We introduce a 'tremor' deformation on strata of translation surfaces. Using it, we give new examples of behaviors of horo cycle flow orbits Uq in strata of translation surfaces. In the genus 2 stratum H(1, 1) we find orbits Uq which are generic for a measure whose support is strictly contained in Uq and find orbits which are not generic for any measure. We also describe a horo cycle orbit-closure whose Hausdorff dimension is not an integer.
Let G be a simple group over a global function field K, and let it be a cuspidal automorphic representation of G. Suppose K has two places u and v (satisfying a mild restriction on the residue field cardinality), at which the group G is quasi-split, such that ituis tempered and itvis unramified and generic. We prove that itw is tempered at all unramified places Kw at which G is unramified quasi-split. More generally, the set of unitary spherical representations is partitioned according to nilpotent conjugacy classes in the Lie algebra of G. We show that if itvis in the set corresponding to the nilpotent class N, and if itusat-isfies an analogous hypothesis, then itw belongs to the same class N, where w is as above. These results are consistent with conjectures of Shahidi and Arthur. The proofs use the Galois parametrization of cuspidal representations due to V. Lafforgue to relate the local Satake parameters of it to Deligne's theory of Frobenius weights. The main observation is that, in view of the classification of unitary spherical representations, due to Barbasch and the first-named author, the theory of weights excludes almost all complementary series as possible local components of it. This in turn determines the local Frobenius weights at all unramified places. In order to apply this observation in practice, we need a result of the second-named author with Gan and Sawin on the weights of discrete series representations.
We study intrinsic diophantine approximation on Grassmann varieties. Using a new correspondence between the diophantine properties of a linear subspace x in Rd and certain diagonal orbits in the space of lattices, we are able to solve some problems suggested by Schmidt in 1967. In particular, we obtain a version of Dirichlet's principle in this setting with an optimal exponent.
We give the first examples of a pair of knots $K_1$, $K_2$ in the $3$-sphere for which their unknotting numbers satisfy $u(K_1 # K_2) < u(K_1)+u(K_2)$. This answers question 1.69(B) from Kirby's problem list, in the negative.
Let $E$ be a CM elliptic curve defined over $\mathbb{Q}$ and $p$ a prime. We show that $${\mathrm corank}_{\mathbb{Z}_{p}} {\mathrm Sel}_{p^{\infty}}(E_{/\mathbb{Q}})=0 \implies {\mathrm ord}_{s=1}L(s,E_{/\mathbb{Q}})=0 $$ for the $p^{\infty}$-Selmer group ${\mathrm Sel}_{p^{\infty}}(E_{/\mathbb{Q}})$ and the complex $L$-function $L(s,E_{/\mathbb{Q}})$. Along with Smith's work on the distribution of $2^\infty$-Selmer groups, this leads to the first instance of the even parity Goldfeld conjecture: For $50\%$ of the positive square-free integers $n$, we have $ {\mathrm ord}_{s=1}L(s,E^{(n)}_{/\mathbb{Q}})=0, $ where $E^{(n)}: ny^{2}=x^{3}-x $ is a quadratic twist of the congruent number elliptic curve $E: y^{2}=x^{3}-x$.
