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arXiv2004

Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik

In math.NT/0307308 we defined the irrationality base of an irrational number and, assuming a stronger hypothesis than the irrationality of Euler's constant, gave a conditional upper bound on its irrationality base. Here we develop the general theory of the irrationality exponent and base, giving formulas and bounds for them using continued fractions and the Fibonacci sequence. A theorem of Jarnik on Diophantine approximation yields numbers with prescribed irrationality measure. By another method we explicitly construct series with prescribed irrationality base. Many examples are given.

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arXiv2005

pi-adic approach of p-class group and unit group of p-cyclotomic fields

Let p > 2 be a prime. Let Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of Q(zeta) lying over p. This article aims to describe some pi-adic congruences characterizing the structure of the p-class group and of the unit group of the field Q(zeta). For the unit group, this article supplements the 1954 and 1956 papers of Denes on this topic. A complete summarizing of the results obtained follows in the Introduction section of the paper (pages 3 to 6). This new version of the article with the same title, submitted with reference math.NT/0407430 25 Jul 2004: - corrects several typing errors in the introduction and in the paper, - simplifies some proofs of pi-adic congruences connected to p-class group, - removes the section dealing of the singular group foreseen in an independant paper.

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arXiv2005

Monodromy calculatons of fourth order equations of Calabi-Yau type

This paper contains a preliminary study of the monodromy of certain fourth order differential equations, that were called of Calabi-Yau type in math.NT/0402386. Some of these equations can be interpreted as the Picard-Fuchs equations of a Calabi-Yau manifold with one complex modulus, which links up the observed integrality to the conjectured integrality of the Gopakumar-Vafa invariants. A natural question is if in the other cases such a geometrical interpretation is also possible. Our investigations of the monodromies are intended as a first step in answering this question. We use a numerical approach combined with some ideas from homological mirror symmetry to determine the monodromy for some further one-parameter models. Furthermore, we present a conjectural identification of the Picard-Fuchs equation for 5 new examples from Borcea's list and one constructed by Tonoli and conjecture the existence of some new Calabi-Yau three folds. The paper does not contain any theorems or proofs but is, we think, nevertheless of interest.

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arXiv2005

Supertopes

A perfect (Delaunay) ellipsoid is an ellipsoid in n-dimensional Euclidean space that does not contain integral points in its interior, but is uniquely defined by integral points that lie on its surface. A perfect Delaunay polytope with respect to a positive quadratic form f() is a polytope with integral vertices that is circumscribed by a perfect Delaunay ellipsoid with an equation whose quadratic part is f(). This document has been corrected on January 15, 2005. Note that it represents the state of the area as of the end of 2002. For recent research on perfect Delaunay polytopes see my recent preprint, with Erdahl and Ordine, math.NT/0408122 on ArXiv.org .

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arXiv2006

Sur l'effectivite du theoreme de Siegel et la conjecture abc

Nous montrons qu'un raffinement du théorème de Siegel sur les points entiers de courbes algébriques impliquerait la conjecture abc de Masser-Oesterlé. Nous formulons une hypothèse "Siegel uniforme" qui est une majoration de la hauteur des points S-entiers de la courbe, en termes du corps de rationalité et de l'ensemble de places S. La validité de l'hypothèse pour une quelconque courbe algébrique de caractéristique d'Euler-Poincaré strictement négative, impliquerait une version de la conjecture abc. Ceci étend aux points S-entiers des résultats précédents de L. Moret-Bailly, et est en quelque sorte, un énoncé réciproque de ce que nous avons montré dans math.NT/0408168, en suivant les idées proposées par N. Elkies. Le principal outil géométrique employé est un théorème de G.V. Belyi. Nous montrons aussi quelques versions inconditionnelles de ces énoncés : un résultat allant dans le sens de la conjecture abc, valable sur tout corps de nombres, ainsi que des bornes pour la hauteur des solutions en S-entiers de certaines équations diophantiennes classiques.

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arXiv2005

Schmidt's theorem, Hausdorff measures and Slicing

A Hausdorff measure version of W.M. Schmidt's inhomogeneous, linear forms theorem in metric number theory is established. The key ingredient is a `slicing' technique motivated by a standard result in geometric measure theory. In short, `slicing' together with the Mass Transference Principle [3] allows us to transfer Lebesgue measure theoretic statements for limsup sets associated with linear forms to Hausdorff measure theoretic statements. This extends the approach developed in [3] for simultaneous approximation. Furthermore, we establish a new Mass Transference Principle which incorporates both forms of approximation. As an application we obtain a complete metric theory for a `fully' non-linear Diophantine problem within the linear forms setup. [3] V. Beresnevich and S. Velani : A Mass Transference Principle and the Duffin--Schaeffer conjecture for Hausdorff measures, Pre-print (22pp): arkiv:math.NT/0401118. To appear: Annals of Math.

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arXiv2005

Evaluating Two Determinants

This article evaluates the determinants of two classes of special matrices, which are both from a number theory problem. Applications of the evaluated determinants can be found in [arXiv:math.NT/0509523]. Note that the two determinants are actually special cases of Theorems 20 and 23 in [arXiv:math.CO/9902004], respectively. Since this paper does not provide any new results, it will not be published anywhere.

