Invariant Functions on $p$-divisible Groups and the $p$-adic Corona Problem II
This note removes the dimension one restriction of the main results in the paper with the same title by the second author.
math.AG↗
arXiv subjects
Christopher Deninger
Publications and source records attributed to Christopher Deninger.
A remark on the vanishing of Higgs fields in the $p$-adic Simpson correspondence
We give a condition on a $p$-adic representation of the fundamental group of a curve over $\overline{\mathbb{Q}}_p$ which ensures that under the $p$-adic Simpson correspondence the Higgs field vanishes.
math.AG↗
Rational Witt vectors and associated sheaves
We study the sheafification of $W_{\mathrm{rat}} (\mathcal{O})$ and of the maps $\underline{\mathbb{Z}} \mathcal{O} \to W_{\mathrm{rat}} (\mathcal{O})$ and $W_{\mathrm{rat}} (\mathcal{O}) \to W_J (\mathcal{O})$ in various Grothendieck topologies, both subcanonical and non-subcanonical. Here, for a commutative ring $A$, $\underline{\mathbb{Z}} A$ is the reduced monoid algebra on $(A , \cdot)$ and $W_{\mathrm{rat}} (A)$ is the subring of rational functions in the big Witt ring $W (A)$. Moreover, $W_J$ is the ind-scheme representing $W_{\mathrm{rat}}$ on Fatou rings which was introduced by Hazewinkel and which we prove to be an ind-ring scheme. It turns out, for example that for any field $K$, we have $W_{\mathrm{rat}} (K) = \Gamma (\mathrm{spec}\, K , (\underline{\mathbb{Z}}\mathcal{O})^{\sharp})$ where $\sharp$ denotes the associated sheaf in the finite flat topology. More generally, this is true for Dedekind rings. By comparing our results with work of Suslin and Voevodsky we found an isomorphism of $W_{\mathrm{rat}} (A)$ for normal domains $A$ with a ring of universally integral finite relative correspondences. This gives a new geometric interpretation of Almkvist's theorem on cyclic $K$-theory for such rings and suggests a number of interesting questions.
math.AC↗
Is there a Birch and Swinnerton-Dyer conjecture for Dedekind zeta functions?
A Birch and Swinnerton-Dyer conjecture for number fields $K / \mathbb{Q}$ would assert that $dim V_K = ord_{s = 1/2} \zeta_K (s)$ for some vector space functorially attached to $K$. Presently there is no natural candidate for the $V_K$'s. However, assuming $V_K$ is of a cohomological nature and assuming a conjecture of Serre on the vanishing order of $\zeta_K (s)$ at $s = 1/2$ we show that such functors $K \mapsto V_K$ (with natural extra structures) exist and are all isomorphic. Their common automorphism group is $2$-torsion and abelian.
math.NT↗
Dynamical systems for arithmetic schemes
Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vectors to attach a new ringed space $W_{\mathrm{rat}} (X)$ to every scheme $X$. We also define $R$-valued points $W_{\mathrm{rat}} (X) (R)$ of $W_{\mathrm{rat}} (X)$ for every commutative ring $R$. For normal schemes $X$ of finite type over spec $\mathbb{Z}$, using $W_{\mathrm{rat}} (X) (\mathbb{C})$ we construct infinite dimensional $\mathbb{R}$-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 $W_{\mathrm{rat}} (X)$ for $X$ the spectrum of a $p$-adic local number ring are related to the points of the Fargues-Fontaine curve. The new intrinsic construction of the dynamical systems generalizes and clarifies the original extrinsic construction in v.1 and v.2. Many further results have been added.
math.DS↗
Weyl Tensors, Strongly Regular Graphs, Multiplicative Characters, and a Quadratic Matrix Equation
We study solutions of a quadratic matrix equation arising in Riemannian geometry. Let $S$ be a real symmetric $n\times n$-matrix with zeros on the diagonal and let $θ$ be a real number. We construct nonzero solutions $(S,θ)$ of the set of quadratic equations \[\sum_kS_{i,k}=0\quad\text{ and }\quad\sum_{k}S_{i,k}S_{k,j}+S_{i,j}^2=θS_{i,j}\text { for }i<j.\] Our solutions relate the equations to strongly regular graphs, to group rings, and to multiplicative characters of finite fields.
