Let p be a prime number. Roughly speaking, rigid analytic geometry is a counterpart of complex analysis where one replaces the field ℂ of complex numbers by the field ℚp of p-adic rational numbers (or some extension thereof).
In this talk, I'll try to explain some of the motivations which led to the development of this theory, and to give some flavour of the recent progress which make it a timely subject for our Special Year. No prior knowledge of p-adic geometry will be assumed.
One of the most famous – and still not fully understood – objects in mathematics is the Mandelbrot set. It is defined as the set of complex numbers c for which the polynomial fc(z)=z2+c has a connected Julia set. But the Mandelbrot set turns out to be related to many different areas of mathematics. Inspired by recent results in arithmetic geometry, I will describe how the tools of arithmetic intersection theory can be applied in the setting of these complex dynamical systems to give new information about the Mandelbrot set.
p-adic non-abelian Hodge theory, also known as the p-adic Simpson correspondence, aims at describing p-adic local systems on a smooth rigid analytic variety in terms of Higgs bundles. I will explain in this talk why the 'Hodge-Tate stacks' recently introduced by Bhatt-Lurie and Drinfeld in their work on prismatic cohomology can be useful to study this kind of questions.
Let X be a smooth projective variety over the field of complex numbers. The classical Riemann-Hilbert correspondence supplies a fully faithful embedding from the category of perverse sheaves on X to the category of algebraic DX-modules. In this talk, I'll describe joint work with Bhargav Bhatt on an analogous construction in the setting of non-archimedean geometry.
Let X be a smooth projective variety over the complex numbers. Let M be the moduli space of irreducible representations of the topological fundamental group of X of a fixed rank r. Then M is a finite type scheme over the spectrum of the integers ℤ. One may ask if the irreducible components of M surject onto Spec(ℤ) or whether M is pure over ℤ in some sense. We give a weak answer to these questions and we discuss what other phenomena can be studied using the method of proof.
Many cohomology theories in algebraic geometry, such as crystalline and syntomic cohomology, are not homotopy invariant. This is a shame, because it means that the stable motivic homotopy theory of Morel-Voevodsky cannot be employed in studying the deeper aspects of such theories, such as cohomology operations that act on the cohomology groups. In this talk, I will discuss ongoing efforts, joint with Ryomei Iwasa and Marc Hoyois, to set up a workable theory of non-homotopy invariant stable motivic homotopy theory, with the goal of providing effective tools of studying cohomology theories in algebraic geometry by geometric means.
I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organised and reproved using the theory of graded Lie algebras, following earlier work of Thorne. This gives a uniform proof of these results and yields new theorems for certain families of non-hyperelliptic curves.
The study of moduli spaces, and special points in moduli spaces, has been of arithmetic interest. In this talk, I will speak about results pertaining to the algebraic and analytic distribution of special points, and touch upon topics such as the Andre-Oort conjecture, the p-adic distribution of special points, and the p-adic and mod p distribution of isogeny-orbits in these moduli spaces.
Given an elliptic curve E, Kolyvagin used CM points on modular curves to construct a system of classes valued in the Galois cohomology of the torsion points of E. Under the conjecture that not all of these classes vanish, he gave a description for the Selmer group of E. This talk will report on recent work proving new cases of Kolyvagin's conjecture. The proof builds on work of Wei Zhang, who used congruences between modular forms to prove Kolyvagin's conjecture under some technical hypotheses. We remove many of these hypotheses by considering congruences modulo higher powers of p. The talk will explain the difficulties associated with higher congruences of modular forms and how they can be overcome.
I will explain some recent work on special cases of the Bloch-Kato conjecture for the symmetric cube of certain modular Galois representations. Under certain standard conjectures, this work constructs non-trivial elements in the Selmer groups of these symmetric cube Galois representations; this works by p-adically deforming critical Eisenstein series in a generically cuspidal family of automorphic representations, and then constructing a lattice in the associated family of Galois representations, all for the exceptional group G2. While I will touch on all of these aspects of the construction, I will mainly focus on the Galois side in this talk.
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