Alexandru Buium

Arithmetic differential equations

Main concepts and results: Preliminaries from algebraic geometry Outline of

Differential algebraic groups of finite dimension

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Differential function fields and moduli of algebraic varieties

-geometry General theory: Global theory Local theory Birational theory Applications: Spherical correspondences Flat correspondences Hyperbolic correspondences List of results Bibliography Index.

Independence of points on elliptic curves arising from special points on modular and Shimura curves, II: local results

Terminology and conventions.- First properties.- Affine D-group schemes.- Commutative algebraic D-groups.- General algebraic D-groups.- Applications to differential algebraic groups.

Geometry of Fermat adeles

Preliminaries.- Differential descent theory.- Normality in differential galois theory.- Complements.

Differential orbit spaces of discrete dynamical systems

In the predecessor to this article, we used global equidistribution theorems to prove that given a correspondence between a modular curve and an elliptic curve A , the intersection of any finite-rank subgroup of A with the set of CM-points of A is finite. In this article we apply local methods, involving the theory of arithmetic differential equations, to prove quantitative versions of a similar statement. The new methods apply also to certain infinite-rank subgroups, as well as to the situation where the set of CM-points is replaced by certain isogeny classes of points on the modular curve. Finally, we prove Shimura-curve analogues of these results.

Independence of points on elliptic curves arising from special points on modular and Shimura curves, I: Global results

If L(a, s) := Σ n c(n,a)n -s is a family of geometric L-functions depending on a parameter a, then the function (p, a) → c(p, a), where p runs through the set of prime integers, is not a rational function and hence is not a function belonging to algebraic geometry. The aim of the paper is to show that if one enlarges algebraic geometry by adjoining a Fermat quotient operation, then the functions c(p, a) become functions in the enlarged geometry at least for L-functions of curves and Abelian varieties.

Differential modular forms on Shimura curves, II: Serre operators

Abstract Regular self maps of algebraic varieties do not admit, in general, regular functions as semi-invariants. In this paper we show that if one replaces regular functions by δ-functions (in the sense of [3]) then one can get non-trivial semi-invariants (and one can often compute all of them) for a remarkable class of regular self maps of the projective line.

Arithmetic and Geometry: Differential calculus with integers

Given a correspondence between a modular curve and an elliptic curve A we study the group of relations among the CM points of A. In particular we prove that the intersection of any finite rank subgroup of A with the set of CM points of A is finite. We also prove a local version of this global result with an effective bound valid also for certain infinite rank subgroups. We deduce the local result from a “reciprocity” theorem for CL (canonical lift) points on A. Furthermore we prove similar global and local results for intersections between subgroups of A and isogeny classes in A. Finally we prove Shimura curve analogues and, in some cases, higher-dimensional versions of these results.

Differential characters on curves

One of the main results announced in Part I (Compositio Math. 139 (2003), 197–237) is proved. The main technique consists in developing a coordinate free version of some of the theory of Serre operators on differential modular forms introduced by Barcau.