This paper proposes a tangential version of the theory of Selmer varieties together with a formulation of cohomological duality in families of Lie algebras indexed by nonabelian cohomology. This theory allows one to consider deformations of cohomology classes as one moves over the Selmer variety and suggests an approach for generalizing to number fields the homotopical techniques for proving Diophantine finiteness that were developed over $\mathbb{Q}$. The utility of this perspective is demonstrated by way of a new proof of Siegel’s theorem on finiteness of $S$-integral points for the projective line minus three points over a totally real field.

Let $I$ be an open interval containing zero, let $M$ be a real manifold, let ${\mathrm{Ṫ}}^{\ast }M$ be its cotangent bundle with the zero-section removed, and let $\Phi =\{{\phi }_t{\}}_{t\in I}$ be a homogeneous Hamiltonian isotopy of ${\mathrm{Ṫ}}^{\ast }M$ with ${\phi }_0=id$. Let $\Lambda \subset {\mathrm{Ṫ}}^{\ast }M\times {\mathrm{Ṫ}}^{\ast }M\times T^{\ast }I$ be the conic Lagrangian submanifold associated with $\Phi$. We prove the existence and unicity of a sheaf $K$ on $M\times M\times I$ whose microsupport is contained in the union of $\Lambda$ and the zero-section and whose restriction to $t=0$ is the constant sheaf on the diagonal of $M\times M$. We give applications of this result to problems of nondisplaceability in contact and symplectic topology. In particular we prove that some strong Morse inequalities are stable by Hamiltonian isotopies, and we also give results of nondisplaceability for nonnegative isotopies in the contact setting.

We prove a companion forms theorem for ordinary $n$-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author and a technique for deducing companion forms theorems developed by the first author. We deduce results about the possible Serre weights of $modl$ Galois representations corresponding to automorphic representations on unitary groups. We then use functoriality to prove similar results for automorphic representations of ${GSp}_4$ over totally real fields.

We derive the quantum Teichmüller space, previously constructed by Kashaev and by Fock and Chekhov, from tensor products of a single canonical representation of the modular double of the quantum plane. We show that the quantum dilogarithm function appears naturally in the decomposition of the tensor square, the quantum mutation operator arises from the tensor cube, the pentagon identity from the tensor fourth power of the canonical representation, and an operator of order three from isomorphisms between canonical representation and its left and right duals. We also show that the quantum universal Teichmüller space is realized in the infinite tensor power of the canonical representation naturally indexed by rational numbers including infinity. This suggests a relation to the same index set in the classification of projective modules over the quantum torus, the unitary counterpart of the quantum plane, and points to a new quantization of the universal Teichmüller space.