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Alexandru Buium

Publications and source records attributed to Alexandru Buium.

31 records · Page 2Linked to original sources
arXiv2014

Transcendental numbers as solutions to arithmetic differential equations

Arithmetic differential equations are analogues of algebraic differential equations in which derivative operators acting on functions are replaced by Fermat quotient operators acting on numbers. Now, various remarkable transcendental functions are solutions to algebraic differential equations; in this note we show that, in a similar way, some remarkable transcendental numbers (including certain "periods") are solutions to arithmetic differential equations. Inspired by a recent paper of Manin, we then speculate on the possibility of understanding the algebraic relations among periods via Galois groups of arithmetic differential equations.

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arXiv2013

Arithmetic Differential Equations of Painleve' VI Type

Using the description of Paileve' VI family of differential equations in terms of a universal elliptic curve, going back to R. Fuchs (cf. [Ma96]), we translate it into the realm of Arithmetic Differential Equations (cf. [Bu05]), where the role of derivative "in the $p$--adic direction" is played by a version of Fermat quotient

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arXiv2013

Differential calculus with integers

Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main motivations, constructions, results, applications, and open problems of the theory.

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arXiv2013

Arithmetic differential equations on $GL_n$, I: differential cocycles

The theory of differential equations has an arithmetic analogue in which derivatives are replaced by Fermat quotients. One can then ask what is the arithmetic analogue of a linear differential equation. The study of usual linear differential equations is the same as the study of the differential cocycle from $GL_n$ into its Lie algebra given by the logarithmic derivative. However we prove here that there are no such cocycles in the context of arithmetic differential equations. In sequels of this paper we will remedy the situation by introducing arithmetic analogues of Lie algebras and a skew version of differential cocycles; this will lead to a theory of linear arithmetic differential equations.

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arXiv2011

p-jets Of p-isogenies

p-jets of finite flat maps of schemes are generally neither finite nor flat. This phenomenon can be seen already in the case of p-isogenies of group schemes. However, for p-divisible groups, this pathology tends to disappear mod p "in the limit". We illustrate this in the case of the p-divisible groups of elliptic curves and of formal groups of finite height; in the case of ordinary elliptic curves the structure in the limit depends on the Serre-Tate parameter.

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arXiv2009

Differential forms on arithmetic jet spaces

We study derivations and differential forms on the arithmetic jet spaces of smooth schemes, relative to several primes. As applications we give a new interpretation of arithmetic Laplacians and we discuss the de Rham cohomology of some specific arithmetic jet spaces.

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arXiv2008

Arithmetic Laplacians

We develop an arithmetic analogue of elliptic partial differential equations. The role of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat quotient operators subjected to certain commutation relations. This leads to arithmetic linear partial differential equations on algebraic groups that are analogues of certain operators in analysis constructed from Laplacians. We classify all such equations on one dimensional groups, and analyze their spaces of solutions.

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arXiv2008

Arithmetic partial differential equations, II: modular curves

We classify ``arithmetic convection equations'' on modular curves, and describe their space of solutions. Certain of these solutions involve the Fourier expansions of the Eisenstein modular forms of weight 4 and 6, while others involve the Serre-Tate expansions of the same modular forms; in this sense, our arithmetic convection equations can be seen as "unifying" the two types of expansions. The theory can be generalized to one of ``arithmetic heat equations'' on modular curves, but we prove that modular curves do not carry ``arithmetic wave equations.'' Finally, we prove an instability result for families of arithmetic heat equations converging to an arithmetic convection equation.

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arXiv2007

Relations among modular points on elliptic curves

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.

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arXiv2006

Differential eigenforms

The aim of this paper is to show how differential characters of Abelian varieties can be used to construct differential modular forms of weight 0 and order 2 which are eigenvectors of Hecke operators. These differential modular forms have "essentially the same" eigenvalues as certain classical complex eigenforms of weight 2.

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arXiv2006

Arithmetic partial differential equations

We develop an arithmetic analogue of linear partial differential equations in two independent ``space-time'' variables. The spatial derivative is a Fermat quotient operator, while the time derivative is the usual derivation. This allows us to ``flow'' integers or, more generally, points on algebraic groups with coordinates in rings with arithmetic flavor. In particular, we show that elliptic curves have certain canonical ``flows'' on them that are the arithmetic analogues of the heat and wave equations. The same is true for the additive and the multiplicative group.

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arXiv2005

Complex dynamics and invariant forms mod p

Complex dynamical systems on the Riemann sphere do not possess ``invariant forms''. However there exist non-trivial examples of dynamical systems, defined over number fields, satisfying the property that their reduction modulo $\wp$ possesses ``invariant forms'' for all but finitely many places $\wp$. The paper completely characterizes those dynamical systems possessing the latter property.

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