math.GR↗
A pro-algebraic fundamental group for topological spaces
Consider a connected topological space $X$ with a point $x \in X$ and let $K$ be a field with the discrete topology. We study the Tannakian category of finite dimensional (flat) vector bundles on $X$ and its Tannakian dual $π_K (X,x)$ with respect to the fibre functor in $x$. The maximal pro-étale quotient of $π_K (X,x)$ is the étale fundamental group of $X$ studied by Kucharczyk and Scholze. For well behaved topological spaces, $π_K (X,x)$ is the pro-algebraic completion of the ordinary fundamental group $π_1 (X,x)$. We obtain some structural results on $π_K (X,x)$ by studying (pseudo-)torsors attached to its quotients. This approach uses ideas of Nori in algebraic geometry and a result of Deligne on Tannakian categories. We also calculate $π_K (X,x)$ for some generalized solenoids.
math.AT↗
On the proalgebraic fundamental group of topological spaces and amalgamated products of affine group schemes
The proalgebraic fundamental group of a connected topological space $X$, recently introduced by the first author, is an affine group scheme whose representations classify local systems of finite-dimensional vector spaces on $X$. In this article, we further develop the theory of the proalgebraic fundamental group, in particular, we establish homotopy invariance and a Seifert-van Kampen theorem. To facilitate the latter, we study amalgamated free product of affine group schemes. We also compute the proalgebraic fundamental group of the arithmetically relevant Kucharcyzk-Scholze spaces and compare it to the motivic Galois group.
math.AG↗
There is no "Weil-"cohomology theory with real coefficients for arithmetic curves
A well known argument by Serre shows that there is no Weil cohomology theory with real coefficients for smooth projective varieties over $\bar{\mathbb{F}}_p$. In this note we explain why no "Weil-"cohomology theory with real coefficients can exist for arithmetic schemes over spec $\mathbb{Z}$, even for spectra of number rings.
math.AG↗
Primes, knots and periodic orbits
This is a text written for the Ennio De Giorgi Colloquio volume. It covers analogies between algebraic number theory and knot theory, analogies between analytic number theory and certain dynamical systems, and a report on our construction of dynamical systems for arithmetic schemes which realize some of these analogies.
math.NT↗
$\mathbb{Z} R$ and rings of Witt vectors $W_S (R)$
Using $λ$ operations, we give some results on the kernel of the natural map from the monoid algebra $\mathbb{Z} R$ of a commutative ring $R$ to the ring of $S$-Witt vectors of $R$. As a byproduct we obtain a very natural interpretation of a power series used by Dwork in his proof of the rationality of zeta functions for varieties over finite fields.
math.AC↗
p-adic limits of renormalized logarithmic Euler characteristics
Given a countable residually finite group $Γ$, we write $Γ_n \to e$ if $(Γ_n)$ is a sequence of normal subgroups of finite index such that any infinite intersection of $Γ_n$'s contains only the unit element $e$ of $Γ$. Given a $Γ$-module $M$ we are interested in the multiplicative Euler characteristics \begin{equation} χ(Γ_n , M) = \prod_i |H_i (Γ_n , M)|^{(-1)^i} \end{equation} and the limit in the field $\mathbb{Q}_p$ of $p$-adic numbers \begin{equation} h_p := \lim_{n\to\infty} (Γ: Γ_n)^{-1} \log_p χ(Γ_n , M) \; . \end{equation} Here $\log_p : \mathbb{Q}^{\times}_p \to \mathbb{Z}_p$ is the branch of the $p$-adic logarithm with $\log_p (p) = 0$. Of course, neither expression will exist in general. We isolate conditions on $M$, in particular $p$-adic expansiveness which guarantee that the Euler characteristics $χ(Γ_n , M)$ are well defined. That notion is a $p$-adic analogue of expansiveness of the dynamical system given by the $Γ$-action on the compact Pontrjagin dual $X = M^*$ of $M$. Under further conditions on $Γ$ we also show that the renormalized $p$-adic limit in the second formula exists and equals the $p$-adic $R$-torsion of $M$. The latter is a $p$-adic analogue of the Li--Thom $L^2$ $R$-torsion of a $Γ$-module $M$ which they related to the entropy $h$ of the $Γ$-action on $X$. We view the limit $h_p$ as a version of entropy which values in the $p$-adic numbers and the equality with $p$-adic $R$-torsion as an analogue of the Li--Thom formula in the expansive case. We discuss the case $Γ= \mathbb{Z}^N$ in more detail where our theory is related to Serre's intersection numbers on arithmetic schemes.
