We develop a fairly explicit Kuznetsov formula on (3) and discuss the analytic behavior of the test functions on both sides. Applications to Weyl's law, exceptional eigenvalues, a large sieve and -functions are given.

We construct a self-affine sponge in whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge, implying that sponges with a dimension gap represent a nonempty open subset of the parameter space.

We prove that for every Bushnell-Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. Moreover, we show that every irreducible smooth -representation contains a rigid type. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors.

We prove a descent criterion for certain families of smooth representations of ( a -adic field) in terms of the -factors of pairs constructed in Moss (Int Math Res Not 2016(16):4903-4936, 2016). We then use this descent criterion, together with a theory of -factors for families of representations of the Weil group  (Helm and Moss in Deligne-Langlands gamma factors in families, arXiv:1510.08743v3, 2015), to prove a series of conjectures, due to the first author, that give a complete description of the center of the category of smooth -modules (the so-called "integral Bernstein center") in terms of Galois theory and the local Langlands correspondence. An immediate consequence is the conjectural "local Langlands correspondence in families" of Emerton and Helm (Ann Sci Éc Norm Supér (4) 47(4):655-722, 2014).

Block and Göttsche have defined a -number refinement of counts of tropical curves in . Under the change of variables , we show that the result is a generating series of higher genus log Gromov-Witten invariants with insertion of a lambda class. This gives a geometric interpretation of the Block-Göttsche invariants and makes their deformation invariance manifest.