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Reductions of 2-dimensional crystalline representations with large slope

Algebra and Number Theory Seminar

Reductions of 2-dimensional crystalline representations with large slope
Series: Algebra and Number Theory Seminar
Location: ENR2 S395
Presenter: Brandon Levin, University of Arizona

For any k >= 2 and p-adic integer a, there is a two-dimensional crystalline p-adic representation V_{k,a} of the Galois group of Q_p with weight k and slope a.   It has been a long-standing question to determine the reduction mod p of V_{k, a} as k and a vary.  A powerful tool in addressing this question is the p-adic local Langlands correspondence which has led to significant breakthroughs when the p-adic valuation of a is small.

On other hand, the correspondence seems to say less about the behavior when valuation of a is large.  For fixed k, it is known that when a is sufficiently close to 0, the reduction is the same as that of the central point V_{k, 0}.   However, how close one needs to be remains somewhat mysterious.   In this talk, I will discuss work with John Bergdall which improves on a 2004 result of Berger-Li-Zhu on the question of the central radius.   Our method uses the theory of Kisin modules.