Witt Vectors Research Articles

Lifting globally 𝐹-split surfaces to characteristic zero

Abstract We prove that every globally 𝐹-split surface admits an equisingular lifting over the ring of Witt vectors.

Dynamical systems for arithmetic schemes

Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vector rings to attach a new ringed space Wrat(X) to every scheme X. We also define R-valued points Wrat(X)(R) of Wrat(X) for every commutative ring R. For normal schemes X of finite type over specZ, using Wrat(X)(ℂ) we construct infinite dimensional R-dynamical systems whose periodic orbits are related to the closed points of X. Various aspects of these topological dynamical systems are studied. We also explain how certain p-adic points of Wrat(X) for X the spectrum of a p-adic local number ring are related to the points of the Fargues–Fontaine curve.

Spectral equivalence of smooth group schemes over principal ideal local rings

Let G be a smooth linear group scheme of finite type. For any positive integer k and a finite field F, let Wk(F) be the ring of Witt vectors of length k over F. We show that the group algebras of G(F[t]/(tk)) and G(Wk(F)) are isomorphic (i.e. the multi-sets of the dimensions of the irreducible representations are equal) for any positive integer k and finite field F with large enough characteristic. We also prove that if charF is large enough, then the cardinality of the set {dim⁡ρ|ρ∈irr(G(F))} is bounded uniformly in F.

Relative Fontaine–Messing Theory Over Power Series Rings

Abstract Let $k$ be a perfect field of characteristic $p>2$, $R := W(k)[\![t_{1}, \dots , t_{d}]\!]$ be the power series ring over the Witt vectors, and $X$ be a smooth proper scheme over $R$. The main goal of this article is to extend classical Fontaine–Messing theory [ 13] to the setting where the base ring is $R$. In particular, we obtain comparison theorems between torsion crystalline cohomology of $X/R$ and torsion étale cohomology in this setting.

Lifts of Supersingular Abelian Varieties With Small Mumford–Tate Groups

Abstract We investigate to what extent an abelian variety over a finite field can be lifted to one in characteristic zero with small Mumford–Tate group. We prove that supersingular abelian surfaces, respectively three-folds, can be lifted to ones isogenous to a square, respectively product, of elliptic curves. On the other hand, we show that supersingular abelian three-folds cannot be lifted to one isogenous to the cube of an elliptic curve over the Witt vectors.

On the strict Picard spectrum of commutative ring spectra

We compute the connective spectra of maps from $\mathbb {Z}$ to the Picard spectra of the spherical Witt vectors associated with perfect rings of characteristic $p$. As an application, we determine the connective spectrum of maps from $\mathbb {Z}$ to the Picard spectrum of the sphere spectrum.

Divisibility on point counting over finite Witt rings

Let Fq denote the finite field of q elements with characteristic p. Let Zq denote the unramified extension of the p-adic integers Zp with residue field Fq. In this paper, we investigate the q-divisibility for the number of solutions of a polynomial system in n variables over the finite Witt ring Zq/pmZq, where the n variables of the polynomials are restricted to run through a box lifting Fqn. It turns out that in general the answers do depend upon the box chosen. Based on the addition operation of Witt vectors, we prove a q-divisibility theorem for any box of low algebraic complexity, including the simplest Teichmüller box. This extends the classical Ax-Katz theorem over finite field Fq (the case m=1). Taking q=p to be a prime, our result extends and improves a recent theorem of Grynkiewicz for the unweighted case.

Witt Vectors and Separably Closed Fields with Higher Derivations

Witt Vectors and Separably Closed Fields with Higher Derivations

Canonical Witt formal scheme extensions and p-torsion groups

We study the nth arithmetic jet space of the p-torsion subgroup attached to a smooth commutative formal group scheme. We show that the nth jet space above fits in the middle of a canonical short exact sequence between a power of the formal scheme of Witt vectors of length n and the p-torsion subgroup we started with. This result generalizes a result of Buium on roots of unity.

