arXiv subjects
Publications and source records attributed to Kiran S. Kedlaya.
We describe a class of multivariate series rings generalizing the usual Robba ring over a p-adic field, and give a basic development of phi-modules over such rings. This makes it possible to give a unified survey of a number of recent developments in p-adic Hodge theory.
In a previous paper, we constructed a category of (phi, Gamma)-modules associated to any adic space over Q_p with the property that the etale (phi, Gamma)-modules correspond to etale Q_p-local systems; these involve sheaves of period rings for Scholze's pro-etale topology. In this paper, we first extend Kiehl's theory of coherent sheaves on rigid analytic spaces to a theory of pseudocoherent sheaves on adic spaces, then construct a corresponding theory of pseudocoherent (phi, Gamma)-modules. We then relate these objects to a more explicit construction in case the space comes equipped with a suitable infinite etale cover; in this case, one can decomplete the period sheaves and establish an analogue of the theorem of Cherbonnier-Colmez on the overconvergence of p-adic Galois representations. As an application, we show that relative (phi, Gamma)-modules in our sense coincide with the relative (phi, Gamma)-modules constructed by Andreatta and Brinon in the geometric setting where the latter can be constructed. As another application, we establish that the category of pseudocoherent (phi, Gamma)-modules on an arbitrary rigid analytic space over a p-adic field is abelian, satisfies the ascending chain condition, and is stable under various natural derived functors (including Hom, tensor product, and pullback). Applications to the etale cohomology of pro-etale local systems will be given in a subsequent paper.
We study the problem of whether a commutative nonarchimedean Banach ring which is algebraically a field can be topologized by a multiplicative norm. This can fail in general, but it holds for uniform Banach rings under some mild extra conditions. Notably, any perfectoid ring whose underlying ring is a field is a perfectoid field.
The ring of Witt vectors over a perfect valuation ring of characteristic p, often denoted A_inf, plays a pivotal role in p-adic Hodge theory; for instance, Bhatt, Morrow, and Scholze have recently reinterpreted and refined the crystalline comparison isomorphism by relating it to a certain A_inf-valued cohomology theory. We address some basic ring-theoretic questions about A_inf motivated by analogies with two-dimensional regular local rings. For example, we show that in most cases A_inf, which is manifestly not noetherian, is also not coherent. On the other hand, it does have the property that vector bundles over the complement of the closed point in Spec A_inf do extend uniquely over the puncture; moreover, a similar statement holds in Huber's category of adic spaces.
Given a formal flat meromorphic connection over an excellent scheme over a field of characteristic zero, in a previous paper we established existence of good formal structures and a good Deligne-Malgrange lattice after suitably blowing up. In this paper, we reinterpret and refine these results by introducing some related structures. We consider the turning locus, which is the set of points at which one cannot achieve a good formal structure without blowing up. We show that when the polar divisor has normal crossings, the turning locus is of pure codimension 1 within the polar divisor, and hence of pure codimension 2 within the full space; this had been previously established by Andre in the case of a smooth polar divisor. We also construct an irregularity sheaf and its associated b-divisor, which measure irregularity along divisors on blowups of the original space; this generalizes another result of Andre on the semicontinuity of irregularity in a curve fibration. One concrete consequence of these refinements is a process for resolution of turning points which is functorial with respect to regular morphisms of excellent schemes; this allows us to transfer the result from schemes to formal schemes, complex analytic varieties, and nonarchimedean analytic varieties.
We give an interim report on some improvements and generalizations of the Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time in $p$. These are illustrated with a number of examples including K3 surfaces, Calabi-Yau threefolds, and a cubic fourfold. The latter example is a non-special cubic fourfold appearing in the Ranestad-Voisin coplanar divisor on moduli space; this verifies that the coplanar divisor is not a Noether-Lefschetz divisor in the sense of Hassett.
For all odd primes N up to 500000, we compute the action of the Hecke operator T_2 on the space S_2(Gamma_0(N), Q) and determine whether or not the reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as eigenvalues. We then partially explain the results in terms of class field theory and modular mod-2 Galois representations. As a byproduct, we obtain some nonexistence results on elliptic curves and modular forms with certain mod-2 reductions, extending prior results of Setzer, Hadano, and Kida.
Let R be a perfect ring of characteristic p. We show that the group of continuous R-linear automorphisms of the perfect power series ring over R is generated by the automorphisms of the ordinary power series ring together with Frobenius; this answers a question of Jared Weinstein.
Let K be a complete discretely valued field of mixed characteristics (0, p) with perfect residue field. One of the central objects of study in p-adic Hodge theory is the category of continuous representations of the absolute Galois group of K on finite-dimensional Qp-vector spaces. In recent years, it has become clear that this category can be studied more effectively by embedding it into the larger category of (phi, Gamma)-modules; this larger category plays a role analogous to that played by the category of vector bundles on a compact Riemann surface in the Narasimhan-Seshadri theorem on unitary representations of the fundamental group of said surface. This category turns out to have a number of distinct natural descriptions, which on one hand suggests the naturality of the construction, but on the other hand forces one to use different descriptions for different applications. We provide several of these descriptions and indicate how to translate certain key constructions, which were originally given in the context of modules over power series rings, to the more modern context of perfectoid algebras and spaces.
