Sean |

Degree committee member | Bump, Daniel, 1952- |

Degree committee member | Howe, Sean |

Associated with | Stanford University, Department of Mathematics. |

## Subjects

Genre | Theses |
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Genre | Text |

## Bibliographic information

Statement of responsibility | David Alfred Sherman. |
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Note | Submitted to the Department of Mathematics. |

Thesis | Thesis Ph.D. Stanford University 2018. |

Location | electronic resource |

## Access conditions

- Copyright
- © 2018 by David Alfred Sherman
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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