Complete Valued Field Research Articles

A Result of Krasner in Categorial Form

In 1957, M. Krasner described a complete valued field (K,v) as the inverse limit of a system of certain structures, called hyperfields, associated with (K,v). We put this result in purely category-theoretic terms by translating it into a limit construction in certain slice categories of the category of valued hyperfields and their homomorphisms. We replace the original metric-dependent arguments employed by Krasner with a clean and elegant transition to certain slice categories.

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.

Dual finite frames for vector spaces over an arbitrary field with applications

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry

Let L be an ample line bundle on a (geometrically reduced) projective variety X over any complete valued field. Our main result describes the leading asymptotics of the determinant of cohomology of large powers of L, with respect to the supnorm of a continuous metric on the Berkovich analytification of L. As a consequence, we establish in this setting the existence of transfinite diameters and equidistribution of Fekete points, following a strategy going back to Berman, Witt Nyström and the first author for complex manifolds. In the non-Archimedean case, our approach relies on a version of the Knudsen–Mumford expansion for the determinant of cohomology on models over the (possibly non-Noetherian) valuation ring, as a replacement for the asymptotic expansion of Bergman kernels in the complex case, and on the reduced fiber theorem, as a replacement for the Bernstein–Markov inequalities. Along the way, a systematic study of spaces of norms and the associated Fubini–Study type metrics is undertaken.

On the structure of certain valued fields

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields K1 and K2 of mixed characteristic with perfect residue fields, we show that if the n-th residue rings are isomorphic for each n≥1, then K1 and K2 are isometric and isomorphic. More generally, for n1≥1, there is n2 depending only on the ramification indices of K1 and K2 such that any homomorphism from the n1-th residue ring of K1 to the n2-th residue ring of K2 can be lifted to a homomorphism between the valuation rings. Moreover, we get a functor from the category of certain principal Artinian local rings of length n to the category of certain complete discrete valuation rings of mixed characteristic with perfect residue fields, which naturally generalizes the functorial property of unramified complete discrete valuation rings. Our lifting result improves Basarab's relative completeness theorem for finitely ramified henselian valued fields, which solves a question posed by Basarab, in the case of perfect residue fields.

A note on C0-groups and C-groups on non-archimedean Banach spaces

In this paper, we introduce new classes of linear operators so called [Formula: see text]-groups, [Formula: see text]-groups and cosine families of bounded linear operators on non-archimedean Banach spaces over non-archimedean complete valued field [Formula: see text]. We show some results about it.

Notions of Stein spaces in non-Archimedean geometry

Let k k be a non-Archimedean complete valued field and let X X be a k k -analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: (1) for every complete valued extension k ′ k’ of k k , every coherent sheaf on X × k k ′ X \times _{k} k’ is acyclic; (2) X X is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex), and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, X X is compact); (3) X X admits a suitable exhaustion by compact analytic domains considered by Liu in his counter-example to the cohomological criterion for affinoidicity. When X X has no boundary the characterization is simpler: in (2) the vanishing of higher cohomology groups of the structure sheaf is no longer needed, so that we recover the usual notion of Stein space in complex geometry; in (3) the domains considered by Liu can be replaced by affinoid domains, which leads us back to Kiehl’s definition of Stein space.

A Survey and New Results on Banach Algebras of Ultrametric Continuous Functions

Let $${ \rm I\!K}$$ be an ultrametric complete valued field and $${ \rm I\!E}$$ be an ultrametric space. We examine some Banach algebras $$S$$ of bounded continuous functions from $${ \rm I\!E}$$ to $${ \rm I\!K}$$ with the use of ultrafilters, particularly the relation of stickness. We recall and deepen results obtained in a previous paper by N. Mainetti and the third author concerning the whole algebra $$\cal A$$ of all bounded continuous functions from $${ \rm I\!E}$$ to $${ \rm I\!K}$$ . Every maximal ideal of finite codimension of $$\cal A$$ is of codimension $$1$$ and we show that this property also holds for every algebra $$S$$ , provided $${ \rm I\!K}$$ is perfect. If $$S$$ admits the uniform norm on $${ \rm I\!E}$$ as its spectral norm, then every maximal ideal is the kernel of only one multiplicative semi-norm, the Shilov boundary is equal to the whole multiplicative spectrum and the Banaschewski compactification of $${ \rm I\!E}$$ is homeomorphic to the multiplicative spectrum of $$S$$ .

