##
**Arithmetic differential equations.**
*(English)*
Zbl 1088.14001

Mathematical Surveys and Monographs 118. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3862-8/hbk). xxxii, 310 p. (2005).

The main purpose of this book is to develop an arithmetic analogue of the theory of ordinary differential equations. In arithmetic theory, replace “time variable” \(t\) by a fixed prime number \(p\), smooth real functions \(x\to x(t)\) by integers \(a\in{\mathbb{Z}}\) (or more generally by algebraic integers), the derivative \(x(t)\to {dx\over {dt}}(t)\) by a “Fermat quotient operator” \(\delta: {\mathbb{Z}}\to{\mathbb{Z}}: a\mapsto \delta\,a:={{a-a^p}\over{p}}\). Smooth manifolds are replaced by algebraic varieties over number fields. Jet spaces are replaced by algebraic jet spaces constructed using \(\delta\). Differential equations are replaced by “arithmetic differential equations”. In particular, differential equations that are invariant under certain group actions are replaced by arithmetic differential equations which are invariant under various correspondences on the varieties in question. The main goal of the book is to construct new quotient spaces which have no counterparts in the classical algebraic geometry. This is done by introducing a new geometry called \(\delta\)-geometry, where the categorical quotients are non-trivial (contrary to the situation in classical algebraic geometry). Three classes of examples are discussed in detail illustrating the new \(\delta\)-geometry, namely, (1) Spherical case: quotients of the \({\mathbb{P}}^1\) by actions of \(\text{SL}_2({\mathbb{Z}})\), (2) Flat case: quotients of \({\mathbb{P}}^1\) by actions of postcritically finite maps \({\mathbb{P}}^1\to{\mathbb{P}}^1\) with (orbifold) Euler characteristic zero, and (3) Hyperbolic case: quotients of modular or Shimura curves by actions of Hecke correspondences.

A conjecture is formulated which characterizes the existence of non-trivial categorical quotients.

Conjecture. The quotient of a curve over a number field by a correspondence is non-trivial for almost all primes \(p\) if and only if the correspondence has a complex “analytic uniformization”.

The main results of the book assert that the “if”part of the conjecture holds under some mild assumptions in the three cases (1),(2) and (3). The “only if” part of the conjecture is proved in the “dynamical system case”.

Let \(\mathcal{C}\) be an arbitrary category. A correspondence in \(\mathcal{C}\) is a tuple \({\mathbb{X}}=(X, \check{X}, \sigma_1,\sigma_2)\) where \(X\) and \(\check{X}\) are objects of \(\mathcal{C}\) and \(\sigma_1,\,\sigma_2\,:\, \check{X}\to X\) are morphisms of \(\mathcal{C}\). A categorical quotient for \({\mathbb{X}}\) is defined to be a pair \((Y,\pi)\) where \(Y\) is an object of \(\mathcal{C}\) and \(\pi:X\to Y\) is a morphism in \(\mathcal{C}\) satisfying the following properties: (1) \(\pi\circ \sigma_1=\pi\circ\sigma_2\); (2) For any pair \((Y^{\prime},\pi^{\prime})\) where \(Y^{\prime}\) is an object of \(\mathcal{C}\) and \(\pi^{\prime}: X\to Y^{\prime}\) is a morphism such that \(\pi^{\prime}\circ\sigma_1=\pi^{\prime}\circ \sigma_2\) there exists a unique morphism \(\gamma: Y\to Y^{\prime}\) such that \(\gamma\circ \pi=\pi^{\prime}\). If a categorical quotient exists, it is unique up to isomorphism, and denoted by \(Y=X/\sigma\). In the classical algebraic geometry, there are interesting correspondences but in many cases their categorical quotients turn out to be trivial. This is the “basic pathology”. The motivation of the book stems from the attempt to remedy this pathological situation. The author’s approach is to enlarge algebraic geometry so that the categorical quotients would become non-trivial. For this, there are at least two approaches, that is, invariant theoretic method, and groupoid theoretic method. The author takes the viewpoint of invariant theoretic approach, and tries to enlarge algebraic geometry by adjoining some new functions \(\delta\), satisfying certain “polynomial compatibility conditions” with respect to addition and multiplication. Locally, there are four types of such \(\delta\): derivation operators, difference operators, \(p\)-derivation operators and \(p\)-difference operators. When \(\delta\) are \(p\)-derivation operators attached to various prime numbers \(p\) (via “Fermat quotient operators”), this leads to the arithmetic differential algebra, arithmetic differential equation and its corresponding \(\delta\)-geometry, and the book develop some of the basic elements of \(\delta\)-geometry and then construct and study interesting categorical quotients whose categorical quotients in the usual algebraic geometry are trivial.

