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On the local mod p representation attached to a modular form
If you have a question about this talk, please contact Teruyoshi Yoshida.
There is a 2-dimensional p-adic Galois representation attached to any modular form which is an eigenform for the Hecke operators. One can reduce it mod p and get an even simpler object – a mod p Galois representation, which will have finite image. One would have thought that nowadays essentially everything was known about this representation, but actually there are still some dangling issues at p. For example – if I give you an explicit modular form (e.g. via its q-expansion), is the associated local mod p representation reducible or irreducible? This question is local, but still poorly understood. I will explain what little I know, most of which is joint work with Toby Gee. In what looks like a sledgehammer-cracking-a-nut approach, we invoke recent deep work of Breuil-Berger and others on the p-adic and mod p Langlands philosophy for GL(2) to get some concrete down-to-earth results.
This talk is part of the Number Theory Seminar series.
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