Hyperfields, truncated DVRs, and valued fields

For any two complete discrete valued fields K1 and K2 of mixed characteristic with perfect residue fields, we show that if the n-th valued hyperfields of K1 and K2 are isomorphic over p for each n≥1, then K1 and K2 are isomorphic. More generally, for n1,n2≥1, if n2 is large enough, then any homomorphism, which is over p, from the n1-th valued hyperfield of K1 to the n2-th valued hyperfield of K2 can be lifted to a homomorphism from K1 to K2. We compute such n2 effectively, which depends only on the ramification indices of K1 and K2. Moreover, if K1 is tamely ramified, then any homomorphism over p between the first valued hyperfields is induced from a unique homomorphism of valued fields. Using this lifting result, we deduce a relative completeness theorem of AKE-style in terms of valued hyperfields.We also study some relationships between valued hyperfields, truncated discrete valuation rings, and complete discrete valued fields of mixed characteristic. Specially, in the tamely ramified case, we show that a category of valued hyperfields is equivalent to the category of complete discrete valued rings of mixed characteristic (0,p) having perfect residue fields.

Berkovich spectra of elements in Banach rings

Extending the notion of the spectrum Σf of an element f in an ultrametric Banach algebra over a complete valued field (as defined by Berkovich), we introduce and briefly study the Berkovich spectrum σR,SBer(a) of an element a in a unital Banach algebra R over a commutative unital Banach ring S (in particular, when R is a Banach ring). This spectrum is a compact subset of the affine analytic line AS1 over S, and the latter can be identified with the suitable equivalence classes of all elements in all complete valued fields that are Banach S-algebras. If R is generated by a as a unital Banach S-algebra, then σR,SBer(a) coincides with the spectrum M(R) of R. If R is a unital complex Banach algebra but we regard it as a Banach ring only, then σR,Z1Ber(a) is the quotient of the ordinary spectrum σRC(a) by complex conjugation.For a non-Archimedean complete valued field k and an infinite dimensional ultrametric Banach k-space E with an orthogonal basis, if u∈L(E) is a completely continuous operator, we show that several different ways to define the spectrum of u give the same compact set σL(E),kBer(u). As an application, we give a relation between σL(E),kBer(u) and zeros of the Fredholm determinant det⁡(1−tu) (as defined by Serre) in complete valued field extensions of k.

Norms and gauges on Clifford Algebra

ABSTRACTFrom a normed quadratic space (V,q), we construct a norm on the Clifford algebra C(V,q). We describe the associated graded form of this norm and give a condition for this norm to be a gauge. Then, we apply our results to prove that for a complete discrete valued field, an anisotropic quadratic form q with dimq = 0 mod 8 and nonsplit Clifford algebra cannot be at the same time a transfer of a K-hermitian form with K∕F an inertial quadratic field extension and a transfer of a T-hermitian form with T∕F a ramified quadratic field extension.

Almost algebraic actions of algebraic groups and applications to algebraic representations

Let G be an algebraic group over a complete separable valued field k . We discuss the dynamics of the G -action on spaces of probability measures on algebraic G -varieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis–Zimmer super-rigidity phenomenon [2].

On standard extensions of local fields

Let $L/K$ be any separable extension of complete discrete valued fields of degree $p$. This work, is a study of some “standard over-extensions” of $L/K$, with the description of their Galois groups. The second target, which is the aim of this work, concerns the Galois closure of $L/K$. The study of the normal case has been done in some former work.

