The recent proofs of the Tate conjecture for K3 surfaces over finite fields start by lifting the surface to characteristic 0. Serre showed in the sixties that not every variety can be lifted, but the question whether every motive lifts to characteristic 0 is open. We give a negative answer to a geometric version of this question, by constructing a smooth projective variety that cannot be dominated by a smooth projective variety that lifts to characteristic 0.
In this talk, I will discuss how recent developments in arithmetic geometry (for example pertaining to perfectoid spaces) led to significant new discoveries in commutative algebra and algebraic geometry in mixed characteristic.
We aim to develop ramification theory for arbitrary valuation fields, extending the classical theory of complete discrete valuation fields with perfect residue fields. By studying wild ramification, we hope to understand the mysterious phenomenon of the defect (or ramification deficiency) unique to the positive residue characteristic case and is one of the main obstacles in obtaining resolution of singularities.
Extensions of degree p in residue characteristic p>0 are building blocks of the general case. We present a generalization of ramification invariants for such extensions. These results enable us to construct an upper ramification filtration of the absolute Galois group of Henselian valuation fields (joint with K.Kato).
Counting non-isomorphic finite nilpotent groups of order n is a very hard problem. One way to approach this problem is to count finite nilpotent groups of fixed nilpotency class c on d generators. The enumeration of such isomorphism classes of objects involves number theory and the theory of algebraic groups. However, very little is known about the explicit generating functions of these sequences of numbers when c > 2 or d > 2. We use a direct enumeration of such groups that began in the works of M. Bacon, L. Kappe, et al, to provide a natural multivariable extension of the generating function counting such groups. Then we rederive the explicit formulas that are known so far.
The Gauss composition law famously describes the class group of an order in a quadratic number field by an operation on binary quadratic forms up to matrix transformation. Using a stricter notion of equivalence, we describe ray class groups of a quadratic order in terms of quadratic forms. We explore applications to representing primes by binary quadratic forms, and we describe leading coefficients of Hecke series for real quadratic fields as twisted traces of cycle integrals of polyharmonic Maass forms. This is ongoing joint work with Gene Kopp.
A local-global principle is a result that allows us to deduce global information about an object from local information. A well-known example is the Hasse-Minkowski theorem, which asserts that a quadratic form represents a number if and only if it does so everywhere locally. In this talk, we'll discuss certain local-global principles in arithmetic geometry, highlighting two that are related to elliptic curves, one for torsion and one for isogenies. In contrast to the Hasse-Minkowski theorem, we'll see that these two results exhibit considerable rigidity in the sense that a failure of either of their corresponding everywhere local conditions must be rather significant.