q-ENGEL SERIES EXPANSIONS AND SLATER'S IDENTITIES

Abstract

Dedicated to the memory of John Knopfmacher (1937–1999)

We describe the q-Engel series expansion for Laurent series discovered by John Knopfmacher and use this algorithm to shed new light on partition identities related to two entries from Slater's list. In our study Al-Salam/Ismail and Santos polynomials play a crucial r?ole.     

Practical numbers in Lucas sequences

Abstract

A practical number is a positive integer n such that all the positive integers m ≤ n can be written as a sum of distinct divisors of n. Let (u_n)_n≥0 be the Lucas sequence satisfying u₀ = 0, u₁ = 1, and u_n+2 = au_n+1 + bu_n for all integers n ≥ 0, where a and b are fixed nonzero integers. Assume a(b + 1) even and a² + 4b > 0. Also, let be the set of all positive integers n such that |u_n| is a practical number. Melfi proved that is infinite. We improve this result by showing that #(x) ? x/log x for all x ≥ 2, where the implied constant depends on a and b. We also pose some open questions regarding .     

Uniform dilations

Every sufficiently large finite setX in [0,1) has a dilationnX mod 1 with small maximal gap and even small discrepancy. We establish a sharp quantitative version of this principle, which puts into a broader perspective some classical results on the distribution of power residues. The proof is based on a second-moment argument which reduces the problem to an estimate on the number of edges in a certain graph. Cycles in this graph correspond to solutions of a simple Diophantine equation: The growth asymptotics of these solutions, which can be determined from properties of lattices in Euclidean space, yield the required estimate.N.A.-Research supported in part by a U.S.A.-Israel BSF grant.Y.P.-Partially supported by a Weizmann Postdoctoral Fellowship.     

LLL & ABC

This note is an observation that the LLL algorithm applied to prime powers can be used to find “good” examples for the ABC and Szpiro conjectures.     

On divisibility properties of certain multinomial coefficients—II

A recent conjecture of Myerson and Sander concerns divisibility properties of certain multinomial coefficients. We obtain results in this direction by further pursuing a line of attack developed earlier by the first author.     

The distribution of patterns of inverses modulo a prime

We investigate the distribution of the numbers x∈[1,p] for which all lie in a subset of the set of multiplicative inverses modulo a prime p. Here the a_i are integers coprime to p and the numbers are distinct .     

Vector-valued Jacobi-like forms

We introduce vector-valued Jacobi-like forms associated to a representation of a discrete subgroup in and establish a correspondence between such vector-valued Jacobi-like forms and sequences of vector-valued modular forms of different weights with respect to ρ. We determine a lifting of vector-valued modular forms to vector-valued Jacobi-like forms as well as a lifting of scalar-valued Jacobi-like forms to vector-valued Jacobi-like forms. We also construct Rankin-Cohen brackets for vector-valued modular forms.     

Links between associated additive Galois modules and computation of H for local formal group modules

Using the methods described in the papers (Documenta Math. 5 (2000) 657; Local Leopoldt's problem for ideals in p-extensions of complete discrete valuation fields, to appear), we prove that a cocycle for a formal group in a Galois p-extension of a complete discrete valuation field is a coboundary if and only if the corresponding group algebra elements increase valuations by a number that is sufficiently large. We also calculate the valuation of the splitting element of a coboundary. A special case of the main theorem allows us to determine when a p-extension of a complete discrete valuation fields contains a root of a Kummer equation for a formal group. The theorem of Coates-Greenberg for formal group modules in deeply ramified extensions is generalized to noncommutative formal groups. Some results concerning finite torsion modules for formal groups are obtained.     

A Short Remark on Weierstrass Points at Infinity on <Emphasis Type="Italic">X</Emphasis><Subscript>0</Subscript>(<Emphasis Type="Italic">N</Emphasis>)

We point out that the formalism of the trace map and reduction modulo p can be used to give a short proof for the fact first proved by Ogg that is not a Weierstrass point on X₀(pM) where p is a prime not dividing M and the genus of X₀(M) is zero.     