The theory of arithmetic differential algebras, arithmetic differential equations and the corresponding \(\delta\)-geometry is then compared to other theories. When \(\delta\) is derivation, this leads to differential algebra of J. F. Ritt [“Differential Algebra.” Colloquium Publications, 33. New York: American Mathematical Society (AMS). (1950; Zbl 0037.18402)] and E. R. Kolchin [“Differential algebra and algebraic groups.” Pure and Applied Mathematics, 54. New York-London: Academic Press. (1973; Zbl 0264.12102)]. If \(\delta\) is a difference operator, this gives rise to the difference algebra of R. Cohn [“Difference Algebra.” New York-London-Sydney: Interscience Publishers. (1965; Zbl 0127.26402)]. The case when \(\delta\) is \(p\)-difference operator seems to lead to a less interesting theory. Only when \(\delta\) is \(p\)-derivation, it gives rise to an arithmetically futile theory of arithmetic differential algebra and its \(\delta\)-geometry. Comparisons to Connes noncommutative geometry, Dwork’s theory, Drinfeld modules, Mochizuki’s \(p\)-adic Teichmüller theory, Ihara’s congruence relations, among others, are also briefly discussed.

A conjecture is formulated which characterizes the existence of non-trivial categorical quotients.

Conjecture. The quotient of a curve over a number field by a correspondence is non-trivial for almost all primes \(p\) if and only if the correspondence has a complex “analytic uniformization”.

The main results of the book assert that the “if”part of the conjecture holds under some mild assumptions in the three cases (1),(2) and (3). The “only if” part of the conjecture is proved in the “dynamical system case”.

Let \(\mathcal{C}\) be an arbitrary category. A correspondence in \(\mathcal{C}\) is a tuple \({\mathbb{X}}=(X, \check{X}, \sigma_1,\sigma_2)\) where \(X\) and \(\check{X}\) are objects of \(\mathcal{C}\) and \(\sigma_1,\,\sigma_2\,:\, \check{X}\to X\) are morphisms of \(\mathcal{C}\). A categorical quotient for \({\mathbb{X}}\) is defined to be a pair \((Y,\pi)\) where \(Y\) is an object of \(\mathcal{C}\) and \(\pi:X\to Y\) is a morphism in \(\mathcal{C}\) satisfying the following properties: (1) \(\pi\circ \sigma_1=\pi\circ\sigma_2\); (2) For any pair \((Y^{\prime},\pi^{\prime})\) where \(Y^{\prime}\) is an object of \(\mathcal{C}\) and \(\pi^{\prime}: X\to Y^{\prime}\) is a morphism such that \(\pi^{\prime}\circ\sigma_1=\pi^{\prime}\circ \sigma_2\) there exists a unique morphism \(\gamma: Y\to Y^{\prime}\) such that \(\gamma\circ \pi=\pi^{\prime}\). If a categorical quotient exists, it is unique up to isomorphism, and denoted by \(Y=X/\sigma\). In the classical algebraic geometry, there are interesting correspondences but in many cases their categorical quotients turn out to be trivial. This is the “basic pathology”. The motivation of the book stems from the attempt to remedy this pathological situation. The author’s approach is to enlarge algebraic geometry so that the categorical quotients would become non-trivial. For this, there are at least two approaches, that is, invariant theoretic method, and groupoid theoretic method. The author takes the viewpoint of invariant theoretic approach, and tries to enlarge algebraic geometry by adjoining some new functions \(\delta\), satisfying certain “polynomial compatibility conditions” with respect to addition and multiplication. Locally, there are four types of such \(\delta\): derivation operators, difference operators, \(p\)-derivation operators and \(p\)-difference operators. When \(\delta\) are \(p\)-derivation operators attached to various prime numbers \(p\) (via “Fermat quotient operators”), this leads to the arithmetic differential algebra, arithmetic differential equation and its corresponding \(\delta\)-geometry, and the book develop some of the basic elements of \(\delta\)-geometry and then construct and study interesting categorical quotients whose categorical quotients in the usual algebraic geometry are trivial.

The theory of arithmetic differential algebras, arithmetic differential equations and the corresponding \(\delta\)-geometry is then compared to other theories. When \(\delta\) is derivation, this leads to differential algebra of J. F. Ritt [“Differential Algebra.” Colloquium Publications, 33. New York: American Mathematical Society (AMS). (1950; Zbl 0037.18402)] and E. R. Kolchin [“Differential algebra and algebraic groups.” Pure and Applied Mathematics, 54. New York-London: Academic Press. (1973; Zbl 0264.12102)]. If \(\delta\) is a difference operator, this gives rise to the difference algebra of R. Cohn [“Difference Algebra.” New York-London-Sydney: Interscience Publishers. (1965; Zbl 0127.26402)]. The case when \(\delta\) is \(p\)-difference operator seems to lead to a less interesting theory. Only when \(\delta\) is \(p\)-derivation, it gives rise to an arithmetically futile theory of arithmetic differential algebra and its \(\delta\)-geometry. Comparisons to Connes noncommutative geometry, Dwork’s theory, Drinfeld modules, Mochizuki’s \(p\)-adic Teichmüller theory, Ihara’s congruence relations, among others, are also briefly discussed.

Reviewer: Noriko Yui (Kingston)

### MSC:

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14G20 | Local ground fields in algebraic geometry |

14G35 | Modular and Shimura varieties |

14L24 | Geometric invariant theory |