Discrete Valuation Field Research Articles

On 𝑝-adic Simpson and Riemann–Hilbert correspondences in the imperfect residue field case

Let K K be a mixed characteristic complete discrete valuation field with residue field admitting a finite p p -basis, and let G K G_K be the Galois group. Inspired by Liu and Zhu’s construction of p p -adic Simpson and Riemann–Hilbert correspondences over rigid analytic varieties, we construct such correspondences for representations of G K G_K . As an application, we prove a Hodge–Tate (resp. de Rham) “rigidity” theorem for p p -adic representations of G K G_K , generalizing a result of Morita.

Common valuations of division polynomials

In this note, we prove a formula for the cancellation exponent $k_{v,n}$ between division polynomials $\psi _n$ and $\phi _n$ associated with a sequence $\{nP\}_{n\in \mathbb {N}}$ of points on an elliptic curve $E$ defined over a discrete valuation field $K$ . The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.

Graded quotients of ramification groups of local fields with imperfect residue fields

abstract: We prove that the graded quotients of the filtration by ramification groups of any henselian discrete valuation field of residue characteristic $p>0$ are ${\bf F}_p$-vector spaces. We define an injection of the character group of each graded quotient to a twisted cotangent space defined using a cotangent complex.

Stable lattices in p-adic representations III. Indecomposable reduction and holonomy

Let G be a group, V a finite dimensional vector space over a complete discrete valuation field K, and ρ:G⟶GLK(V) an irreducible representation with bounded image (in the metric topology). First, we show that there are many stable lattices with indecomposable reduction, in the sense that their simplicial convex hull in the Bruhat-Tits building of GLK(V) is the set S(ρ) of stable norms.Then we investigate the reductions more closely, in case K has finite residue field and the simplicial set S(ρ) has dimension 2. When the reduction type (defined below) is (2,1), we show that there exists a stable lattice with indecomposable reduction that has at most 2 composition series (uniserial or biserial). When the reduction type is (3), we show that the number of v∈S(ρ)0 with semisimple reduction is bounded above by a constant times |S(ρ)0|.To prove these last two statements, we prove and apply an index theorem (of Gauss-Bonnet type).

Local-global principle for projective homogeneous spaces of unitary groups

Let K be a complete discrete valued field with residue field k and F the function field of a curve over K. Let A∈2Br(F) be a central simple algebra with an involution σ of any kind and F0=Fσ. Let h be an hermitian space over (A,σ) and G=SU(A,σ,h) if σ is of first kind and G=U(A,σ,h) if σ is of second kind. Suppose that char(k)≠2 and ind(A)≤4. Then we prove that projective homogeneous spaces under G over F0 satisfy a local-global principle for rational points with respect to discrete valuations of F.

Ramification filtration and differential forms

Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $\Gamma_{L}^{(v)}$ the ramification subgroups of $\Gamma_{L}=\operatorname{Gal}(L^{\mathrm{sep}}/L)$. We consider the category $\operatorname{M\Gamma}_{L}^{\mathrm{Lie}}$ of finite $\mathbb{Z}_p[\Gamma_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $\Gamma_L$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups $\Gamma_L^{(v)}$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$ can be explicitly extracted from some differential forms $\widetilde{\Omega} [N]$ on the Fontaine etale $\phi $-module $M(H)$ associated with $H$. The forms $\widetilde{\Omega}[N]$ are completely determined by a canonical connection $\nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic, which contain a primitive $p$th root of unity, we show that a similar problem for $\mathbb{F}_p[\Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding $\phi $-module together with the action of the Galois group of a cyclic extension $L_1$ of $L$ of degree $p$. Then our solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of $\operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the $p$-adic period of the formal group $\mathbb{G}_m$.

Ramification filtration and differential forms

Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $\Gamma_{L}^{(v)}$ the ramification subgroups of $\Gamma_{L}=\operatorname{Gal}(L^{\mathrm{sep}}/L)$. We consider the category $\operatorname{M\Gamma}_{L}^{\mathrm{Lie}}$ of finite $\mathbb{Z}_p[\Gamma_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $\Gamma_L$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups $\Gamma_L^{(v)}$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$ can be explicitly extracted from some differential forms $\widetilde{\Omega} [N]$ on the Fontaine etale $\phi $-module $M(H)$ associated with $H$. The forms $\widetilde{\Omega}[N]$ are completely determined by a canonical connection $\nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic, which contain a primitive $p$th root of unity, we show that a similar problem for $\mathbb{F}_p[\Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding $\phi $-module together with the action of the Galois group of a cyclic extension $L_1$ of $L$ of degree $p$. Then our solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of $\operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the $p$-adic period of the formal group $\mathbb{G}_m$. Bibliography: 21 titles.

Duality for local fields and sheaves on the category of fields

Duality for complete discrete valuation fields with perfect residue field with coefficients in (possibly p-torsion) finite flat group schemes was obtained by Begueri, Bester and Kato. In this paper, we give another formulation and proof of this result. We use the category of fields and a Grothendieck topology on it. This simplifies the formulation and proof and reduces the duality to classical results on Galois cohomology. A key point is that the resulting site correctly captures extension groups between algebraic groups.