Indecomposable forms of higher degree

All nondegenerate indecomposable forms of higher degree over a perfect field k can be realized as traces of nondegenerate absolutely indecomposable forms of higher degree over a suitable algebraic field extension of k. With the help of trace forms of certain nonassociative algebras we construct classes of indecomposable forms of degree d≥3.     

Local Galois module structure in positive characteristic and continued fractions

For a Galois extension of degree p of local fields of characteristic p, we express the Galois action on the ring of integers in terms of a combinatorial object: a balanced {0, 1}-valued sequence that only depends on the discriminant and p. We show that the embedding dimension edim(R) of the associated order R is tightly related to the minimal number d of R-module generators of the ring of integers. Moreover, we show how to compute d and edim(R) from p and the discriminant with a continued fraction expansion. We thank Bruno Anglès, Wieb Bosma and Rob Tijdeman for their bibliographic assistance. Received: 19 March 2006     

Computation of Galois groups associated to the 2-class towers of some quadratic fields

The p-group generation algorithm from computational group theory is used to obtain information about large quotients of the pro-2 group for with d=−445,−1015,−1595,−2379. In each case we are able to narrow the identity of G down to one of a finite number of explicitly given finite groups. From this follow several results regarding the corresponding 2-class tower.     

A relation between <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions and the Tamagawa number conjecture for Hecke characters

We prove that the submodule in K-theory which gives the exact value of the L-function by the Beilinson regulator map at non-critical values for Hecke characters of imaginary quadratic fields K with cl (K) = 1(p-local Tamagawa number conjecture) satisfies that the length of its coimage under the local Soulé regulator map is the p-adic valuation of certain special values of p-adic L-functions associated to the Hecke characters. This result yields immediately, up to Jannsens conjecture, an upper bound for in terms of the valuation of these p-adic L-functions, where V_p denotes the p-adic realization of a Hecke motive.Received: 4 June 2003     

On class numbers of quadratic extensions¶over function fields

We consider class numbers of quadratic extensions over a fixed function field. We will show that there exist infinitely many quadratic extensions which have class numbers not being divisible by 3 and satisfy prescribed ramification conditions. Received: 24 October 1997 / Revised version: 26 February 1998     

On primitive roots of tori: The case of function fields

We generalize Bilharz's Theorem for to all one-dimensional tori over global function fields of finite constant field. As an application, we also derive an analogue, in the setting of function fields, of a theorem (Chen-Kitaoka-Yu, Roskam) on the distribution of fundamental units modulo primes. Received: 16 October 2000 / Published online: 2 December 2002 Research partially supported by National Science Council, Rep. of China.     

An Existential Divisibility Lemma for Global Fields

We generalize the Existential Divisibility Lemma by Th. Pheidas [7] to all global fields K of characteristic not 2, and for all sets of primes that are inert in a quadratic extension L of K. We also remove the conditions in real and ramifying primes, which were present in Pheidas’ version. As a Corollary, we recover the known fact that the set of integral elements at a prime in a global field is existentially definable. The first author is a Research Assistant of the Research Foundation – Flanders (FWO – Vlaanderen). Work partially supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00287.     

Problèmes diophantiens simultanés

Simultaneous Diophantine Approximations. We study a mixed simultaneous diophantine problem, with an approximation condition and a divisibility condition. We solve this problem for quadratic numbers.     

Quasi-Elliptic Integrals and Periodic Continued Fractions

 In this report we detail the following story. Several centuries ago, Abel noticed that the well-known elementary integral

On Numbers which are Mahler Measures

We investigate which algebraic numbers can be Mahler measures. Adler and Marcus showed that these must be Perron numbers. We prove that certain integer multiples of every Perron number are Mahler measures. The results of Boyd give some necessary conditions on Perron number to be a measure. These do not include reciprocal algebraic integers, so it would be of interest to find one which is not a Mahler measure. We prove a result in this direction. Finally, we show that for every non-negative integer k there is a cubic algebraic integer having norm 2 such that precisely the kth iteration of its Mahler measure is an integer.