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Foundations of arithmetic differential geometry. (English) Zbl 1388.11001

Mathematical Surveys and Monographs 222. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3623-0/hbk; 978-1-4704-4089-3/ebook). x, 344 p. (2017).
The aim of this book is to introduce and develop an arithmetic analog of (classical) differential geometry; this analogue is referred to by the author as arithmetic differential geometry. In this new geometry, the ring of integers \(\mathbb Z\) is meant to play the role of a ring of functions on an infinite dimensional manifold. The prime numbers \(p\in\mathbb Z\) are meant to play the role of coordinate functions on such a manifold, and for a prime number \(p\in\mathbb Z\), the Fermat quotients \[ \mathbb Z\longrightarrow \mathbb Z,\qquad n\mapsto \delta_pn=\frac{n-n^p}{p}\tag{1} \]
are meant to play the role of partial derivatives (with respect to the coordinates). Symmetric (resp., anti-symmetric) matrices with coefficients in \({\mathbb Z}\) are meant to play the roles of metrics (respectively, \(2\)-forms).
Finally, and most importantly for this book, certain adelic (respectively, global) objects, called adelic connections, attached to matrices as above are meant to play the roles of connections (respectively, curvature) attached to metrics or \(2\)-forms. Let us indicate how these objects are defined. First, the theory is in fact developed not just for the base ring \({\mathbb Z}\) but also for more general base rings \(A={\mathbb Z}[1/M, \zeta_N]\) where \(M\) is an even integer, \(N\) is a positive integer and \(\zeta_N\) is a primitive \(N\)-th root of unity. (Even more general localizations of number rings are allowed.) For a fixed positive integer \(n\) let \(B=A[x,\det (x)^{-1}]\) where \(x=(x_{ij})\) is an \(n\times n\) matrix of indeterminates. Fix a (possibly infinite) set \({\mathcal V}\) of prime numbers in \({\mathbb Z}\) not dividing \(NM\). For \(p\in{\mathcal V}\) let \(\phi_p: A\to A\) be the unique ring automorphism with \(\phi_p(\zeta_N)=\zeta_N^p\). The maps (1) then generalize as \[ A\longrightarrow A,\quad\quad a\mapsto \delta_pa=\frac{ \phi_p(a)-a^p}{p}\tag{2} \] for \(p\in {\mathcal V}\). For \(p\in {\mathcal V}\) denote by \(B^{\hat{p}}=\projlim B/p^nB\) the \(p\)-adic completion of \(B\). An adelic connection on \(G=\text{GL}_n/A=\text{Spec}(B)\) is then defined to be a family \(\delta=(\delta_p)_{p\in{\mathcal V}}\) of \(p\)-derivations \[ \delta_p:B^{\hat{p}}\longrightarrow B^{\hat{p}} \] extending the maps (1) on \(A\), i.e. it is required that the (set theoretic) maps \(\delta_p:B^{\hat{p}}\longrightarrow B^{\hat{p}}\) have the property that the attached map \[ \phi_p:B^{\hat{p}}\longrightarrow B^{\hat{p}},\quad\quad b\mapsto b^p+p\delta_p(b) \] is a ring homomorphism.
A host of various types of adelic connections is introduced, each one of them mimicking a specific type of connections in classical differential geometry. Accordingly, they are labelled
(1) Ehresmann connections
(2) Chern connections
(3) Levi-Cività connections
(4) Fedosov connections
(5) Lax connections
(6) Hamiltonian connections
(7) Cartan connections
(8) Riccati connections
(9) Weierstrass connections
(10) Painlevé connections
and in fact even more types and ramifications of such are considered. Many of them are attached to fixed elements \(q\in\text{GL}_n(A)\) and are characterized by certain equivariance, resp., horizontality properties with respect to self maps on \(B\) (and \(B\otimes_A B\)) attached to \(q\). Thus, they owe their very existence several non trivial existence and uniqueness theorems proved in the book – arguably, the main difficulty probably was to actually identify the correct and by no means straightforward defining conditions (so as to create objects truly deserving to be called arithmetic analogues of their classical counterparts). They display a rich variety of different individual features so that each of them needs a specific treatment and analysis, in particular with respect to the study of curvatures for them.
Curvatures are the basic global (as opposed to adelic) concepts proposed in this book. They are meant to encode the commutators between the maps \(\phi_p:B^{\hat{p}}\longrightarrow B^{\hat{p}}\) appearing above when \(p\in{\mathcal V}\) varies. Ideally, the defining formula for them is \[ \varphi_{p p'}=\frac{1}{p p'}[\phi_p,\phi_{p'}] \] for \(p,p'\in {\mathcal V}\). Here however, one faces the problem that the \(\phi_p\) for the various \(p\) are not acting on the same ring so that the definition a priori does not make much sense. Nevertheless, very often the said connections have the property of being global along the unit section; this allows to give meaning to the above formula as defining self maps on \(A[[T]]\), the completion of \(B\) along the unit section (i.e. \(T=x-1\)). Let \(\operatorname{End}(A[[T]])\) denote the Lie ring of \({\mathbb Z}\)-module endomorphisms of \(A[[T]]\). The holonomy ring \({\mathfrak{hol}}\) of the connection \(\delta=(\delta_p)_{p\in{\mathcal V}}\) is then defined to be the \({\mathbb Z}\)-linear span in \(\operatorname{End}(A[[T]])\) of all the Lie monomials \[ [\phi_{p_1},[\phi_{p_2},\ldots [\phi_{p_{s-1}},\phi_{p_{s}}]\ldots]]:A[[T]]\longrightarrow A[[T]] \] where \(s\ge 2\), \(p_i\in{\mathcal V}\). Next, denoting by \({\mathfrak{hol}}_n\) the image of \({\mathfrak{hol}}\) in \(\operatorname{End}(A[[T]]/(T)^n)\), the completed holonomy ring is \(\widehat{\mathfrak{hol}}=\projlim {\mathfrak{hol}}_n\).
