For a representation of the absolute Galois group of the rationals over a finite field of characteristic p, we would like to know if there exists a lift to characteristic 0 with nice properties. In particular, we would like it to be geometric in the sense of the Fontaine-Mazur conjecture: ramified at finitely many primes and potentially semistable at p. For 2-dimensional representations, Ramakrishna proved that under technical assumptions, odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. The key ingredient is a smooth local deformation condition obtained by analysing unipotent orbits and their centralizers in the relative situation, not just over fields.
I will speak about results contained in my article "G-torseurs en théorie de Hodge p-adique" linked to local class field theory. I will in particular explain the computation of the Brauer group of the curve and why its fundamental class is the one from local class field theory.
In this talk we first introduce a new 'singularity-free' approach to the proof of Seidel's long exact sequence, including the fixed-point version. This conveniently generalizes to Dehn twists along Lagrangian submanifolds which are rank one symmetric spaces and their covers, including ℝPn and ℂPn, matching a mirror prediction due to Huybrechts and Thomas. The idea of the proof can be interpreted as a 'mirror' of the construction in algebraic geometry, realized by a new surgery and cobordism construction.
Given the p-adic Galois representation associated to a regular algebraic polarized cuspidal automorphic representation, one naturally obtains a pure weight zero representation called its adjoint representation. Because it has weight zero, a conjecture of Bloch and Kato says that the only de Rham extension of the trivial representation by this adjoint representation is the split extension. We will discuss a proof of this case of their conjecture, under an assumption on the residual representation. This is done by using the Taylor-Wiles patching method, Kisin's technique of analyzing the generic fibre of deformation rings, and a characterization of smooth closed points in the generic fibre of certain local deformation rings.
Venkatesh has recently proposed a fascinating conjecture relating motivic cohomology with automorphic forms and the cohomology of arithmetic groups. I'll describe this conjecture, and discuss its connections with the local geometry of eigenvarieties and nonabelian analogues of the Leopoldt conjecture. This is joint work with Jack Thorne.
This video is of the London Mathematical Society and European Mathematical Society‘s Joint Mathematical Weekend in 2015.
This video is of the London Mathematical Society and European Mathematical Society‘s Joint Mathematical Weekend in 2015.
This video is of the London Mathematical Society and European Mathematical Society‘s Joint Mathematical Weekend in 2015.
This video is of the London Mathematical Society and European Mathematical Society‘s Joint Mathematical Weekend in 2015.
This video is of the London Mathematical Society and European Mathematical Society‘s Joint Mathematical Weekend in 2015.