**Title:** Mini-course on p-adic integration (for the Hitchin system) I

**Speaker:** Michael Gröchenig

**Speaker Info:** Free University, Berlin

**Brief Description:** First lecture of a four-lecture mini-course

**Special Note**:

**Abstract:**

In this series of talks I will report on joint work with D. Wyss and P. Ziegler. We prove a conjecture by Hausel—Thaddeus which predicts an agreement of appropriately defined Hodge numbers for (complex) moduli spaces of Higgs bundles for the structure groups SL(n) and PGL(n). Despite of the topological nature of the statement our proof is entirely arithmetic. Inspired by a celebrated result by Batyrev, we replace the field of complex numbers by the p-adics, and Hodge numbers by p-adic volumes.Talk 1) We will see the definition of Higgs bundles, their moduli spaces, and the Hitchin fibration. I will explain why the generic fibres of the Hitchin fibration are abelian varieties, and mention that Langlands dual structure groups lead to dual fibres (in the sense of abelian varieties).

Talk 2) We introduce p-adic integration and explain Batyrev’s proof that birational Calabi-Yau varieties have equal Betti numbers.

Talk 3) We will discuss p-adic integration for orbifolds (or rather quotient stacks). In the process we will encounter several arithmetic properties of local fields.

Talk 4) After recalling the basic properties of the Hitchin fibration we will use p-adic integration to prove the topological mirror symmetry conjecture by Hausel—Thaddeus.

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