Connecting interpolation and multiplicity estimates in commutative algebraic groups

The Siegel-Shidlovskii method is one of the main methods in the theory of transcendental numbers. By using this method, it is possible to establish the transcendency and algebraic independence of the values of entire functions belonging to a certain class (so-called E-functions) provided that these functions are algebraically independent over C(z). There are many examples of E-functions used in mathematics: exp(z), sinz, cosz, sinhz, coshz, Bessel functions, Kummer functions, incomplete gamma-function, generalized hypergeometric functions, and some other special functions. By using the Siegel-Shidlovskii method, it is possible to obtain the lower estimates for the moduli of polynomials from the values of E-functions. These estimates are called algebraic independence measures of the numbers. They serve as the quantitative characteristics of algebraic independence. The problem of obtaining the estimates of algebraic independence measures has long been dealt with by many researchers. The first estimates of measures for the values of exp(z) were obtained by E. Borel and K. Mahler, and for the values of Bessel functions such estimates were obtained by C. Siegel. Estimates of algebraic independence measures for the values of general E-functions were established by S. Lang, A.I. Galochkin, A.B. Shidlovskii, and Yu.V. Nesterenko. The constant appearing in the estimate of a measure is called effective if it can be expressed in terms of the parameters characterizing the considered functions and points at which their values are taken. An estimate of a measure is considered to be effective if it contains only effective constants. Generally speaking, the algebraic independence condition of functions is not sufficient for obtaining effective estimates of algebraic independence measures. Some additional conditions must be satisfied for this. The case in which the main totality of functions is algebraically dependent over C(z) is the most complicated one. In a number of the author’s works, the estimates of algebraic independence measures of the values of E-functions were improved. After 1995, no papers have been published on this topic, although it still remains of issue.

We prove certain results pertaining to some isomorphism properties of commutative modular group algebras and briefly review a paper by pointing out some obvious mistakes and essential incorrectness.

Families of translations of commutative algebraic groups

In this article we develop an important tool in transcendental number theory. More precisely, we study multiplicity estimates (or multiplicity lemmas) for analytic functions. Our main theorem reduces multiplicity estimates at zero to the study of ideals in polynomial ring stable under an appropriate map. In particular, in the case of algebraic morphisms this result gives a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. Specialized to the case of differential operators this theorem improves Nesterenko's conditional result on solutions of systems of differential equations. We also deduce an analog of Nesterenko's theorem for Mahler's functions and for solutions of q-difference equations. Further, analyzing stable ideals we prove the unconditional optimal result in the case of linear functional systems of generalized Mahler's type. The latter result generalizes a famous theorem of Nishioka (1986) previously conjectured by Mahler (1969). This new multiplicity estimate allows to prove new results on algebraic independence and on measures of algebraic independence, as done in Zorin (2010 and 2011).

Applications to cryptography of twisting commutative algebraic groups

Let G be an abelian group and let R be a commutative ring with identity. Denote by R t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ℬ(R) and ℬ(R t G) the nil radicals of R and R t G, respectively, by G p the p-component of G and by G 0 the torsion subgroup of G. We prove that: i. If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ℬ(R) = 0, then there exists a twisted group algebra R t 1 (G/G p ) such that R t G/ℬ(R t G) ≅ R t 1 (G/G p ) as R-algebras; ii. If R is a ring of prime characterisitic p and R* is p-divisible, then ℬ(R t G) = 0 if and only if ℬ(R) = 0 and G p = 1; and iii. If B(R) = 0, the orders of the elements of G 0 are not zero divisors in R, H is any group and the commutative twisted group algebra R t G is isomorphic as R-algebra to some twisted group algebra R t 1 H, then R t G 0 ≅ R t 1 H 0 as R-algebras.

The subject of algebraic groups has had a rapid development in recent years. Leaving aside the late research by many people on the Albanese and Picard variety, it has received much substance and impetus from the work of Severi on commutative algebraic groups over the complex number field, that of Kolchin, Chevalley, and Borel on algebraic groups of matrices, and especially Weil's research on abelian varieties and algebraic transformation spaces. The main purpose of the present paper is to give a more or less systematic account of a large part of what is now known about general algebraic groups, which may be abelian varieties, algebraic groups of matrices, or actually of neither of these types.