Bogomolov-Sommese vanishing and liftability for surface pairs in positive characteristic

We show that the Bogomolov-Sommese vanishing theorem holds for a log canonical projective surface (X,B) in large characteristic unless the Iitaka dimension of KX+⌊B⌋ is not equal to two. As an application, we prove that a log resolution of a pair of a normal projective surface and a reduced divisor in large characteristic lifts to the ring of Witt vectors when the Iitaka dimension of the log canonical divisor is less than or equal to zero. Moreover, we give explicit and optimal bounds on the characteristic unless their Iitaka dimensions are equal to zero.

The integral geometric Satake equivalence in mixed characteristic

Let k k be an algebraically closed field of characteristic p p . Denote by W ( k ) W(k) the ring of Witt vectors of k k . Let F F denote a totally ramified finite extension of W ( k ) [ 1 / p ] W(k)[1/p] and O \mathcal {O} its ring of integers. For a connected reductive group scheme G G over O \mathcal {O} , we study the category P L + G ( G r G , Λ ) \mathrm {P}_{L^+G}(Gr_G,\Lambda ) of L + G L^+G -equivariant perverse sheaves in Λ \Lambda -coefficient on the Witt vector affine Grassmannian G r G Gr_G where Λ = Z ℓ \Lambda =\mathbb {Z}_{\ell } and F ℓ ( ℓ ≠ p ) \mathbb {F}_{\ell } \ (\ell \ne p) , and prove that it is equivalent as a tensor category to the category of finitely generated Λ \Lambda -representations of the Langlands dual group of G G .

A universal Hochschild–Kostant–Rosenberg theorem

In this work we study the failure of the HKR theorem over rings of positive and mixed characteristic. For this we construct a filtered circle interpolating between the usual topological circle and a formal version of it. By mapping to schemes we produce this way an interpolation, realized in practice by the existence of a natural filtration, from Hochschild and (a filtered version of) cyclic homology to derived de Rham cohomology. In particular, we show that this recovers the filtration of Antieau and Bhatt-Morrow-Scholze. The construction of our filtered circle is based on the theory of affine stacks and affinization introduced by the third author, together with some facts about schemes of Witt vectors.

Integral $p$-adic Hodge filtrations in low dimension and ramification

Given an integral p -adic variety, we observe that if the integral Hodge–de Rham spectral sequence behaves nicely, then the special fiber knows the Hodge numbers of the generic fiber. Applying recent advancements of integral p -adic Hodge theory, we show that such a nice behavior is guaranteed if the p -adic variety can be lifted to an analogue of second Witt vectors and satisfies some bound on dimension and ramification index. This is a (ramified) mixed characteristic analogue of results due to Deligne–Illusie, Fontaine–Messing, and Kato. Lastly, we discuss an example illustrating the necessity of the aforementioned lifting condition, which is of independent interest.

Automata and finite order elements in the Nottingham group

The Nottingham group at 2 is the group of (formal) power series t+a2t2+a3t3+⋯ in the variable t with coefficients ai from the field with two elements, where the group operation is given by composition of power series. The depth of such a series is the largest d⩾1 for which a2=…=ad=0.Only a handful of power series of finite order (forcedly a power of 2) are explicitly known through a formula for their coefficients. We argue in this paper that it is advantageous to describe such series in closed computational form through automata, based on effective versions of proofs of Christol's theorem identifying algebraic and automatic series.Up to conjugation, there are only finitely many series σ of order 2n with fixed break sequence (i.e. the sequence of depths of σ∘2i). Starting from Witt vector or Carlitz module constructions, we give an explicit automaton-theoretic description of: (a) representatives up to conjugation for all series of order 4 with break sequence (1,m) for m<10; (b) representatives up to conjugation for all series of order 8 with minimal break sequence (1,3,11); and (c) an embedding of the Klein four-group into the Nottingham group at 2.We study the complexity of the new examples from the algebro-geometric properties of the equations they satisfy. For this, we generalise the theory of sparseness of power series to a four-step hierarchy of complexity, for which we give both Galois-theoretic and combinatorial descriptions. We identify where our different series fit into this hierarchy. We construct sparse representatives for the conjugacy class of elements of order two and depth 2μ±1(μ⩾1). Series with small state complexity can end up high in the hierarchy. This is true, for example, for a new automaton we found, representing a series of order 4 with 5 states (the minimal possible number for such a series).

