Mathematical Surveys and Monographs Volume 222 Foundations of Arithmetic Differential Geometry Alexandru Buium American Mathematical Society 10.1090/surv/222 Foundations of Arithmetic Differential Geometry Mathematical Surveys and Monographs Volume 222 Foundations of Arithmetic Differential Geometry Alexandru Buium American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 11E57, 11F85, 14G20, 53C21. For additional information and updates on this book, visit www.ams.org/bookpages/surv-222 Library of Congress Cataloging-in-Publication Data Buium, Alexandru, 1955- Foundations of arithmetic differential geometry / Alexandru Buium. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Mathe- matical surveys and monographs ; volume 222 | Includes bibliographical references and index. Identifiers: LCCN 2016056302 | ISBN 9781470436230 (alk. paper) Subjects: LCSH: Geometry, Differential. | AMS: Number theory – Forms and linear algebraic groups – Classical groups. msc | Number theory – Discontinuous groups and automorphic forms – p-adic theory, local fields. msc | Algebraic geometry – Arithmetic problems. Diophantine geometry – Local ground fields. msc | Differential geometry – Global differential geometry – Methods of Riemannian geometry, including PDE methods; curvature restrictions. msc Classification: LCC QA641 .B774 2017 | DDC 516.3/6–dc23 LC record available at https://lccn.loc.gov/2016056302 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. 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Visit the AMS home page at http://www.ams.org/ 10987654321 222120191817 Contents Preface vii Acknowledgments x Introduction 1 0.1. Outline of the theory 1 0.2. Comparison with other theories 22 Chapter 1. Algebraic background 27 1.1. Algebra 27 1.2. Algebraic geometry 32 1.3. Superalgebra 33 Chapter 2. Classical differential geometry revisited 39 2.1. Connections in principal bundles and curvature 39 2.2. Lie algebra and classical groups 53 2.3. Involutions and symmetric spaces 58 2.4. Logarithmic derivative and differential Galois groups 63 2.5. Chern connections: the symmetric/anti-symmetric case 64 2.6. Chern connections: the hermitian case 70 2.7. Levi-Civit`a connection and Fedosov connection 72 2.8. Locally symmetric connections 78 2.9. Ehresmann connections attached to inner involutions 79 2.10. Connections in vector bundles 80 2.11. Lax connections 82 2.12. Hamiltonian connections 85 2.13. Cartan connection 92 2.14. Weierstrass and Riccati connections 94 2.15. Differential groups: Cassidy and Painlev´e95 Chapter 3. Arithmetic differential geometry: generalities 99 3.1. Global connections and their curvature 99 3.2. Adelic connections 112 3.3. Semiglobal connections and their curvature; Galois connections 115 3.4. Curvature via analytic continuation between primes 119 3.5. Curvature via algebraization by correspondences 125 3.6. Arithmetic jet spaces and the Cartan connection 138 3.7. Arithmetic Lie algebras and arithmetic logarithmic derivative 147 3.8. Compatibility with translations and involutions 152 3.9. Arithmetic Lie brackets and exponential 159 3.10. Hamiltonian formalism and Painlev´e 161 v vi CONTENTS 3.11. p-adic connections on curves: Weierstrass and Riccati 166 Chapter 4. Arithmetic differential geometry: the case of GLn 169 4.1. Arithmetic logarithmic derivative and Ehresmann connections 169 4.2. Existence of Chern connections 178 4.3. Existence of Levi-Civit`a connections 193 4.4. Existence/non-existence of Fedosov connections 198 4.5. Existence/non-existence of Lax-type connections 202 4.6. Existence of special linear connections 214 4.7. Existence of Euler connections 215 4.8. Curvature formalism and gauge action on GLn 218 4.9. Non-existence of classical δ-cocycles on GLn 228 4.10. Non-existence of δ-subgroups of simple groups 236 4.11. Non-existence of invariant adelic connections on GLn 242 Chapter 5. Curvature and Galois groups of Ehresmann connections 245 5.1. Gauge and curvature formulas 245 5.2. Existence, uniqueness, and rationality of solutions 246 5.3. Galois groups: generalities 252 5.4. Galois groups: the generic case 259 Chapter 6. Curvature of Chern connections 267 6.1. Analytic continuation along tori 267 6.2. Non-vanishing/vanishing of curvature via analytic continuation 279 6.3. Convergence estimates 299 6.4. The cases n =1andm = 1 301 6.5. Non-vanishing/vanishing of curvature via correspondences 303 Chapter 7. Curvature of Levi-Civit`a connections 317 7.1. The case m = 1: non-vanishing of curvature mod p 317 7.2. Analytic continuation along the identity 320 7.3. Mixed curvature 323 Chapter 8. Curvature of Lax connections 325 8.1. Analytic continuation along torsion points 325 8.2. Non-vanishing/vanishing of curvature 329 Chapter 9. Open problems 333 9.1. Unifying holQ and ΓQ 333 9.2. Unifying “∂/∂p”and“∂/∂ζp∞ ” 334 9.3. Unifying Sh and GLn 335 9.4. Further concepts and computations 336 9.5. What lies at infinity? 