We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type SLn and P GLn. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for d coprime to n, that the number of rank n Higgs bundles of degree d over a fixed curve defined over a finite field, is independent of d. This proves a conjecture by Mozgovoy-Schiffman in the coprime case.Contents arXiv:1707.06417v2 [math.AG] 8 Jun 2018Proof. We begin the proof by fixing notation. The relative Jacobian of the trivial family of curves X × S A / A will be denoted by J. The relative Jacobian of the universal spectral curve Y / A will be denoted by J. Similarly we denote by J 1 and J 1 the relative moduli spaces of degree 1 line bundles.Henceforth we restrict every A-scheme to the open subset A good . To avoid awkward notation we will omit the corresponding superscript.The relative norm map induces a morphism of abelian A-schemes J Nm / / J. Similarly, pullback of line bundles yields π * : J / / J. We claim that these two morphisms are dual to each other with respect to the canonical isomorphism J ∨ J induced by the Poincaré bundle (and similarly for J). To see this we observe that we have a commutative diagram (the horizontal arrows represent the Abel-Jacobi map)to which we can apply the contravariant Pic 0 (?/ A) functor to obtain the commutative diagram of abelian schemes Furthermore, the top row of the first diagram is the fibre of the vertical arrows and hence the top row of the second diagram is the corresponding cofibre. By explicitly computing the fibres in the second diagram

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In this article we give a new proof of Ngô's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme G to the cohomology of Hitchin fibres for the endoscopy groups Hκ. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results for p-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of G-Higgs bundles in terms of endoscopic data, and extend duality for generic Hitchin fibres of Langlands dual group schemes to the quasi-split case.

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We prove, that Hausel's formula for the number of rational points of a Nakajima quiver variety over a finite field also holds in a suitable localization of the Grothendieck ring of varieties. In order to generalize the arithmetic harmonic analysis in his proof we use Grothendieck rings with exponentials as introduced by Cluckers-Loeser and Hrushovski-Kazhdan.