Complete non-Archimedean Valued Field Research Articles

(Looking for) the heart of abelian Polish groups

We prove that the category M of abelian groups with a Polish cover introduced in collaboration with Bergfalk and Panagiotopoulos is the left heart of (the derived category of) the quasi-abelian category A of abelian Polish groups in the sense of Beilinson–Bernstein–Deligne and Schneiders. Thus, M is an abelian category containing A as a full subcategory such that the inclusion functor A→M is exact and finitely continuous. Furthermore, M is uniquely characterized up to equivalence by the following universal property: for every abelian category B, a functor A→B is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor M→B. In particular, this provides a description of the left heart of A as a concrete category.We provide similar descriptions of the left heart of a number of categories of algebraic structures endowed with a topology, including: non-Archimedean abelian Polish groups; locally compact abelian Polish groups; totally disconnected locally compact abelian Polish groups; Polish R-modules, for a given Polish group or Polish ring R; and separable Banach spaces and separable Fréchet spaces over a separable complete non-Archimedean valued field.

Some integrals for groups of bounded linear operators on finite-dimensional non-Archimedean Banach spaces

In this paper, we extend the Volkenborn integral and Shnirelman integral for groups of bounded linear operators on finite-dimensional non-Archimedean Banach spaces over $\mathbb{Q}_{p}$ and $\mathbb{C}_{p}$ respectively. When the ground field is a complete non-Archimedean valued field, which is also algebraically closed, we give some functional calculus for groups of infinitesimal generator $A$ such that $A$ is a nilpotent operator on finite-dimensional non-Archimedean Banach spaces.

On the analyticity of WLUD∞ functions of one variable and WLUD∞ functions of several variables in a complete non-Archimedean valued field

AbstractLet $\mathcal {N}$ be a non-Archimedean-ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD$^{k}$) functions and WLUD$^{\infty }$ functions at a point or on an open subset of $\mathcal {N}$. Then, we show under what conditions a WLUD$^{\infty }$ function at a point $x_0\in \mathcal {N}$ is analytic in an interval around $x_0$, that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD$^{k}$ and WLUD$^{\infty }$ to functions from $\mathcal {N}^{n}$ to $\mathcal {N}$, for any $n\in \mathbb {N}$. Then, we formulate conditions under which a WLUD$^{\infty }$ function at a point $\boldsymbol {x_0} \in \mathcal {N}^{n}$ is analytic at that point.

Geometry of hyperfields

Given a scheme X over Z and a hyperfield H which is equipped with a topology which satisfies certain conditions, we endow the set X(H) of H-rational points with a natural topology. We then prove that; (1) when H is the Krasner hyperfield, X(H) is homeomorphic to the underlying space of X, (2) when H is the tropical hyperfield and X is of finite type over a complete non-Archimedean valued field k, X(H) is homeomorphic to the underlying space of the Berkovich analytification Xan of X, and (3) when H is the hyperfield of signs, X(H) is homeomorphic to the underlying space of the real scheme Xr associated with X.

Invariant subspace problem and compact operators on non-Archimedean Banach spaces

In this paper, the invariant Subspace Problem is studied for the class of non-Archimedean compact operators on an infinite-dimensional Banach space E over a nontrivial complete non-Archimedean valued field K. Our first main result (Theorem 9) asserts that if K is locally compact, then each compact operator on E possessing a quasi null vector admits a nontrivial hyperinvariant closed subspace. In the second one (Theorem 17), we prove that each bounded operator on E which contains a cyclic quasi null vector can be written as the sum of a triangular operator and a compact shift operator, each one of them possesses a nontrivial invariant closed subspace. Finally, we conclude that if K is algebraically closed, then every compact operator on E either has a nontrivial invariant closed subspace or is a sum of upper triangular operator and shift operator, each of them is compact and has a nontrivial invariant closed subspace.

Extension property of semipositive invertible sheaves over a non-archimedean field

In this article, we prove an extension property of semipositively metrized ample invertible sheaves on a projective scheme over a complete non-archimedean valued field. As an application, we establish a Nakai-Moishezon type criterion for adelically normed graded linear series.