We give a sharp divisibility bound, in terms of g, for the degree of the field extension required to realize the endomorphisms of an abelian variety of dimension g over an arbitrary number field; this refines a result of Silverberg. This follows from a stronger result giving the same bound for the order of the component group of the Sato-Tate group of the abelian variety, which had been proved for abelian surfaces by Fite-Kedlaya-Rotger-Sutherland. The proof uses Minkowski's reduction method, but with some care required in the extremal cases when p equals 2 or a Fermat prime.
We show that for k a perfect field of characteristic p, there exist endomorphisms of the completed algebraic closure of k((t)) which are not bijective. As a corollary, we resolve a question of Fargues and Fontaine by showing that for p a prime and C_p a completed algebraic closure of Q_p, there exist closed points of the Fargues-Fontaine curve associated to C_p whose residue fields are not (even abstractly) isomorphic to C_p as topological fields.
Let K be an algebraically closed field of characteristic p. We exhibit a counterexample against a theorem asserted in one of our earlier papers, which claims to characterize the integral closure of K((t)) within the field of Hahn-Mal'cev-Neumann generalized power series. We then give a corrected characterization, generalizing our earlier description in terms of finite automata in the case where K is the algebraic closure of a finite field. We also characterize the integral closure of K(t), thus generalizing a well-known theorem of Christol and suggesting a possible framework for computing in this integral closure. We recover various corollaries on the structure of algebraic generalized power series; one of these is an extension of Derksen's theorem on the zero sets of linear recurrent sequences in characteristic $p$.
We prove that the cohomology groups of an etale Q_p-local system on a smooth proper rigid analytic space are finite-dimensional Q_p-vector spaces, provided that the base field is either a finite extension of Q_p or an algebraically closed nonarchimedean field containing Q_p. This result manifests as a special case of a more general finiteness result for the higher direct images of a relative (phi, Gamma)-module along a smooth proper morphism of rigid analytic spaces over a mixed-characterstic nonarchimedean field.
We compute the complete set of candidates for the zeta function of a K3 surface over F_2 consistent with the Weil conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over F_2. These sets differ substantially, but we do identify natural subsets which coincide. This gives some numerical evidence towards a Honda-Tate theorem for transcendental zeta functions of K3 surfaces; such a result would refine a recent theorem of Taelman, in which one must allow an uncontrolled base field extension.
We make explicit Serre's generalization of the Sato-Tate conjecture for motives, by expressing the construction in terms of fiber functors from the motivic category of absolute Hodge cycles into a suitable category of Hodge structures of odd weight. This extends the case of abelian varietes, which we treated in a previous paper. That description was used by Fite--Kedlaya--Rotger--Sutherland to classify Sato-Tate groups of abelian surfaces; the present description is used by Fite--Kedlaya--Sutherland to make a similar classification for certain motives of weight 3. We also give conditions under which verification of the Sato-Tate conjecture reduces to the identity connected component of the corresponding Sato-Tate group.
Building on foundations introduced in a previous paper, we give several p-adic analytic descriptions of the categories of etale Zp-local systems and etale Qp-local systems on an affinoid algebra over a finite extension of Qp (or more generally, over the fraction field of the Witt vectors of a perfect field of characteristic p). These include generalizations of Fontaine's theory of (phi, Gamma)-modules, the refinement of Fontaine's construction introduced by Cherbonnier and Colmez, and a recent description by Fargues and Fontaine in terms of semistable vector bundles on a certain scheme. Our descriptions depend on the embedding of the associated affinoid space into an affine toric variety, but there are natural functoriality maps relating the constructions for different choices of the embedding; these may be used to give analogous descriptions over more general rigid or Berkovich analytic spaces.
We establish the group-theoretic classification of Sato-Tate groups of self-dual motives of weight 3 with rational coefficients and Hodge numbers h^{3,0} = h^{2,1} = h^{1,2} = h^{0,3} = 1. We then describe families of motives that realize some of these Sato-Tate groups, and provide numerical evidence supporting equidistribution. One of these families arises in the middle cohomology of certain Calabi-Yau threefolds appearing in the Dwork quintic pencil; for motives in this family, our evidence suggests that the Sato-Tate group is always equal to the full unitary symplectic group USp(4).
We revisit Huber's theory of continuous valuations, which give rise to the adic spectra used in his theory of adic spaces. We instead consider valuations which have been reified, i.e., whose value groups have been forced to contain the real numbers. This yields reified adic spectra which provide a framework for an analogue of Huber's theory compatible with Berkovich's construction of nonarchimedean analytic spaces. As an example, we extend the theory of perfectoid spaces to this setting.