We define an analogue of the `Real' Deligne cohomology group at a prime of semi-stable or good reduction of a variety. We also define regulator maps to this group and formulate a conjecture about the image. This allows us to formulate a non-Archimedian version of Beilinson's Hodge-D-conjecture, S-integral and function field versions of Beilinson's global conjectures as well as a precise special value conjecture in the function field case. Finally we give a few examples where these conjectures are known to be true.
The aim of this note is to give an effective criterion to verify whether a cubic polynomial over a non-Archimedean field has a weak Néron model or not.
We prove non-Archimedean analogs of results of Noguchi and Winkelmann showing algebraic degeneracy of rigid analytic maps to projective varieties omitting an effective divisor with sufficiently many irreducible components relative to the rank of the group they generate in the Neron-Severi group of the variety.
We construct a rigid analytical regulator for the K_2 of Mumford curves, a non-archimedean analogue of the complex analytical Beilinson-Bloch-Deligne regulator.
If $f$ is a non-Archimedean analytic curve in a projective variety $X$ embedded in $P^N$ and if $D_1,\dots,D_q$ are hypersurfaces of $P^N$ in general position with $X,$ then we prove the defect relation: $$ \sum_{j=1}^q δ(f,D_j) \le \dim X. $$
We study the degeneration dimension of non-archimedean analytic maps into the complement of hypersurface divisors of smooth projective varieties. We also show that there exist no non-archimedean analytic maps into $P^n\setminus\cup_{i= 1}^n D_i$ where $D_i, 1\le i\le n$, are hypersurfaces of degree at least 2 in general position and intersecting transversally. Moreover, we prove that there exist no non-archimedean analytic maps into $P^2\setminus\cup_{i=1}^2 D_i$ when $D_1, D_2$ are generic plane curves with deg$D_1+$deg$D_2\ge 4$.
We investigate arithmetic aspects of the middle degree cohomology of compactified Picard modular surfaces $X$ attached to the unitary similitude group $\mathrm{GU}(2,1)$ for an imaginary quadratic extension $E/\mathbf{Q}$. We construct new Beilinson--Flach classes on $X$ and compute their Archimedean regulator. We obtain a special value formula involving a non-critical $L$-value of the degree six standard $L$-function, a Whittaker period, and the regulator. This provides evidence for Beilinson's conjecture in this setting.
We study the dynamics of the Hénon map defined over complete, locally compact non-Archimedean fields of odd residue characteristic. We establish basic properties of its one-sided and two-sided filled Julia sets, and we determine, for each Hénon map, whether these sets are empty or nonempty, whether they are bounded or unbounded, and whether they are equal to the unit ball or not. On a certain region of the parameter space we show that the filled Julia set is an attractor. We prove that, for infinitely many distinct Hénon maps over ${\mathbb Q}_3$, this attractor is infinite and supports an SRB…
Let $K$ be an algebraically closed, complete, non-Archimedean valued field of characteristic zero, and let $\mathscr{X}$ be a $K$-analytic space (in the sense of Huber). In this work, we pursue a non-Archimedean characterization of Campana's notion of specialness. We say $\mathscr{X}$ is $K$-analytically special if there exists a connected, finite type algebraic group $G/K$, a dense open subset $\mathscr{U}\subset G^{\text{an}}$ with $\text{codim}(G^{\text{an}}\setminus \mathscr{U}) \geq 2$, and an analytic morphism $\mathscr{U} \to \mathscr{X}$ which is Zariski dense. With this definition, we…
We develop differential algebraic K-theory of regular arithmetic schemes. Our approach is based on a new construction of a functorial, spectrum level Beilinson regulator using differential forms. We construct a cycle map which represents differential algebraic K-theory classes by geometric vector bundles. As an application we derive Lott's relation between short exact sequences of geometric bundles with a higher analytic torsion form.
We define a regulator map from the weight n polylogarithmic motivic complex to the weight n Deligne complex of an algebraic variety X. The regulator map is constructed explicitly via the classical polylogarithms with some funny combinations of Bernoulli numbers as coefficients. This leads to conjectures on special values of L-functions at s=n which reduce to Zagier's conjecture when X is of dimension zero over Q. The cone of this map, shifted by one, should be called the Arakelov motivic complex of X. Its last cohomology group is identified with a group of codimension n Arakelov cycles on X.