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arXiv2006

Interpolation of numbers of Catalan type in a local field of positive characteristic

Let $k$ be a local field of positive characteristic. Let $L$ be a cocompact discrete subgroup of $k$. Let $U$ be an open compact subgroup of $k$. Let $\ell$, $u$ and $a$ be elements of $k$, with $a$ nonzero. We study the behavior of the product $\prod_{0\neq x\in (\ell+L)\cap a(u+U)}x$ as $a$ varies. Our main result, which blends harmonic analysis and local class field theory, provides local confirmation of a conjecture concerning global function fields recently made by the author (math.NT/0407535).

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arXiv2006

Twisted character of a small representation of PGL(4)

We compute by a purely local method the elliptic, twisted by transpose-inverse, character χ_πof the representation π=I_{(3,1)}(1_3) of PGL(4,F) normalizedly induced from the trivial representation of the maximal parabolic subgroup of type (3,1), where F is a p-adic field. Put C=(GL(2,F)xGL(2,F))'/F^x (F^x embeds diagonally, prime means equal determinants). It is a twisted elliptic endoscopic group of PGL(4). We deduce from the computation that χ_πis an unstable function: its value at one twisted regular elliptic conjugacy class with norm in C is minus its value at the other class within the twisted stable conjugacy class, and zero at the classes without norm in C. Moreover πis the unstable endoscopic lift of the trivial representation of C. Naturally, this computation plays a role in the theory of lifting from C (=``SO(4,F)'') and PGp(2,F) to PGL(4,F) using the trace formula. Our work develops a 4-dimensional analogue of the model of the small representation of PGL(3,F) introduced by the first author with Kazhdan in a 3-dimensional case, and it uses the classification of twisted stable and unstable regular conjugacy classes in PGL(4,F). It extends the local method of computation introduced by us in the 3-dimensional case. An extension math.NT/0606263 of our work here to apply to similar representations of GL(4,F) whose central character is nontrivial will appear in Int. J. Number Theory.

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arXiv2006

Interpolation of hypergeometric ratios in a global field of positive characteristic

In connection with each global field of positive characteristic we exhibit many examples of two-variable algebraic functions possessing properties consistent with a conjectural refinement of the Stark conjecture in the function field case recently proposed by the author (math.NT/0407535). Most notably, all examples are Coleman units. We obtain our results by studying rank one shtukas in which both zero and pole are generic, i.~e., shtukas not associated to any Drinfeld module.

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arXiv2007

The distribution of integers with at least two divisors in a short interval

Let H(x,y,z) be the number of integers $\le x$ with a divisor in (y,z] and let H_1(x,y,z) be the number of integers $\le x$ with exactly one such divisor. When y and z are close, it is expected that H_1(x,y,z) H(x,y,z), that is, an integer with a divisor in (y,z] usually has just one. We determine necessary and sufficient conditions on y and z so that H_1(x,y,z) H(x,y,z). In doing so, we answer an open question from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223.

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arXiv2013

Integers with a divisor in (y,2y]

We give a relatively short proof of one of the central cases of the main theorem from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223. Namely, we determine the order of magnitude of the number of integers <=x with a divisor in (y,2y]. The lower bound uses a different argument than that in the aforementioned paper. As a corollary, we deduce the order of magnitude for the number of distinct products in an N x N multiplication table.

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arXiv2007

Fast Jacobian group operations for C_{3,4} curves over a large finite field

Let C be an arbitrary smooth algebraic curve of genus g over a large finite field K. We revisit fast addition algorithms in the Jacobian of C due to Khuri-Makdisi (math.NT/0409209, to appear in Math. Comp.). The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |K| >> g, and which asymptotically require O(g^{2.376}) field operations using fast linear algebra, are shown to perform efficiently even for certain low genus curves. Specifically, we provide explicit formulae for performing the group law on Jacobians of C_{3,4} curves of genus 3. We show that, typically, the addition of two distinct elements in the Jacobian of a C_{3,4} curve requires 117 multiplications and 2 inversions in K, and an element can be doubled using 129 multiplications and 2 inversions in K. This represents an improvement of approximately 20% over previous methods.

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arXiv2008

On Primes in Quadratic Progressions

We verify the Hardy-Littlewood conjecture on primes in quadratic progressions on average. The results in the present paper significantly improve those of a previous paper of the authors(arXiv:math.NT/0605563).

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arXiv1997

Elementary background in elliptic curves

This paper gives additional background in algebraic geometry as an accompaniment to the article, ``Formal Groups, Elliptic Curves, and some Theorems of Couveignes'' [arXiv:math.NT/9708215]. Section 1 discusses the addition law on elliptic curves, and Sections 2 and 3 explain about function fields, uniformizers, and power series expansions with respect to a uniformizer.

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arXiv1998

Asymptotic sieve for primes

This article develops a new sieve method which by adding an additional axiom to the classical formulation breaks the well-known parity problem and allows one to detect primes in thin, interesting integer sequences. In the accompanying paper [math.NT/9811184] the practicality of the axiom is demonstrated by verifying it (and the other axioms) to produce primes in the sequence a^2+b^4 with the relevant asymptotics.

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