We consider a crystalline universal deformation ring R of an n-dimensional, mod p Galois representation whose semisimplification is the direct sum of two non-isomorphic absolutely irreducible representations. Under some hypotheses, we obtain that R is a discrete valuation ring. The method examines the ideal of reducibility of R, which is used to construct extensions of representations in a Selmer group with specified dimension. This can be used to deduce modularity of representations.
An integral ring R is a ring additively isomorphic to Z^n . The subring zeta function is an important tool in studying subring growth in these rings. One can compute these zeta functions using p-adic integration due to a result of Grunewald, Segal and Smith. I shall talk about computing these zeta functions for Z[t]/(t^n) for small n and describe some results on subring growth and ideal growth for integral rings. This includes joint work with Ramin Takloo-Bighash and Gautam Chinta.
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.
Let k be a perfect field of characteristic p>0 and let X be a proper scheme over W(k) with semistable reduction. I shall define a log-Chow group for the special fibre X_k and give an interpretation in terms of logarithmic Milnor K-theory. Then, by gluing a logarithmic variant of the Suslin-Voevodsky motivic complex to a log-syntomic complex along the logarithmic Hyodo-Kato Hodge-Witt sheaf, I will prove that an element of the r-th log-Chow group of X_k formally lifts to the continuous log-Chow group of X if and only if it is “Hodge” (i.e. its log-crystalline Chern class lands in the r-th step of the Hodge filtration of the generic fibre of X under the Hyodo-Kato isomorphism). This simultaneously generalises a result of Yamashita (which is the case r=1), and of Bloch-Esnault-Kerz (which is the case of good reduction). This is joint work with Andreas Langer.
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.