Finite morphisms between projective varieties and Skeleta

In this paper we study finite morphisms between irreducible projective varieties in terms of the morphisms they induce between the respective analytifications. The background for the principal result is as follows. Let $V'$ and $V$ be irreducible, projective varieties over an algebraically closed, non- archimedean valued field $k$ and $\phi$ be a finite morphism $\phi : V' \to V$. Let $x \in V^{an}(L)$, where $L/k$ is an algebraically closed complete non-archimedean valued field extension. We associate canonically to $x$ an $L$-point of the space $(V \times_k L)^{an}$ which lies on the fiber over $x$ and denote this point $x_L$. The embedding of $V$ into some $n$-dimensional projective space defines in a natural way a family of open neighbourhoods $\mathcal{O}_{x_L}$ in $(V \times_k L)^{an}$ of $x_L$. Each element of this family is parametrized by an $(n + 1)^2$-tuple which quantifies its size. Of particular interest to us will be those elements $O$ of the set $\mathcal{O}_{x_L}$ whose preimage for the morphism $(\phi \times id_L)^{an}$ decomposes into the disjoint union of homeomorphic copies of $O$ via $(\phi \times id_L)^{an}$. Let $\mathcal{G}_{x_L} \subset \mathcal{O}_{x_L}$ denote the sub collection of elements of this form. Theorem 1.3 shows that there exists a deformation retraction of the space $V$ onto a finite simplicial complex such that along the fibers of the retraction the size of the largest element belonging to $\mathcal{G}_{x_L}$ is constant.

A combinatorial Li–Yau inequality and rational points on curves

We present a method to control gonality of nonarchimedean curves based on graph theory. Let $$k$$ denote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a curve over the algebraic closure of $$k$$ in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some “volume” of the original graph; this can be seen as a substitute for graphs of the Li–Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally, we apply the results to give a lower bound for the gonality of arbitrary Drinfeld modular curves over finite fields and for general congruence subgroups $$\varGamma $$ of $$\varGamma (1)$$ that is linear in the index $$[\varGamma (1):\varGamma ]$$ , with a constant that only depends on the residue field degree and the degree of the chosen “infinite” place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian.

On orthogonal properties of immediate extensions of [formula omitted

Let K be a non-spherically complete non-Archimedean valued field. We prove that there exist normed spaces over K for which every finite-dimensional linear subspace has an orthogonal base and which possess one-dimensional linear subspaces without orthogonal complements.

Examples of non-archimedean Fréchet spaces without nuclear Köthe quotients

Let K be a spherically complete non-archimedean valued field. We prove that the dual space l ∞ of the Banach space c 0 has a total strongly non-norming subspace M. Using this subspace M we construct a non-normable Fréchet space F of countable type with a continuous norm such that its strong dual F b ′ is a strict LB-space. Next we show that F has no nuclear Köthe quotient.

Complete Spaces of p-adic Measures

Let $\mathbb K$ be a complete non-Archimedean valued field and let $C(X,E)$ be the space of all continuous functions from a zero-dimensional Hausdorff topological space $X$ to a non-Archimedean Hausdorff locally convex space $E$. We will denote by $C_{b}(X,E)$ (resp. by $C_{rc}(X,E)$) the space of all $f\in C(X,E)$ for which $f(X)$ is a bounded (resp. relatively compact) subset of $E$. The dual space of $C_{rc}(X,E)$, under the topology $t_{u}$ of uniform convergence, is a space $M(X,E')$ of finitely-additive $E'$-valued measures on the algebra $K(X)$ of all clopen , i.e. both closed and open, subsets of $X$. Some subspaces of $M(X,E')$ turn out to be the duals of $C(X,E)$ or of $C_{b}(X,E)$ under certain locally convex topologies.\\ In this paper we continue with the investigation of certain subspaces of $M(X,E')$. Among other results we show that, if $E$ is a polar Fréchet space, then :\\ 1. The space $\mathcal{M}_{\theta_{o}}(X,E')$, of all $m\in M(X,E')$ for which the support of the corresponding measure $m^{\beta_{o}}$, on the Banaschewski compactification of $X$, is contained in the $\theta_{o}$-repletion of $X$, is complete under the topology of uniform convergence on the family $\mathcal{E}$ of all equicontinuous subsets $B$ of $C(X,E)$ for which $B(x)$ is a compactoid subset of $E$ for all $x\in X$.\\ 2. The space $M_{bs}(X,E')$, of all the so called strongly-separable members of $M(X,E')$ is complete under the topology of uniform convergence on the family of all uniformly bounded members of $\mathcal{E}$.\\ 3. The space $M_{s}(X,E')$ of all $m\in M(X,E')$, for which $ms$ is separable for all $s\in E$, is complete under the topology of uniform convergence on the family of all $B \in \mathcal{E}$ for which the set $B(X)$ is compactoid.

