Volume 5 (2005), Number 4. Abstracts

Y. Bugeaud and M. Laurent. On Exponents of Homogeneous and Inhomogeneous Diophantine Approximation [PDF]

In Diophantine Approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the inhomogeneous exponent of approximation to a generic point in Rⁿ by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent associated to the system of dual linear forms.

Keywords. Diophantine approximation, measures of homogeneous and inhomogeneous approximation, uniform exponents, spectra

2000 Mathematics Subject Classification. 11J20, 11J13, 11J82

A. Garcia and H. Stichtenoth. Some Artin–Schreier Towers Are Easy [PDF]

Towers of function fields (resp., of algebraic curves) with positive limit provide examples of curves with large genus having many rational points over a finite field. It is in general a difficult task to calculate the genus of a wild tower. In this paper, we present a method for calculating the genus of certain Artin–Schreier towers. As an illustration of our method, we obtain a very simple and unified proof for the limits of some towers that attain the Drinfeld–Vlăduţ bound or the Zink bound.

Keywords. Tower of function fields, finite field, Artin–Schreier extension, genus, rational place, limit of towers

2000 Mathematics Subject Classification. 11R58, 14H05, 11D59, 14G15

G. van der Geer and T. Katsura. Note on Tautological Classes of Moduli of K3 Surfaces [PDF]

In this note, we prove some cycle class relations on moduli of K3 surfaces.

Keywords. K3 surface, moduli space, tautological class

2000 Mathematics Subject Classification. 14K10

A. Glutsyuk. Upper Bounds of Topology of Complex Polynomials in Two Variables [PDF]

It is well known that the roots and critical points of a complex polynomial in one variable admit an explicit upper bound in terms of the highest coefficient and the maximal modulus of critical values. In the present paper, we prove similar bounds for generic complex polynomials in two variables. In particular, we give an upper bound of the radii of bidisks that contain all the nontrivial topology of level curves. These results were used in studying the restricted version of Hilbert's 16th problem in the joint paper of the author with Yu. S. Ilyashenko.

Keywords. Complex polynomial in two variables, level curve, bidisk containing the topology of level curve, critical point, critical value, vanishing cycle, abelian integral, period determinant

2000 Mathematics Subject Classification. 14D05, 30C15, 30C10

G. Lachaud. Ramanujan Modular Forms and the Klein Quartic [PDF]

In one of his notebooks, Ramanujan gave some algebraic relations between three theta functions of order 7. We describe the automorphic character of a vector-valued mapping constructed from these theta series. This provides a systematic way to establish old and new identities on modular forms for the congruence subgroup of level 7, above all, a parametrization of the Klein quartic. From a historical point of view, this shows that Ramanujan discovered the main properties of this curve with his own means. As an application, we introduce an L-series in four different ways, generating the number of points of the Klein quartic over finite fields. From this, we derive the structure of the Jacobian of a suitable form of the Klein quartic over finite fields and some congruence properties on the number of its points.

Keywords. Ramanujan, Klein quartic, modular form, theta series, curve over a finite field, L-series, Jacobian, zeta function

2000 Mathematics Subject Classification. 11G25, 11M38

G. Lachaud. The Klein Quartic as a Cyclic Group Generator [PDF]

Let k be a field, and let a, b, c be three elements in k^×. The nonsingular projective plane curve X defined over k with equation

a x³ y + b y³ z + c z³ x = 0

has genus 3 and is reduced to the familiar Klein quartic when a = b = c = 1. The Jacobian J_X of X is a three-dimensional abelian variety, defined over k as well. The aim of this article is to give some formulas for the number of points of the group J_X(k) of rational points of J_X if k = F_q is a finite field.

We assume that the full group of seventh roots of unity is contained in k; this amounts to saying that q ≡ 1 (mod 7). If q is a prime number and the coefficients a, b, c are appropriately chosen, we noticed that the number of points of the group J_X(k) is prime in a significant number of occurences. This provides cyclic groups which seem to be accurate for cryptographic applications.

Keywords. Jacobi sum, Faddeev curve, Klein quartic, curve over a finite field, Jacobian, zeta function

2000 Mathematics Subject Classification. 11G25

Yu. Manin. Iterated Shimura Integrals [PDF]

In this paper, I continue the study of iterated integrals of modular forms and noncommutative modular symbols for Γ ⊂ SL(2,Z) started in my paper math.NT/0502576. The main new results involve a description of the iterated Shimura cohomology and the image of the iterated Shimura cocycle class inside it. The concluding section of the paper contains a concise review of the classical modular symbols for SL(2) and a discussion of open problems.

