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Arithmetic version of Deligne’s semisimplicity theorem, and beyond.

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KAH - K-theory, algebraic cycles and motivic homotopy theory

A fundamental result in the theory of variation of Hodge structure is the Deligne’s semisimplicity theorem. In this talk, I am going to present an arithmetic version of this theorem. The novel thing is the introduction of the notion of periodic logarithmic de Rham/Higgs bundles. A basic result, which underlies the arithmetic semisimplicity theorem, is that a geometric logarithmic de Rham/Higgs bundle is periodic. We conjecture the converse, and in particular we shall propose the Semisimplicity conjecture: a periodic logarithmic de Rham/Higgs bundle is semisimple. I shall explain an unexpected relation between a very special case of the Semisimplicity conjecture with a basic result of N. Elkies: there exist infinitely many supersingular primes for any elliptic curve defined over $\mathbb Q$. This is a joint work with Raju Krishnamoorthy.

This talk is part of the Isaac Newton Institute Seminar Series series.

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