ON THE ALGEBRAIC DIMENSION OF BANACH SPACES OVER NON-ARCHIMEDEAN VALUED FIELDS OR ARBITRARY RANK

Let K be a complete non-archimedean valued field of any rank, and let E be a K-Banach space with a countable topological base. We determine the algebraic dimension of E (2.3, 2.4, 3.1).

Mumford coverings of the projective line

A Mumford covering of the projective line over a complete non-archimedean valued field K is a Galois covering \( X\rightarrow {\bf P}^1_K \) such that X is a Mumford curve over K. The question which finite groups do occur as Galois group is answered in this paper. This result is extended to the case where \( {\bf P}^1_K \) is replaced by any Mumford curve over K.

P-Adic almost periodic functions

A bounded function f from an abelian group G into a complete nonarchimedean valued field K is said to be almost periodic ( fϵ AP( G→ K)) if its translates form a compactoid in the uniform topology. The goal of this paper is the theorem below.

On Spaces of Compact Operators in Non-Archimedean Banach Spaces

AbstractLet K be a non-trivial complete non-Archimedean valued field and let E be an infinite-dimensional Banach space over K. Some of the main results are:(1) K is spherically complete if and only if every weakly convergent sequence in l∞ is norm-convergent.(2) If the valuation of K is dense, then C0 is complemented in E if and only if C(E,c0) is n o t complemented in L(E,c0), where L(E,c0) is the space of all continuous linear operators from E to c0 and C(E,c0) is the subspace of L(E, c0) consisting of all compact linear operators.

Automatic continuity of certain linear isomorphisms

Let S and T be compact, 0-dimensional, Hausdorff spaces and let C(S) and C(T) denote the sup-normed Banach spaces of continuous maps on S. and T, respectively, into a complete nonarchimedean valued field F. If F were the reals, then S and T would be homeomorphic if and only if C(S) and C(T) were isometrically isomorphic ; moreover, the isometric isomorphism A : C(T) C(S) must be of the following type : For each x in C(T) and any s in S, Ax(s) = a(s)x(h(s)) for some homeomorphism A of S onto T and some nonva-nishing continuous map a defined on S. As we have shown earlier, there are other kinds of isometric isomorphisms when dealing with F-valued functions. In this paper we focus attention on "separating" linear maps A, maps with the property that the cozero sets of Ax and Ay must be disjoint if the cozero sets of x and y are. We develop criteria for separating maps A to be continuous, for S and T to be homeomorphic, and for such maps to be of the "change of variables" type of the Stone-Banach theorem.

Duality of projective limit spaces and inductive limit spaces over a nonspherically complete non-Archimedean field

