Higgs bundles and p-adic integration
-- Michael Groechenig, September 7, 2018

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In the first half of the talk I will introduce the theory of Higgs bundles as a relative version of Lie theory over a Riemann surface. The moduli space of Higgs bundles is the total space of an integrable system, known as the Hitchin system. Its generic fibre is a complex torus, but the singular fibres exhibit a more intricate geometry which is reflected in rich arithmetic properties. The latter has been exemplified by Ng&#244;'s proof of the fundamental lemma. In the second half of the talk I will report on joint work with Wyss and Ziegler which studies moduli spaces of Higgs bundles over local fields (such as the p-adic numbers). Combining the arithmetics of p-adic fields with basic measure theory we can compare invariants of Hitchin fibres for Langlands dual structure groups. This yields a proof of a conjecture by Hausel-Thaddeus on the complex analytic Hodge numbers of moduli spaces of Higgs bundles. Furthermore, our methods shed new light on the geometric part of Ng&#244;s proof of the fundamental lemma.