Keywords. Iterated integrals, modular forms, modular symbols, multiple zeta values

2000 Mathematics Subject Classification. Primary 11F67, 11M41; Secondary 11G55

A. Panchishkin. The Maass–Shimura Differential Operators and Congruences between Arithmetical Siegel Modular Forms [PDF]

We extend further a new method for constructing p-adic L-functions associated with modular forms. For this purpose, we study congruences between nearly holomorphic Siegel modular forms using an explicit action of the Maass–Shimura arithmetical differential operators. We view nearly holomorphic arithmetical Siegel modular forms as certain formal expansions over A = C_p. The important property of these arithmetical differential operators is their commutation with the Hecke operators (under an appropriate normalization).

We show in Section 5 that a fine combinatorial structure of the action of these arithmetical differential operators on the A-module M = M(A) of nearly holomorphic Siegel modular forms produces new congruences between nearly holomorphic Siegel modular forms inside a formal q-expansion ring of the form A[[q^B]][R_ij] where B = B_m = {ξ= ^tξ ∈ M_m(Q): ξ ≥ 0, ξ half-integral} is the semi-group, important for the theory of Siegel modular forms), and the nearly holomorphic parameters (R_ij) = R correspond to the matrix R = (4π Im(z))₋₁ in the Siegel modular case. These congruences produce various p-adic L-functions attached to modular forms using a general method of canonical projection. We give in Theorem 5.1 a general construction of h-admissible measures attached to sequences of special modular distributions. Our construction generalizes at the same time the following two cases:

(1) the standard L-function of a Siegel cusp eigenform, [20, Chapter 4];

(2) the Mellin transform of an elliptic cusp eigenform of weight k ≥ 2, see [58], [55].

Keywords. Siegel modular forms, Hecke operators, Shimura differential operators, Siegel–Eisenstein series

2000 Mathematics Subject Classification. 11F60, 11F67, 11F85, 11F46

S. Rybakov. Zeta Functions of Conic Bundles and Del Pezzo Surfaces of Degree 4 over Finite Fields [PDF]

First of all, we construct a conic bundle with a prescribed zeta function. This is a key step to classify Del Pezzo surfaces of degree 4 over a finite field. In particular, we see that the zeta function determines the combinatorics of a Del Pezzo surface.

Keywords. Zeta function, conic bundle, surface over a finite field, Del Pezzo surface

2000 Mathematics Subject Classification. 11R58, 14G15, 11M38, 14G05

A. Rybko and S. Shlosman. Poisson Hypothesis for Information Networks. II [PDF]

This is the second part of our paper. We study the Poisson Hypothesis, which is a device to analyze approximately the behavior of large queuing networks. We prove it in some simple limiting cases. We show in particular that the corresponding dynamical system, defined by the non-linear Markov process, has a line of fixed points which are global attractors. To do this we derive the corresponding non-linear equation and we explore its self-averaging properties. We also argue that in cases of heavy-tail service times the PH can be violated.

Keywords. Mean-field models, server, waiting time, phase transition, limit theorem, self-averaging property, attractor

2000 Mathematics Subject Classification. Primary: 82C20; Secondary: 60J25

A. Zykin. The Brauer–Siegel and Tsfasman–Vlăduţ Theorems for Almost Normal Extensions of Number Fields [PDF]

The classical Brauer–Siegel theorem states that if k runs through the sequence of normal extensions of Q such that n_k/log|D_k| → 0, then log h_kR_k/log \sqrt{|D_k|} → 1. First, in this paper we obtain the generalization of the Brauer–Siegel and Tsfasman–Vlăduţ theorems to the case of almost normal number fields. Second, using the approach of Hajir and Maire, we construct several new examples concerning the Brauer–Siegel ratio in asymptotically good towers of number fields. These examples give smaller values of the Brauer–Siegel ratio than those given by Tsfasman and Vlăduţ.

Keywords. Global field, Brauer–Siegel theorem, asymptotically good tower, asymptotically bad tower

2000 Mathematics Subject Classification. 11R29, 11R42

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