Let $K$ be a complete, algebraically closed, non-Archimedean valued field, and let $\textbf{P}^1$ denote the Berkovich projective line over $K$. The Lyapunov exponent for a rational map $ϕ\in K(z)$ of degree $d\geq 2$ measures the exponential rate of growth along a typical orbit of $ϕ$. When $ϕ$ is defined over $\mathbb{C}$, the Lyapunov exponent is bounded below by $\frac{1}{2}\log d$. In this article, we give a lower bound for $L(ϕ)$ for maps $ϕ$ defined over non-Archimedean fields $K$. The bound depends only on the degree $d$ and the Lipschitz constant of $ϕ$. For maps $ϕ$ whose Julia sets …
Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map $f$ of $\mathbb{P}^1$ of degree $d>1$ over $K$, in terms of the multipliers of $n$-periodic points of $f$, with an explicit control in terms of $n$, $f$ and $K$. As an immediate consequence, we obtain an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of rational maps over $K$. Combined with our former archimedean version, this non-…
Let L be an ample line bundle on a smooth projective variety X over a non-archimedean field K. For a continuous metric on L, we show in the following two cases that the semipositive envelope is a continuous semipositive metric on L and that the non-archimedean Monge-Ampère equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that X is a surface defined geometrically over the function field of a curve over a perfect field k of positive characteristic. The second case holds in higher dimensions if we assume resolution of singu…
Let k be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let D_1,...,D_n be effective nef divisors intersecting transversally in an n-dimensional nonsingular projective variety X. We study the degeneracy of non-Archimedean analytic maps from k into $X\setminus \cup_{i=1}^nD_i$ under various geometric conditions. When X is a rational ruled surface and D_1 and D_2 are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from k into $X\setminus D_1 \cup D_2$. Using a dictionary…
We construct the first examples of rational functions defined over a non-archimedean field with certain dynamical properties. In particular, we find such functions whose Julia sets, in the Berkovich projective line, are connected but not contained in a line segment. We also show how to compute the measure-theoretic and topological entropy of such maps. In particular, we show for some of our examples that the measure-theoretic entropy is strictly smaller than the topological entropy, thus answering a question of Favre and Rivera-Letelier.
We consider the degree 4 L-function associated to an automorphic representation of the symplectic group GSp(4). Starting with Beilinson's Eisenstien symbol we construct some motivic cohomology classes on the Shimura variety of GSp(4). We show that the image of these classes under the absolute Hodge regulator vanishes on the boundary of the Baily-Borel compactification of the Shimura variety. This allows to relate these classes to the product of an archimedean integral, Harris' occcult period invariant, a Deligne period and the special value of the L-function predicted by Beilinson's conjecture…
This paper is a sequel to our earlier paper "Wach modules and Iwasawa theory for modular forms" (arXiv: 0912.1263), where we defined a family of Coleman maps for a crystalline representation of the Galois group of Qp with nonnegative Hodge-Tate weights. In this paper, we study these Coleman maps using Perrin-Riou's p-adic regulator L_V. Denote by H(Γ) the algebra of Qp-valued distributions on Γ= Gal(Qp(μ(p^\infty) / Qp). Our first result determines the H(Γ)-elementary divisors of the quotient of D_{cris}(V) \otimes H(Γ) by the H(Γ)-submodule generated by (ϕ* N(V))^{ψ= 0}, where N(V) is the Wac…
We consider a class of equations with the fractional differentiation operator $D^α$, $α>0$, for complex-valued functions $x\mapsto f(|x|_K)$ on a non-Archimedean local field $K$ depending only on the absolute value $|\cdot |_K$. We introduce a right inverse $I^α$ to $D^α$, such that the change of an unknown function $u=I^αv$ reduces the Cauchy problem for an equation with $D^α$ (for radial functions) to an integral equation whose properties resemble those of classical Volterra equations. This contrasts much more complicated behavior of $D^α$ on other classes of functions.