As the main global results obtained in the book, several vanishing and non vanishing results on the curvatures \(\varphi_{p p'}\) are proven. Again, there is no universal approach for computing (and not even for defining) them for the various types of connections; rather, each of them needs to be studied individually.
For example, the results obtained for the so called real Chern connections (where the label real only indicates an analogy with a real-number concept) are then interpreted as saying that \(\text{Spec}({\mathbb Z})\) is curved (the \(\varphi_{p p'}\) do not vanish for \(p\neq p'\)), but it is only mildy curved (we have \(\varphi_{p p'}(T)\equiv0\) mod \((T)^3\)). For the holonomy rings of such connections these results imply: \(\widehat{\mathfrak{hol}}\) is non-zero and pronilpotent, and \({\mathfrak{hol}}_{\mathbb Q}\) is not spanned over \({\mathbb Q}\) by the components of the curvature. The first one of these latter results is then seen in stark contrast with the fact that holonomy algebras arising from Galois theory are never nilpotent unless they vanish.
Another important topic addressed is the problem of attaching Galois groups to the connections studied, and of developing a Galois theory. This is begun here only for the class of so called Ehresmann connections.
The main purpose of this book is to present the foundations of arithmetic differential geometry, which, as the author writes, is still in its infancy. Many paths for a further development of the theory are indicated. In fact, the book closes with a long list of intriguing problems whose solutions would give enormous additional momentum to the theory.
For example, the first problem suggests to look for a unification of the holonomy algebra \({\mathfrak{hol}}_{\mathbb Q}\) with the absolute Galois group \(\Gamma_{\mathbb Q}=\mathrm{Gal}({\mathbb Q}^a/{\mathbb Q})\) of \({\mathbb Q}\): The Lie algebra \({\mathfrak{hol}}_{\mathbb Q}\) should be to \(\Gamma_{\mathbb Q}\) what the identity component \(\operatorname{Hol}^0\) of the holonomy group \(\operatorname{Hol}\) is to the monodromy group \(\operatorname{Hol}/\operatorname{Hol}^0\) in classical differential geometry. As such the Lie algebra \({\mathfrak{hol}}_{\mathbb Q}\) could be viewed as an infinitesimal analog of \(\Gamma_{\mathbb Q}\) and, as the author writes, as an object of study in its own right, as fundamental, perhaps, as \(\Gamma_{\mathbb Q}\) itself.
Another circle of problems suggested aims at building bridges between the arithmetic differential geometry developed in the present book and the arithmetic differential calculus developed in a long series of papers of the author, starting at about 1995 and to a large extent covered by the author’s monograph [Arithmetic differential equations. Providence, RI: AMS (2005; Zbl 1088.14001)]. Logically the present book is independent of this previous work of the author. The latter theory, similarly centered around the study of the Fermat quotients (1), (2), was directed towards the study of Shimura varieties, modular forms, abelian varieties.
More intrinsically, the theory as developed in the present book treats connections on principal \(G\)-bundles for \(G=\mathrm{GL}_n\) (and accordingly, it rests on matrix calculations): more general reductive groups should \(G\) be studied in the future.
At present, arithmetic differential geometry appears to be a theory logically still somewhat isolated from main stream number theory. In fact, this impression is substantiated by the extensive comparisons, ideological as well as straight to the point, with other lines of research in current number theory which can be found in the introduction of the book. On the other hand, it so convincingly incorporates central and deep concepts from classical differential geometry into a highly non trivial arithmetic theory that one can hardly doubt its potential significance for number theory.
The book is written with great care. Great emphasis is given on motivating the various concepts introduced. This happens by extensive comparisons with the corresponding concepts from classical differential geometry. In fact, the (long) second chapter of this book is entirely devoted to revisiting classical differential geometry from an algebraic standpoint. That some of the classical differential geometric facts are presented here, as the author writes, in a somewhat non-conventional way in order to facilitate and motivate the transition to the arithmetic setting will certainly much appreciated by the (more algebraically oriented) reader.
The table of contents of this highly original book reads as follows:
Preface
Introduction
Chapter 1: Algebraic background
Chapter 2: Classical differential geometry revisited
Chapter 3: Arithmetic differential geometry: generalities
Chapter 4: Arithmetic differential geometry: the case of \(\mathrm{GL}_n\)
Chapter 5: Curvatures and Galois groups of Ehresmann connections
Chapter 6: Curvature of Chern connections
Chapter 7: Curvature of Levi-Cività connections
Chapter 8: Curvature of Lax connections
Chapter 9: Open problems

MSC:

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11E57 Classical groups
11E08 Quadratic forms over local rings and fields
11E95 \(p\)-adic theory
11F85 \(p\)-adic theory, local fields
14G20 Local ground fields in algebraic geometry
53C05 Connections (general theory)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Citations:

Zbl 1088.14001
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