Internal indecomposability of profinite groups

It is well-known that various profinite groups appearing in anabelian geometry — e.g., the absolute Galois groups of p-adic local fields or number fields — satisfy distinctive group-theoretic properties such as slimness [i.e., the property that every open subgroup is center-free] and strong indecomposability [i.e., the property that every open subgroup has no nontrivial product decomposition]. In the present paper, we consider another group-theoretic property on profinite groups, which we shall refer to as strong internal indecomposability. This is a stronger property than both slimness and strong indecomposability. In the present paper, we examine basic properties of strong internal indecomposability and prove that the absolute Galois groups of Henselian discrete valuation fields with positive characteristic residue fields or Hilbertian fields [which may be regarded as generalizations of p-adic local fields or number fields] satisfy strong internal indecomposability.

On the Brauer p-dimension of Henselian discrete valued fields of residual characteristic p > 0

Let $(K, v)$ be a Henselian discrete valued field with residue field $\widehat K$ of characteristic $p$, and Brd$_{p}(K)$ be the Brauer $p$-dimension of $K$. This paper shows that Brd$_{p}(K) \ge n$, if $[\widehat K\colon \widehat K ^{p}] = p ^{n}$, for some $n \in \mathbb{N}$. It proves that Brd$_{p}(K) = \infty $ if and only if $[\widehat K\colon \widehat K ^{p}] = \infty $.

Kurihara Invariants and Elimination of Wild Ramification

The paper continues a series of papers devoted to study of the connection between two approaches to classification of complete discrete valuation fields with imperfect residue fields and, in particular, 2-dimensional local fields in the case of mixed characteristic. One of these approaches was introduced by Masato Kurihara (1987) in terms of the module of differentials. Another one is based on Epp’s theory of elimination of wild ramification. A lower bound for the degree of a constant field extension that makes a given field into an almost standard one is established. This bound is expressed in terms of the invariant introduced in Kurihara’s paper.

Is there an infinite field whose multiplicative group is indecomposable?

In [2], we determined the finite fields with indecomposable multiplicative groups and conjectured that there is no infinite field whose multiplicative group is indecomposable. In this paper, we prove this conjecture for several popular classes of fields, including finitely generated fields, discrete valued fields, fields of Hahn series, local fields, global fields, and function fields.

Anabelian group-theoretic properties of the absolute Galois groups of discrete valuation fields

Let K be a field. Write GK for the absolute Galois group of K. In the present paper, we discuss the slimness [i.e., the property that every open subgroup is center-free] and the elasticity [i.e., the property that every nontrivial topologically finitely generated normal closed subgroup of an open subgroup is open] of GK. These two group-theoretic properties are closely related to [various versions of] the Grothendieck Conjecture in anabelian geometry. For instance, with regard to the slimness, Mochizuki proved that GK is slim if K is a subfield of a finitely generated extension of the field of fractions of the Witt ring W(F‾p) as a consequence of a [highly nontrivial] Grothendieck Conjecture-type result. In the present paper, we generalize this result to the case where K is a subfield of the field of fractions of an arbitrary mixed characteristic Noetherian local domain. Our proof is based on elementary field theories such as Kummer theory. On the other hand, with regard to the elasticity, Mochizuki proved that GK is elastic if K is a finite extension of the field of p-adic numbers. In the present paper, we generalize this result to the case where K is an arbitrary mixed characteristic Henselian discrete valuation field. As a corollary of this generalization, we prove the semi-absoluteness of isomorphisms between the étale fundamental groups of smooth varieties over mixed characteristic Henselian discrete valuation fields. Moreover, we also prove the weak version of the Grothendieck Conjecture for hyperbolic curves of genus 0 over subfields of finitely generated extensions of mixed characteristic higher local fields.

Henselian discrete valued stable fields

Let $(K, v)$ be a Henselian discrete valued field with residue field $\widehat K$ of characteristic $q \ge 0$, and Brd$_{p}(K)$ be the Brauer $p$-dimension of $K$, for each prime $p$. The present paper shows that if $p = q$, then Brd$_{p}(K) \le 1$ if and only if $\widehat K$ is a $p$-quasilocal field and the degree $[\widehat K\colon \widehat K ^{p}]$ is $\le p$. This complements our earlier result that, in case $p \neq q$, we have Brd$_{p}(K) \le 1$ if and only if $\widehat K$ is $p$-quasilocal and Brd$_{p}(\widehat K) \le 1$.

Reduction type of smooth plane quartics

Let $C/K$ be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e. smooth plane quartic, hyperelliptic genus 3 curve or bad) over $K$ in terms of the existence of a special plane quartic model and, over $\bar{K}$, in terms of the valuations of certain algebraic invariants of $C$ when the characteristic of the residue field is not $2,\,3,\,5$ or $7$. On the way, we gather several results of general interest on geometric invariant theory over an arbitrary ring $R$ in the spirit of (Seshadri 1977). For instance when $R$ is a discrete valuation ring, we show the existence of a homogeneous system of parameters over $R$. We exhibit explicit ones for ternary quartic forms under the action of $\textrm{SL}_{3,R}$ depending only on the characteristic $p$ of the residue field. We illustrate our results with the case of Picard curves for which we give simple criteria for the type of reduction.