We propose a functional integral representation for Archimedean L-factors given by products of Gamma-functions. The corresponding functional integral arises in the description of type A equivariant topological linear sigma model on a disk. The functional integral representation provides in particular an interpretation of the Gamma-function as an equivariant symplectic volume of an infinite-dimensional space of holomorphic maps of the disk to C. This should be considered as a mirror-dual to the classical Euler integral representation of the Gamma-function. We give an analogous functional integr…
Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying \begin{align*} \left|\sum_{j=1}^{n}λ_j^2\right|=\max_{1\leq j \leq n}|λ_j|^2, \quad \forall λ_j \in \mathbb{K}, 1\leq j \leq n, \forall n \in \mathbb{N}. \end{align*} For $d\in \mathbb{N}$, let $\mathbb{K}^d$ be the standard $d$-dimensional non-Archimedean Hilbert space. Let $m \in \mathbb{N}$ and $\text{Sym}^m(\mathbb{K}^d)$ be the non-Archimedean Hilbert space of symmetric m-tensors. We prove the following result. If $\{τ_j\}_{j=1}^n$ is a collection in $\mathbb{K}^d$ satisfying $\langle τ_j, τ_j\rangle =1$ for all $1\l…
A main theme of the paper is a conjecture of Bloch-Kato on the image of $p$-adic regulator maps for a proper smooth variety $X$ over an algebraic number field $k$. The conjecture for a regulator map of particular degree and weight is related to finiteness of two arithmetic objects: One is the $p$-primary torsion part of the Chow group in codimension 2 of $X$. Another is an unramified cohomology group of $X$. As an application, for a regular model ${\mathscr X}$ of $X$ over the integer ring of $k$, we show an injectivity result on torsion of a cycle class map from the Chow group in codimension …
Let G be an absolutely almost simple algebraic group defined over a non-archimedean local field K. Let X be a projective homogeneous variety for G and let L be an ample line bundle on X. Then there exists a unique G-linearisation of L. We regard X as a rigid analytic variety. We consider the open rigid analytic subspace Y (resp. Y') of X that consists of the points x in X that are stable (resp. semistable) for all maximal K-split tori in G. Here we take for each maximal K-split torus S in G the S-linearisation of L obtained by restricting the G-linearisation of L to S. Then G(K) acts on both Y…
We prove the local epsilon-isomorphism conjecture of Fukaya and Kato [FK06] for certain crystalline families of G_Qp-representations. This conjecture can be regarded as a local analogue of the Iwasawa main conjecture for families. Our work extends earlier work of Kato for rank-1 modules (cf. [Ven13]), of Benois and Berger for crystalline G_Qp-representations with respect to the cyclotomic extension (cf. [BB08]), as well as of Loeffler, Venjakob, and Zerbes (cf. [LVZ13]) for crystalline G_Qp- representations with respect to abelian p-adic Lie extensions of Qp. Nakamura [Nak13, Nak14] has also f…
We consider three examples of families of curves over a non-archimedean valued field which admit a non-trivial group action. These equivariant deformation spaces can be described by algebraic parameters (in the equation of the curve), or by rigid-analytic parameters (in the Schottky group of the curve). We study the relation between these parameters as rigid-analytic self-maps of the disk.
We prove the non-Archimedean (resp. p-adic) Banach space version of non-Archimedean (resp. p-adic) Welch bounds recently obtained by M. Krishna. More precisely, we prove following results. 1. Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying $\left|\sum_{j=1}^{n}λ_j^2\right|=\max_{1\leq j \leq n}|λ_j|^2$ for all $ λ_j \in \mathbb{K}, 1\leq j \leq n$, for all $n \in \mathbb{N}.$ Let $\mathcal{X}$ be a $d$-dimensional non-Archimedean Banach space over $\mathbb{K}$. If $\{τ_j\}_{j=1}^n$ is any collection in $\mathcal{X}$ and $\{f_j\}_{j=1}^n$ is any collection in $\mathcal{…
We prove that if $f$ is a polynomial over a number field $K$ with a finite superattracting periodic point and a non-archimedean place of bad reduction, then there is an $ε>0$ such that only finitely many $P\in K^{\text{ab}}$ have canonical height less than $ε$ with respect to $f$. The key ingredient is the geometry of the filled Julia set at a place of bad reduction. We also prove a conditional uniform boundedness result for the $K$-rational preperiodic points of such polynomials, as well as a uniform lower bound on the canonical height of non-preperiodic points in $K$. We further prove uncond…
Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic different from $2$. Let $A$ be an $F$-central simple algebra of even dimension so that it contains $E$ as a subfield, set $G=A^\times$ and $H$ for the centralizer of $E^\times$ in $G$. Using a Galois descent argument, we prove that all double cosets $H g H\subset G$ are stable under the anti-involution $g\mapsto g^{-1}$, reducing to Guo's result for $F$-split $G$ which we extend to fields of positive characteristic different from $2$. We then show, combining global and local results, that $H$-distinguished irr…
McMullen proved the moduli space of complex rational maps can be parametrized by the spectrum of all periodic-point multipliers up to a finite amount of data, with the well-understood exception of Lattès maps. We generalize his method to large positive characteristic. McMullen's method is analytic; a modified version of the method using rigid analysis works over a function field over a finite field of characteristic larger than the degree of the map. Over a finite field with such characteristic it implies that, generically, rational maps can indeed be parametrized by their multiplier spectra u…
A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the…
Let $M(α)$ denote the Mahler measure of the algebraic number $α$. In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure on the multiplicative group of algebraic numbers. Later, Fili and the author used similar techniques to study a non-Archimedean version. We show how to generalize the above constructions in order to associate, to each point in $(0,\infty]$, a metric version $M_x$ of the Mahler measure, each having a triangle inequality of a different strength. We are able to compute $M_x(α)$ for sufficiently small $x$, identifying, in the process, a function…