math.GR↗
Parallel transport for vector bundles on p-adic varieties
We develop a theory of étale parallel transport for vector bundles with numerically flat reduction on a $p$-adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous $p$-adic representation of the étale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a $p$-adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings' $p$-adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.
math.AG↗
The Universal Deformation of the Witt Ring Scheme
We determine the universal deformation over reduced base rings of the Witt ring scheme enhanced by a Frobenius lift and Verschiebung. It agrees with a q-deformation earlier introduced by the second author, for which we also give a simpler description. In the appendix we discuss a Witt vector theory for ind-rings which may be of independent interest.
math.RA↗
Witt Vector Rings and the Relative de Rham Witt Complex
In this paper we develop a novel approach to Witt vector rings and to the (relative) de Rham Witt complex. We do this in the generality of arbitrary commutative algebras and arbitrary truncation sets. In our construction of Witt vector rings the ring structure is obvious and there is no need for universal polynomials. Moreover a natural generalization of the construction easily leads to the relative de Rham Witt complex. Our approach is based on the use of free or at least torsion free presentations of a given commutative ring $R$ and it is an important fact that the resulting objects are independent of all choices. The approach via presentations also sheds new light on our previous description of the ring of $p$-typical Witt vectors of a perfect $\mathbb{F}_p$-algebra as a completion of a semigroup algebra. We develop this description in different directions. For example, we show that the semigroup algebra can be replaced by any free presentation of $R$ equipped with a linear lift of the Frobenius automorphism. Using the result in the appendix by Umberto Zannier we also extend the description of the Witt vector ring as a completion to all $\bar{\mathbb{F}}_p$-algebras with injective Frobenius map.
math.RA↗
An alternative to Witt vectors
The ring of Witt vectors associated to a ring R is a classical tool in algebra. We introduce a ring C(R) which is more easily constructed and which is isomorphic to the ring of Witt vectors W(R) for a perfect F_p-algebra R. It is obtained as the completion of the monoid ring ZR, for the multiplicative monoid R, with respect to the powers of the kernel of the natural map from ZR to R.
math.NT↗
C*-algebras of Toeplitz type associated with algebraic number fields
We associate with the ring $R$ of algebraic integers in a number field a C*-algebra $\cT[R]$. It is an extension of the ring C*-algebra $\cA[R]$ studied previously by the first named author in collaboration with X.Li. In contrast to $\cA[R]$, it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $ax+b$-semigroup $R\rtimes R^\times$ on $\ell^2 (R\rtimes R^\times)$. The algebra $\cT[R]$ carries a natural one-parameter automorphism group $(σ_t)_{t\in\Rz}$. We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where $R$ is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The "partition functions" are partial Dedekind $ζ$-functions. We prove a result characterizing the asymptotic behavior of quotients of such partial $ζ$-functions, which we then use to show uniqueness of the $β$-KMS state for each inverse temperature $β\in(1,2]$.
math.OA↗
Horizontal factorizations of certain Hasse--Weil zeta functions - a remark on a paper by Taniyama
In one of his papers, using arguments about l-adic representations, Taniyama expresses the zeta function of an abelian variety over a number field as an infinite product of modified Artin L-functions. The latter can be further decomposed as products of modified Dedekind zeta functions. After recalling Taniyama's work, we give a simple geometric proof of the resulting product formula for abelian and more general group schemes.
math.AG↗