337 Bibliography 339 Index 343 Preface Aim of the book. The aim of this book is to introduce and develop an arithmetic analogue of classical differential geometry; this analogue will be referred to as arithmetic differential geometry. In this new geometry the ring of integers Z will play the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold will be played by the prime numbers p ∈ Z. The role of partial derivatives of functions with respect to the coordinates n−np ∈ Z ∈ Z will be played by the Fermat quotients, δpn := p , of integers n with respect to the primes p. The role of metrics (respectively 2-forms) will be played by symmetric (respectively anti-symmetric) matrices with coefficients in Z.Theroleof connections (respectively curvature) attached to metrics or 2-forms will be played by certain adelic (respectively global) objects attached to matrices as above. One of the main conclusions of our theory will be that (the “manifold” corresponding to) Z is “intrinsically curved;” this curvature of Z (and higher versions of it) will be encoded into a Q-Lie algebra holQ, which we refer to as the holonomy algebra, and the study of this algebra is, essentially, the main task of the theory. Needless to say, arithmetic differential geometry is still in its infancy. However, its foundations, which we present here, seem to form a solid platform upon which one could further build. Indeed, the main differential geometric concepts of this theory turn out to be related to classical number theoretic concepts (e.g., Christoffel symbols are related to Legendre symbols); existence and uniqueness results for the main objects (such as the arithmetic analogues of Ehresmann, Chern, Levi-Civit`a, and Lax connections) are being proved; the problem of defining curvature (which in arithmetic turns out to be non-trivial) is solved in some important cases (via our method of analytic continuation between primes and, alternatively, via alge- braization by correspondences); and some basic vanishing/non-vanishing theorems are being proved for various types of curvature. It is hoped that all of the above will create a momentum for further investigation and further discovery. Immediate context. A starting point for this circle of ideas can be found in our paper [23] where we showed how to construct arithmetic analogues of the classical jet spaces of Lie and Cartan; these new spaces were referred to as arithmetic jet spaces. The main idea, in this construction, was to replace classical derivation operators acting on functions with Fermat quotient operators acting on numbers and to develop an arithmetic differential calculus that parallels classical calculus. There were two directions of further development: one towards a theory of arithmetic differential equations and another one towards an arithmetic differential geometry. Atheoryofarithmetic differential equations was developed in a series of papers [23]-[42], [7] and was partly summarized in our monograph [35](cf.alsothe survey papers [43, 102]); this theory led to a series of applications to invariant vii viii PREFACE theory [27, 7, 28, 35], congruences between modular forms [27, 37, 38], and Diophantine geometry of Abelian and Shimura varieties [24, 36]. On the other hand an arithmetic differential geometry was developed in a series of papers [42]- [47], [8]; the present book follows, and further develops, the theory in this latter series of papers. We should note that our book [35]onarithmetic differential equations and the present book on arithmetic differential geometry, although both based on the same conceptual framework introduced in [23], are concerned with rather different objects. In particular the two books are independent of each other and the over- lap between them is minimal. Indeed the book [35] was mainly concerned with arithmetic differential calculus on Abelian and Shimura varieties.