Preface The main purpose of this research monograph is to develop an arithmetic ana- logue of the theory of ordinary differential equations. In our arithmetic theory the "time variable" t is replaced by a fixed prime integer p. Smooth real functions, t i— • x(t), are replaced by integer numbers a G Z or, more generally, by integers in various (completions of) number fields. The derivative operator on functions, *(*)~ f (*), is replaced by a "Fermat quotient operator" 5 which, on integer numbers, acts as * : Z - Z , a-aP a i— da := . P Smooth manifolds (configuration spaces) are replaced by algebraic varieties defined over number fields. Jet spaces (higher order phase spaces) of manifolds are replaced by what can be called "arithmetic jet spaces" which we construct using 5 in place of d/dt. Usual differential equations (viewed as functions on usual jet spaces) are replaced by "arithmetic differential equations" (defined as functions on our "arith- metic jet spaces"). Differential equations (Lagrangians) that are invariant under certain group actions on the configuration space are replaced by "arithmetic differ- ential equations" that are invariant under the action of various correspondences on our varieties. As our main application we will use the above invariant "arithmetic differen- tial equations" to construct new quotient spaces that "do not exist" in algebraic geometry. To explain this we start with the remark that (categorical) quotients of algebraic curves by correspondences that possess infinite orbits reduce to a point in algebraic geometry. In order to address the above basic pathology we propose to "enlarge" algebraic geometry by replacing its algebraic equations with our more general arithmetic differential equations. The resulting new geometry is referred to as S—geometry. It then turns out that certain quotients that reduce to a point in algebraic geometry become interesting objects in S—geometry this is because there are more invariant "arithmetic differential equations" than invariant algebraic equa- tions. Here are 3 classes of examples for which this strategy works: 1) Spherical case. Quotients of the projective line P 1 by actions of certain finitely generated groups (such as SL2(Z)) 2) Flat case. Quotients of P 1 by actions of postcritically finite maps P 1 — * P 1 with (orbifold) Euler characteristic zero 3) Hyperbolic case. Quotients of modular or Shimura curves (e.g. of P 1 ) by actions of Hecke correspondences.

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