Arithmetic Differential Equations

by Buium, Alexandru

• ISBN13: 9780821838624
• ISBN10: 0821838628
• Format: Hardcover
• Copyright: 2005-09-28
• Publisher: Amer Mathematical Society

Summary

This monograph contains exciting original mathematics that will inspire new directions of research in algebraic geometry. Developed here is an arithmetic analog of the theory of ordinary differential equations, where functions are replaced by integer numbers, the derivative operator is replaced by a "Fermat quotient operator", and differential equations (viewed as functions on jet spaces) are replaced by "arithmetic differential equations". The main application of this theory concerns the construction and study of quotients of algebraic curves by correspondence with infinite orbits. Any such quotient usually reduces to a point in algebraic geometry. But many of the above quotients cease to be trivial (and become quite interesting) if one enlarges algebraic geometry by using arithmetic differential equations in place of algebraic equations. This book, in part, follows a series of papers written by the author. However, a substantial amount of the material has never been published before. For most of the book, the only prerequisites are the basic facts of algebraic geometry and algebraic number theory. It is suitable for graduate students and researchers interested in algebraic geometry.

Table of Contents

Preface	vii
Introduction	xi
Motivation and strategy	xi
Rough outline of the theory	xiv
Comparison with other theories	xix
Part 1. Main concepts and results	1	(68)
Preliminaries from algebraic geometry	3	(28)
Algebro-geometric terminology	3	(8)
Categorical quotients in algebraic geometry	11	(4)
Analytic uniformization and critical finiteness	15	(16)
Outline of δ-geometry	31	(38)
Main concepts of δ-geometry	31	(24)
Main conjectures	55	(1)
Main results: a sample	56	(5)
Appendix: Axiomatic characterization of δ--geometries	61	(8)
Part 2. General theory	69	(90)
Global theory	71	(36)
p--jet spaces of schemes	71	(4)
Behavior with respect to etale maps	75	(3)
Link between p--jet spaces and δ-functions	78	(4)
Galois covers	82	(3)
δ--base loci versus postcritical loci	85	(3)
δ--tangent maps and δ--differentials of δ--functions	88	(19)
Local theory	107	(34)
Local analogue of the main conjectures	107	(12)
Bounding the rank of the module of δ--invariants	119	(2)
δ--invariants for the formal multiplicative group	121	(2)
p--jets of formal group laws	123	(3)
Local versus global picture: fixed points and cycles	126	(6)
Some general global converse results	132	(9)
Birational theory	141	(18)
The basic graded rings	141	(4)
δ--Galois groups	145	(2)
δ--invariants for subgroups of PGL2(Zp)	147	(12)
Part 3. Applications	159	(140)
Spherical correspondences	161	(24)
Spherical correspondences over R and their cycles	161	(4)
δ--sections of bundles on the projective line	165	(2)
Case Γ trivial: δ--invariants and δ--cohomology	167	(6)
Case Γ non-trivial: δ--invariants, δ--cohomology and δ--fiber	173	(5)
Case <Γ, δ> solvable	178	(3)
A converse theorem: biquadratic correspondences	181	(4)
Flat correspondences	185	(42)
Flat correspondences over R: δ--line bundles and cycles	185	(5)
δ--characters	190	(15)
δ--invariants	205	(5)
δ--base loci	210	(2)
δ--cohomology	212	(9)
The relative generic δ--fiber	221	(2)
Converse theorems: quadratic dynamical systems	223	(4)
Hyperbolic correspondences	227	(72)
Review of Abelian schemes and their crystals	228	(12)
Hecke correspondences over R: δ--line bundles and cycles	240	(11)
δ--Serre-Tate expansion maps and δ--Serre operators	251	(9)
Constructions of δ--invariants	260	(15)
δ--invariants in the ordinary case	275	(4)
δ--invariants in the non-ordinary case	279	(6)
δ--cohomology	285	(3)
The relative generic δ--fiber	288	(4)
Hecke correspondences with a regular Hecke n--cycle	292	(3)
A converse theorem: non-rational hyperbolic uniformization	295	(4)
List of Results	299	(2)
Bibliography	301	(6)
Index	307