Lifting Divisors on a Generic Chain of Loops

AbstractLet C be a curve over a complete valued field having an infinite residue field and whose skeleton is a chain of loops with generic edge lengths. We prove that any divisor on the chain of loops that is rational over the value group lifts to a divisor of the same rank on C, confirming a conjecture of Cools, Draisma, Robeva, and the third author.

The convergence Newton polygon of a p-adic differential equation II: Continuity and finiteness on Berkovich curves

We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Relying on work of the second author who investigated its properties on affinoid domains of the affine line, we prove that its slopes give rise to continuous functions that factorize by the retraction through a locally finite subgraph of the curve.

Some subgroups of the general linear group of order two over the ring of p-adic integers

Let p be a prime number, ℤ p the ring of p-adic integers and ℚ p its field of fractions. In this paper we are mainly concerned with the profinite structure of the general linear group GL(2, ℤ p ) and some of its subgroups. In particular, we study the subgroup SL(2, ℤ p ), the special linear group with p-adic integer entries. We determine the p-adic completion of the general linear group GL(2, ℤ)with integer entries. In contrastwith the padic completion of SL(2, ℤ) which is equal to SL(2, ℤ p ) that of GL(2, ℤ) is far to be equal to GL(2, ℤ p ). It is the semi-direct product {−1, 1} ⋉ SL(2, ℤ p ) and is a normal subgroup of GL(2, ℤ p ). It is well known that the special linear group SL(2, ℤ) is generated by two transvections. Hence SL(2, ℤ p ) is topological generated by these transvections. Moreover since ℤ p is a local ring, one can apply O. Litoff’s theorem to get short decomposition of any element of SL(2, ℤ p ). We revisit here the algebraic and profinite structures of these groups. The explicit description of these structures will be applied elsewhere to their continuous linear representations in ultrametric Banach spaces. In particular from the profinite structure of the general linear group as well as the special linear group, one can state when there exists or not a Haar measure on these groups with values in a given complete valued field. As already said, the special linear group SL(2, ℤ p ) has two topological generators, we shall prove that the general linear group GL(2, ℤ p ) admits four topological generators. As a first consequence we deduce here that any subgroup of these groups that is of finite index is open. On the other hand, the fact that these groups are finitely generated profinite groups will be applied elsewhere to their ultrametric continuous linear representations.

Sums of squares in algebraic function fields over a complete discretely valued field

A recently found local-global principle for quadratic forms over function fields of curves over a complete discrete valued field is applied to the study of quadratic forms, sums of squares, and related field invariants.

Sous-groupes de [formula omitted] et arbres

In this paper, we first characterize the subsets of the Bruhat–Tits tree of PGL2(K), K a complete valued field, that are the sets of fixed points C(G) of a subgroup G of GL2(K). When G is irreducible, the C(G) are the “strips” in the tree. We then evaluate the form of the strip C(G), in particular in terms of algebraic invariants of the group G. We give two applications to representation theory, one about a new generalization of Ribet's lemma on extensions, the other about the trace-convergence of a sequence of representations.

On the topology on group cohomology of algebraic groups over complete valued fields

We introduce some topologies on the group cohomology of algebraic groups over complete valued fields and consider some applications.

Continuity and finiteness of the radius of convergence of a p-adic differential equation via potential theory

Abstract We study the radius of convergence of a differential equation on a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Several properties of this function are known: F. Baldassarri proved that it is continuous (see [Invent. Math. 182 (2010), no. 3, 513–584]) and the authors showed that it factorizes by the retraction through a locally finite graph (see [`The convergence Newton polygon of a p-adic differential equation I: Affinoid domains of the Berkovich affine line', preprint 2012] and [`The convergence Newton polygon of a p-adic differential equation II: Continuity and finiteness on Berkovich curves', preprint 2012]). Here, assuming that the curve has no boundary or that the differential equation is overconvergent, we provide a shorter proof of both results by using potential theory on Berkovich curves.