Fields of dimension one algebraic over a global or local field need not be of type C1

Let (K,v) be a Henselian discrete valued field with a quasifinite residue field. This paper proves the existence of an algebraic extension E/K satisfying the following: (i) E has dimension dim(E)≤1, i.e. the Brauer group Br(E′) is trivial, for every algebraic extension E′/E; (ii) finite extensions of E are not C1-fields. This, applied to the maximal algebraic extension K of the field Q of rational numbers in the field Qp of p-adic numbers, for a given prime p, proves the existence of an algebraic extension Ep/Q, such that dim(Ep)≤1, Ep is not a C1-field, and Ep has a Henselian valuation of residual characteristic p.

RECIPROCITY SHEAVES AND THEIR RAMIFICATION FILTRATIONS

AbstractWe define a motivic conductor for any presheaf with transfersFusing the categorical framework developed for the theory of motives with modulus by Kahn, Miyazaki, Saito and Yamazaki. IfFis a reciprocity sheaf, this conductor yields an increasing and exhaustive filtration on$F(L)$, whereLis any henselian discrete valuation field of geometric type over the perfect ground field. We show that ifFis a smooth group scheme, then the motivic conductor extends the Rosenlicht–Serre conductor; ifFassigns toXthe group of finite characters on the abelianised étale fundamental group ofX, then the motivic conductor agrees with the Artin conductor defined by Kato and Matsuda; and ifFassigns toXthe group of integrable rank$1$connections (in characteristic$0$), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf, the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type withperfectresidue field can be uniquely extended to all such fields without any restriction on the residue field. For example, the Kato–Matsuda Artin conductor is characterised as the canonical extension of the classical Artin conductor defined in the case of a perfect residue field.

Ramification of wild automorphisms of Laurent series fields

Let K K be a complete discrete valuation field with residue class field k k , where both are of positive characteristic p p . Then the group of wild automorphisms of K K can be identified with the group under composition of formal power series over k k with no constant term and X X -coefficient 1 1 . Under the hypothesis that p > b 2 p > b^2 , we compute the first nontrivial coefficient of the p p th iterate of a power series over k k of the form f = X + ∑ i ≥ 1 a i X b + i f = X + \sum _{i \geq 1} a_iX^{b+i} . As a result, we obtain a necessary and sufficient condition for an automorphism to be “ b b -ramified”, having lower ramification numbers of the form i n ( f ) = b ( 1 + ⋯ + p n ) i_n(f) = b(1 + \cdots + p^n) . This is a vast generalization of Nordqvist’s 2017 theorem on 2 2 -ramified power series, as well as the analogous result for minimally ramified power series which proved to be useful for arithmetic dynamics in a 2013 paper of Lindahl on linearization discs in C p \mathbf {C}_p and a 2015 result of Lindahl–Rivera-Letelier on optimal cycles over nonarchimedean fields of positive residue characteristic. The success of our computation is also promising progress towards a generalization of Lindahl–Nordqvist’s 2018 theorem bounding the norm of periodic points of 2 2 -ramified power series.

Extending tamely ramified strict 1-motives into két log 1-motives

AbstractWe define két abelian schemes, két 1-motives and két log 1-motives and formulate duality theory for these objects. Then we show that tamely ramified strict 1-motives over a discrete valuation field can be extended uniquely to két log 1-motives over the corresponding discrete valuation ring. As an application, we present a proof to a result of Kato stated in [12, §4.3] without proof. To a tamely ramified strict 1-motive over a discrete valuation field, we associate a monodromy pairing and compare it with Raynaud’s geometric monodromy.

On the Rost divisibility of henselian discrete valuation fields of cohomological dimension 3

Let $F$ be a field, $\ell$ a prime and $D$ a central division $F$-algebra of $\ell$-power degree. By the Rost kernel of $D$ we mean the subgroup of $F^*$ consisting of elements $\lambda$ such that the cohomology class $(D)\cup (\lambda)\in H^3(F,\,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}(2))$ vanishes. In 1985, Suslin conjectured that the Rost kernel is generated by $i$-th powers of reduced norms from $D^{\otimes i},\,\forall i\ge 1$. Despite of known counterexamples, we prove some new cases of Suslin's conjecture. We assume $F$ is a henselian discrete valuation field with residue field $k$ of characteristic different from $\ell$. When $D$ has period $\ell$, we show that Suslin's conjecture holds if either $k$ is a $2$-local field or the cohomological $\ell$-dimension $\mathrm{cd}_{\ell}(k)$ of $k$ is $\le 2$. When the period is arbitrary, we prove the same result when $k$ itself is a henselian discrete valuation field with $\mathrm{cd}_{\ell}(k)\le 2$. In the case $\ell=\text{char}(k)$ an analog is obtained for tamely ramified algebras. We conjecture that Suslin's conjecture holds for all fields of cohomological dimension 3.