This exploration of the relation between periods and transcendental numbers brings Baker's theory of linear forms in logarithms into its most general framework, the theory of 1-motives. Written by leading experts in the field, it contains original results and finalises the theory of linear relations of 1-periods, answering long-standing questions in transcendence theory. It provides a complete exposition of the new theory for researchers, but also serves as an introduction to transcendence for graduate students and newcomers. It begins with foundational material, including a review of the theory of commutative algebraic groups and the analytic subgroup theorem as well as the basics of singular homology and de Rham cohomology. Part II addresses periods of 1-motives, linking back to classical examples like the transcendence of π, before the authors turn to periods of algebraic varieties in Part III. Finally, Part IV aims at a dimension formula for the space of periods of a 1-motive in terms of its data.

This volume consists of a collection of papers devoted primarily to transcendental number theory and diophantine approximations written by the author. Most of the materials included in this volume are English translations of the author's Russian manuscripts, extensively rewritten and brought entirely up to date. These papers and other papers included in this volume were available to specialists in manuscript form, but this is the first time that they have been collected and published.Though the earlier papers have been preserved in the form in which they were prepared initially, the volume is organized in such a way as to reflect recent progress and to allow readers to follow recent developments in the field. As an introductory guide to the volume, the author included an expanded and updated text of his invited address on his work on the theory of transcendental numbers to the 1978 International Congress of Mathematicians in Helsinki. The appendix contains a paper on the extremality of certain multidimensional manifolds prepared by A. I. Vinogradov and the author in 1976. Chudnovsky received a MacArthur Foundation Fellowship in 1981.

We establish new measures of linear independence of logarithms on a commutative algebraic group. Let G be a connected commutative algebraic group over Q−− and let tG be the tangent space at the origin. We consider a vector u∈tG⊗Q−C such that its image by the exponential map of the Lie group G(C) is an algebraic point p∈G(Q−−). Let V be a hyperplane in tG. We obtain lower bounds for the distance d(u,V) between u and V⊗Q−C in the rational case, where V=tH is the tangent space at the origin of an algebraic connected subgroup of G. These lower bounds are the best currently known in terms of the height h(p) of p. They generalize measures of linear forms in logarithms previously obtained by Gaudron. Our approach is based on new arguments which allow us to exclude the so-called periodic case in the demonstration, by revisiting previous work of Bertrand and Philippon. Our proofs also rely on tools from Bost’s slope theory of hermitian vector bundles. Moreover, we present ultrametric analogues of our results, and we deal with the case where V=tH is a linear subspace of any dimension.

Existence of abelian group code partitions

Siegel’s Lemma is concerned with finding a “small” nontrivial integer solution of a large system of homogeneous linear equations with integer coefficients, where the number of variables substantially exceeds the number of equations (for example, n equations and N variables with N ≥ 2n), and “small” means small in the maximum norm. Siegel’s Lemma is a clever application of the Pigeonhole Principle, and it is a pure existence argument. The basically combinatorial Siegel’s Lemma is a key tool in transcendental number theory and diophantine approximation. David Masser (a leading expert in transcendental number theory) asked the question whether or not the Siegel’s Lemma is best possible. Here we prove that the so-called “Third Version of Siegel’s Lemma” is best possible apart from an absolute constant factor. In other words, we show that no other argument can beat the Pigeonhole Principle proof of Siegel’s Lemma (apart from an absolute constant factor). To prove this, we combine a concentration inequality (i.e., Fourier analysis) with combinatorics.

This work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group, defined over the field of algebraic numbers $\overline{\mathbb{Q}}$ . Let G be such a group. Let W be a hyperplane of the tangent space at the origin of G, defined over $\overline{\mathbb{Q}}$ , and u be a complex point of this tangent space, such that the image of u by the exponential map of the Lie group G(ℂ) is an algebraic point. Then we obtain a lower bound for the distance between u and W⊗ℂ, which improves the results known before and which is, in particular, the best possible for the height of the hyperplane W. The proof rests on Baker’s method and Hirata’s reduction as well as a new arithmetic argument (Chudnovsky’s process of variable change) which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof.

Let k be a number field, let A be a commutative algebraic group defined over k and let n be a positive integer. We will prove that for almost all primes p ∈ k, the existence of a counterexample to the localglobal divisibility by p in A assures the existence of a counterexample to the local-global divisibility by p in A, for all positive integers s. In particular, we will prove the existence of counterexamples to the localglobal divisibility by 2 and 3, for every n ≥ 2, in the case when k = Q and A is an elliptic curve. Furthermore, we will show a method to find numerical examples. We will use this method to produce numerical counterexamples to the local-global divisibility by 2, for every n ≥ 2, in elliptic curves defined over Q.

Mesure d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif