## Arithmetic Differential EquationsThis 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. |

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adic affine analytic uniformization arithmetic differential assertion Assume automorphism basis canonical bundle categorical quotient claim coefficients commutative conjecture consider Corollary cycles defined DEFINITION deg(w degree denote derivation differential algebra element elliptic curve endomorphism Equation equivalent exists false elliptic curve fibers finite type flat formal group formal schemes formula functions Galois group global Hecke correspondence Hecke fixed point hence hyperbolic implies induced infinite injective integer invariant invertible irreducible isogeny isomorphism jet spaces Jr(X Lemma lift of Frobenius line bundle linear modular curves modular forms module morphism multiplier non-zero Note number field Or(X ord(w p—adic p—ring particular polynomial postcritically finite prime principal homogeneous space PROOF Proposition prove Recall reduction mod regular Hecke REMARK respectively ring homomorphism root of unity sequence Serre-Tate Shimura curves smooth scheme Spec spherical structure subgroup Theorem theory trivial tuple type and smooth Zariski