We study large groups of birational transformations Bir(X), where X is a variety of dimension at least 3, defined over C or a subfield of C. Two prominent cases are when X is the projective space P^n, in which case Bir(X) is the Cremona group of rank n, or when X⊂P^{n+1} is a smooth cubic hypersurface. In both cases, and more generally when X is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from Bir(X) to Z/2, showing in particular that the group Bir(X) is not perfect, and thus not simple. As a consequence, we also obtain that the Cremona group of rank n⩾3 is not generated by linear and Jonquières elements.

This volume contains the written account of the Bonn seminar on arith-metic geometry 2003/2004. It gives a coherent exposition of the theory of intersections of modular correspondences. The focus of the seminar is the formula for the intersection number of arithmetic modular correspondences due to Gross and Keating. Other topics treated are Hurwitz's theorem on the intersection of modular correspondences over the field of complex numbers, and the relation of the arithmetic intersection numbers to Fourier coefficients of Siegel-Eisenstein series. Also included is background material on one-dimensional formal groups and their endomorphisms, and on quadratic forms over the ring of p-adic integers.

Let X_\Sigma be a toric surface and let (\check {X}, W) be its Landau-Ginzburg (LG) mirror where W is the Hori-Vafa potential as shown in their preprint. We apply asymptotic analysis to study the extended deformation theory of the LG model (\check {X}, W), and prove that semi-classical limits of Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to tropical disks in X with Maslov index 0 or 2, the latter of which produces a universal unfolding of W. For X = \mathbb{P}^2, our construction reproduces Gross' perturbed potential W_n [Adv. Math. 224 (2010), pp. 169-245] which was proven to be the universal unfolding of W written in canonical coordinates. We also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of W_n across walls of the scattering diagram formed by Maslov index 0 tropical disks originally observed by Gross in the same work (in the case of X = \mathbb{P}^2).

In this article we propose a geometric description of Arthur packets for p-adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of p-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur's main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan's work, especially the prediction that Arthur packets are ABV-packets for p-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.

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