##### Department of Mathematics,

University of California San Diego

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### Special Colloquium

## Claus Sorensen

#### Princeton University

## Integral Structures in Steinberg Representations and P-Adic Langlands

##### Abstract:

As a vast generalization of quadratic reciprocity, class field theory describes all abelian extensions of a number field. Over Q, they are precisely those contained in cyclotomic fields. However, there are a lot more non-abelian extensions, which arise naturally. The Langlands program attempts to systematize them, by relating Galois representations and automorphic forms; mathematical objects of rather disparate nature. We will illustrate the basic plot for GL(2) through the example of elliptic curves and modular forms - the context of Wiles' proof of Fermat's Last Theorem. The main goal of the talk will be to motivate a ``p-adic" Langlands correspondence, which is at the forefront of contemporary number theory, but still only well-understood for GL(2) over $Q_p$. We will discuss, in some depth, the case of semistable elliptic curves, which provide the first non-trivial example. This leads naturally to a result we proved recently, which shows the existence of (many) integral structures in locally algebraic representations of ``Steinberg" type, for any reductive group G (such as GL(n), symplectic, and orthogonal groups). As a result, there are a host of ways to p-adically complete the Steinberg representation (tensored with an algebraic representation). The ensuing Banach spaces should play a role in a (yet elusive) higher-dimensional p-adic Langlands correspondence. We hope to at least give some idea of the proof, which goes via automorphic representations and the trace formula.

Host: Cristian Popescu

### January 15, 2013

### 3:00 PM

### AP&M 6402

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