Pathologies and liftability of Du Val del Pezzo surfaces in positive characteristic

In this paper, we study pathologies of Du Val del Pezzo surfaces defined over an algebraically closed field of positive characteristic by relating them to their non-liftability to the ring of Witt vectors. More precisely, we investigate the condition (NB): all the anti-canonical divisors are singular, (ND): there are no Du Val del Pezzo surfaces over the field of complex numbers with the same Dynkin type, Picard rank, and anti-canonical degree, (NK): there exists an ample \(\mathbb {Z}\)-divisor which violates the Kodaira vanishing theorem for \(\mathbb {Z}\)-divisors, and (NL): the pair (Y, E) does not lift to the ring of Witt vectors, where Y is the minimal resolution and E is its reduced exceptional divisor. As a result, for each of these conditions, we determine all the Du Val del Pezzo surfaces which satisfy the given one.

Witt vectors with coefficients and characteristic polynomials over non-commutative rings

AbstractFor a not-necessarily commutative ring $R$ we define an abelian group $W(R;M)$ of Witt vectors with coefficients in an $R$-bimodule $M$. These groups generalize the usual big Witt vectors of commutative rings and we prove that they have analogous formal properties and structure. One main result is that $W(R) := W(R;R)$ is Morita invariant in $R$. For an $R$-linear endomorphism $f$ of a finitely generated projective $R$-module we define a characteristic element $\chi _f \in W(R)$. This element is a non-commutative analogue of the classical characteristic polynomial and we show that it has similar properties. The assignment $f \mapsto \chi _f$ induces an isomorphism between a suitable completion of cyclic $K$-theory $K_0^{\mathrm {cyc}}(R)$ and $W(R)$.

Extending torsors on the punctured Spec(A inf)

Abstract We prove that torsors under parahoric group schemes on the punctured spectrum of Fontaine’s ring A inf {A_{\mathrm{inf}}} , extend to the whole spectrum. Using descent we can extend a similar result for the ring 𝔖 {\mathfrak{S}} of Kisin and Pappas to full generality. Moreover, we treat similarly the case of equal characteristic. As applications we extend results of Ivanov on exactness of the loop functor and present the construction of a canonical specialization map from the B dR + {B^{+}_{\mathrm{dR}}} -affine Grassmannian to the Witt vector affine flag variety.

Witt Vectors, Polynomial Maps, and Real Topological Hochschild Homology

We show that various flavors of Witt vectors are functorial with respect to multiplicative polynomial laws of finite degree. We then deduce that the $p$-typical Witt vectors are functorial in multiplicative polynomial maps of degree at most $p-1$. This extra functoriality allows us to extend the $p$-typical Witt vectors functor from commutative rings to $\mathbb{Z}/2$-Tambara functors, for odd primes $p$. We use these Witt vectors for Tambara functors to describe the components of the dihedral fixed-points of the real topological Hochschild homology spectrum at odd primes.

Serre–Tate theory for Calabi–Yau varieties

Abstract Classical Serre–Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the base of its universal deformation with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo p 2 {p^{2}} of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two. If the variety has unobstructed deformations and bijective first higher Hasse–Witt operation, the Frobenius lifting gives rise to canonical coordinates. One of the key features of our liftings is that the crystalline Frobenius preserves the Hodge filtration. We also extend Nygaard’s approach from K3 surfaces to higher dimensions, and show that no non-trivial families of such varieties exist over simply connected bases with no global one-forms.

Equivariant splitting of the Hodge\u2013de Rham exact sequence

Let X be an algebraic curve with a faithful action of a finite group G over a field k. We show that if the Hodge–de Rham short exact sequence of X splits G-equivariantly then the action of G on X is weakly ramified. In particular, this generalizes the result of Köck and Tait for hyperelliptic curves. We discuss also converse statements and tie this problem to lifting coverings of curves to the ring of Witt vectors of length 2.