A duality theorem of projective and inductive limit spaces over a nonspherically complete valued field is obtained under a certain condition, and topologies of spaces of locally analytic functions are ?studied. Introduction. Morita obtained in |5| a duality theorem of projec­ tive limit spaces and inductive limit spaces over a spherically complete nonarchimedean valued field, and Schikhof studied in [8J locally convex spaces over a nonsphorically complete nonarchimedean valued field. In this paper, we use the results of [8j and study the duality of such spaces over a nonspherically complete nonarchimedean valued field. The duality theorem of |5| was obtained as a generalization of the results of Komatsu |8| by Morita using: the theory of van Tiel [10] about locally convex spaces over a spherically complete nonarchimedean valued field. There the following two facts are used essentially: (i) The Mackey topology is the topology of uniform convergence on weakly e-compact sets; (ii) Any absolutely convex weakly c-compact set is strongly closed. Though we can generalize the notion of e-compactness to our ease, it is difficult to obtain good analogues of these two facts over a nonspherically complete valued field. Hence we restrict our attention to a more restricted class than in |5], and prove a duality theorem by making use of van dor Put’s duality theorem of sequence spaces c0 = {(au cl, aa, • • •')} and V {(&„ L, b„ ■••) X m (m Yn (m n(xn), y j m = (xn, vn_m{ym))n holds for any xn e X a and ym e Ym. Let (X, um) be the locally convex projective limit of {Xm, and let (Y, vm) be the locally convex inductive limit of [Ym, vnim}. We assume further that (iv) the projection map uw: X —> Xm has a dense image for each m. By definition, any element x of the projective limit X can be written as x — (xm) with xm e Xm satisfying umi„(x„) — xm for any m and n with m K by (x, y) = (uM(x), ym)m with such a ym 6 Ym. It is easy to see that this pairing ( , ) is iT-bilinear. Since the projection map um: X —> Xm is con­ tinuous, our pairing ( , ) is bicontinuous on X x Y m for each m. Hence, by the universal mapping property of the inductive limit topology, ( , ) is bicontinuous on X x Y . Let x = (xm) be a nonzero element of X Then xm =£ 0 for some m. Since ( , )m is nondegenerate, (xm, ym)m ^ 0 for some ym 6 Yn. Hence (*, y) = (xm, y j m # 0 for some y = vm(ym) e vm( Y J c Y. Let y = vm( y j (■ym e F J be a nonzero element of Y. Then {xm e Xm; (xm, ym)m ^ 0} is a non-empty open subset of Xm. Since the image of the projection map um: X -* X m is dense, there is an element x = (xm) e X such that (x, y) = (xm, ym)m ¥= 0. Therefore our pairing ( , ) is nondegenerate. Hence we have proved the following: DUALITY OVER A COMPLETE NONARCHIMEDEAN FIELD 389 P r o p o s i t i o n 1. Let X = proj lim and Y = ind lim Y m be as before* Then we have a nondegenerate bicontinuous K'-bilinear fo rm ( , ) : X x Y —> K .

On the topology of simple convergence in non-Archimedean function spaces

Let X be an ultraregular Hausdorff space, F a non-trivial complete nonArchimedean valued field and E a non-Archimedean F-locally convex space over F. Let C,(X, E) denote the space of all continuous E-valued functions on X equipped with the topology of simple convergence. In this paper, we study some of the properties of the space C,(X, E). We look at the problems: When is this space barrelled, quasi-barrelled or bornological? For the bornologicity of C,(X, E) we show that, when E is a non-Archimedean normed space, the space C,(X, E) is bornological iff X is N-replete. This is analogous to the result of Govaerts [4] for the space C,(X, F). We also characterize the strongly bounded subsets of the dual space of C,(X, E). In case E is a non-Archimedean normed space, it is shown that:

The strict topology in non-Archimedean vector-valued function spaces

Let F be a non-trivial complete non-Archimedean valued field. We study the strict topology β 0 on the space C b ( X, E) of all bounded continuous functions from a topological space X to a non-Archimedean F-locally convex space E over F. We also show that the dual of the space ( C b ( X, E), β o ) is a certain space of E′-valued measures and we give a characterization of the equicontinuous subsets of this dual space.

Non-archimedean harmonic analysis on topological semigroups

Let K be a complete non-Archimedean valued field, S a commutative topological semigroup (not necessarily locally compact). We study the convolution Banach algebra M( S) of all tight K-valued measures on S and its ideal M a ( S) consisting of all elements μ of M(S) for which the shift map x ↦ x ̄ ∗μ is continuous (the analogue of the group algebra). In particular, we show that under reasonable conditions the Fourier-Stieltjes transform is isometric. Furthermore, we describe the images of M( S) and M a ( S) under the Fourier-Stieltjes transform. The results obtained are known for groups but new for topological (semi) lattices.