Related papers: Local-global questions for tori over $p$-adic func…

The goal of the present paper is to push forward the frontiers of computations on Farrell-Tate cohomology for arithmetic groups. The conjugacy classification of cyclic subgroups is reduced to the classification of modules of group rings…

For a simple algebraic group $G$ we study the space $Q$ of Quasimaps from the projective line $C$ to the flag variety of $G$. We prove that the global Intersection Cohomology of $Q$ carries a natural pure Tate Hodge structure, and compute…

We establish the finiteness of the kernel and cokernel of the restriction map III^{i}(F,M) ---> III^{i}(K,M)^{G} for i=1 and 2, where M is a (Deligne) 1-motive over a global field F and K/F is a finite Galois extension of global fields with…

We construct a new class of three-dimensional topological quantum field theories (3d TQFTs) by considering generalized Argyres-Douglas theories on $S^1 \times M_3$ with a non-trivial holonomy of a discrete global symmetry along the $S^1$.…

Following (and elaborating on) a method of Logan and van Luijk, we exhibit explicit genus-2 curves whose Jacobians have nontrivial 2-torsion in their Tate-Shafarevich groups.

Let $X$ be a smooth, projective, geometrically connected curve over a finite field $\mathbb{F}_q$, and let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Its dual group $\hat{G}$ is a split reductive group over $\mathbb{Z}$.…

We prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects `discrete Selmer groups' and…

For a cuspidal automorphic representation of GL2/Q associated to a modular form, the local and global Langlands correspondences are compatible at all finite places of Q. On the p-adic Coleman-Mazur eigencurve this principle can fail (away…

Building upon our arithmetic duality theorems for 1-motives, we prove that the Manin obstruction related to a finite subquotient $\Be (X)$ of the Brauer group is the only obstruction to the Hasse principle for rational points on torsors…

We establish the Hasse principle (local-global principle) in the context of the Baum-Connes conjecture with coefficients. We illustrate this principle with the discrete group $GL(2,F)$ where $F$ is any global field.

We provide new constraints for algebro-geometric subgroups of mapping class groups, namely images of fundamental groups of curves under complex algebraic maps to the moduli space of smooth curves. Specifically, we prove that the restriction…

We describe functorially the first Galois cohomology set $H^1({\mathbb R},G)$ of a connected reductive algebraic group $G$ over the field $\mathbb R$ of real numbers in terms of a certain action of the Weyl group on the real points of order…

We study the Oort groups for a prime p, i.e. finite groups G such that every G-Galois branched cover of smooth curves over an algebraically closed field of characteristic p lifts to a G-cover of curves in characteristic 0. We prove that all…

Let $\mathcal X$ be a regular variety, flat and proper over a complete regular curve over a finite field, such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of $\mathcal X$ is finite if and…

We construct an arithmetic analogue of the quantum local systems on the moduli of curves, and study its basic structure. Such an arithmetic local system gives rise to a uniform way of assigning a Galois cohomology class of the first…

We establish a ramified class field theory for smooth projective curves over local fields. As key steps in the proof, we obtain new results in the class field theory for 2-dimensional local fields of positive characteristic, and prove a…

Let $(R,\mathfrak{m}, k)$ be a strictly local normal $k$-domain of positive characteristic and $P$ be a prime divisor on $X=\text{Spec } R$. We study the Galois category of finite covers over $X$ that are at worst tamely ramified over $P$…

If T is an algebraic torus defined over a discretely valued field K with perfect residue field k, we relate the K-cohomology of T to the k-cohomology of certain objects associated to T. When k has cohomological dimension <= 1, our results…

We extend the well-known Cassels-Tate dual exact sequence for abelian varieties A over global fields K in two directions: we treat the p-primary component in the function field case, where p is the characteristic of K, and we dispense with…

Let $G$ be a simple algebraic group over an algebraically closed field $k$. Let $\Gamma$ be a finite group acting on $G$. We classify and compute the local types of $(\Gamma, G)$-bundles on a smooth projective $\Gamma$-curve in terms of the…