Normal integral bases for cyclic Kummer extensions

Let K/F be a Kummer cyclic extension of number fields. In the case when the degree is a prime number, Gómez Ayala gave an explicit criterion for the existence of a normal integral basis. More recently Ichimura proposed a generalization of that result for cyclic extensions of arbitrary degree, but we have found that Ichimura’s result is incorrect. In this paper we present a counter-example to Ichimura’s result as well as the correct generalization of Gómez Ayala’s result.     

Indices of Kummer extensions of prime degree

Let p be a prime and let ${\zeta }_p$ be a primitive p-th root of unity. For a finite extension k of $\mathbb{Q}$ containing ${\zeta }_p$, we consider a Kummer extension $L/k$ of degree p. In this paper, we show that if $k=\mathbb{Q}({\zeta }_p)$ and the class number of k is one, the index of $L/k$ is one. We also show that if $L/k$ is tamely ramified with a normal integral basis, the index is at most a power of p. In the last section, we show that there exist infinitely many cubic Kummer extensions of $\mathbb{Q}({\zeta }_3)$ for both wildly and tamely ramified cases, whose integer rings do not have a power integral basis over that of $\mathbb{Q}({\zeta }_3)$.     

Rational places in extensions and sequences of function fields of Kummer type

In this paper we prove results on the number of rational places in extensions of Kummer type over finite fields and give sufficient conditions for non-trivial lower bounds on the number of rational places at each step of sequences of function fields over a finite field, that we call (a, b)-sequences. In the case of a prime field, we apply these results to the study of rational places in certain sequences of function fields of Kummer type.     

Applications of the classical umbral calculus

We describe applications of the classical umbral calculus to bilinear generating functions for polynomial sequences, identities for Bernoulli and related numbers, and Kummer congruences.Dedicated to the Memory of Gian-Carlo Rota     

<Emphasis Type="Italic">p</Emphasis>-adic interpolation of Taylor coefficients of modular forms

In this paper we show that the Taylor coefficients of a Hecke eigenform at a CM-point, suitably modified, form a sequence of algebraic numbers that satisfy the Kubota–Leopoldt generalization of the Kummer congruences for primes that split in the imaginary quadratic field associated with a CM-point. More generally, we show that these numbers are moments of a certain p-adic measure. In addition, we write down explicitly the “Euler factor” at p in terms of the p th Hecke eigenvalue of the modular form in question and certain data of the CM-point. P. Guerzhoy is supported by NSF grant DMS-0700933.     

Rational classes and divisors on curves of genus 2

We describe a method of looking for rational divisor classes on a curve of genus 2. We have an algorithm to decide if a given class of divisors of degree 3 contains a rational divisor. It is known that the shape of the kernel of Cassel’s morphism (X − T) is related to the existence of rational classes of degree 1. Our key tool is the dual Kummer surface.V. G. L. Neumann supported by CNPq, Brazil     

Determination of conductors from Galois module structure

Let E denote an unramified extension of , and set for an odd prime p and . We determine the conductors of the Kummer extensions of F by those elements such that is Galois. This follows from a comparison of the Galois module structure of with the unit filtration of F. Received: 28 August 2000; in final form: 11 October 2001 / Published online: 4 April 2002     

Generalised Kummer constructions and Weil restrictions

We use a generalised Kummer construction to realise all but one known weight four newforms with complex multiplication and rational Fourier coefficients in Calabi-Yau threefolds defined over Q. The Calabi-Yau manifolds are smooth models of quotients of the Weil restrictions of elliptic curves with CM of class number three.     

Average values of L-functions in characteristic two

Gauss made two conjectures about average values of class numbers of orders in quadratic number fields, later on proven by Lipschitz and Siegel. A version for function fields of odd characteristic was established by Hoffstein and Rosen. In this paper, we extend their results to the case of even characteristic. More precisely, we obtain formulas of average values of L-functions associated to orders in quadratic function fields over a constant field of characteristic two, and then derive formulas of average class numbers of these orders.     

On the Class-number of the Maximal Real Subfield of a Cyclotomic Field

For any square-free positive integer m, let H(m) be the class-number of the field , where ζ_m is a primitive m-th root of unity. We show that if m = {3(8 g + 5)}² ? 2 is a square-free integer, where g is a positive integer, then H(4 m) > 1. Similar result holds for a square-free integer m = {3(8 g +7)}² ?2, where g is a positive integer. We also show that n|H(4 m) for certain positive integers m and n.     

A complete determination of Rabinowitsch polynomials

Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x²+x−m. We call a polynomial fm(x) a Rabinowitsch polynomial if for and consecutive integers x=x₀,x₀+1,…,x₀+s−1, |fm(x)| is either 1 or prime. In this paper, we show that there are exactly 14 Rabinowitsch polynomials fm(x).     

Universal Kummer congruences mod prime powers

We have previously proved Kummer congruences mod primes p such that for the universal divided Bernoulli numbers . In this paper we strengthen these congruences to hold mod powers of p.     

Class number 2 problem for certain real quadratic fields of Richaud-Degert type

In this paper we will apply Biró's method in [A. Biró, Yokoi's conjecture, Acta Arith. 106 (2003) 85-104; A. Biró, Chowla's conjecture, Acta Arith. 107 (2003) 179-194] to class number 2 problem of real quadratic fields of Richaud-Degert type and will show that there are exactly 4 real quadratic fields of the form with class number 2, where n²+1 is a even square free integer.     

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.     

Real Quadratic Fields with Class Number Divisible by 5 or 7

We shall show that the number of real quadratic fields whose absolute discriminant is ≤ x and whose class number is divisible by 5 or 7 is improving the existing best known bound for g = 5 and for g = 7 of Yu (J Number Theory 97:35–44, 2002).This work was supported by KRF-R08-2003-000-10243-0 and partially by KRF-2005-070-C00004.     

Maximal class numbers of CM number fields

Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis and Artin's conjecture on the entirety of Artin L-functions, we derive an upper bound (in terms of the discriminant) on the class number of any CM number field with maximal real subfield F. This bound is a refinement of a bound established by Duke in 2001. Under the same hypotheses, we go on to prove that there exist infinitely many CM-extensions of F whose class numbers essentially meet this improved bound and whose Galois groups are as large as possible.     

Class groups of quadratic fields of 3-rank at least 2: Effective bounds

In this paper, we give parametric families of both real and complex quadratic number fields whose class group has 3-rank at least 2. As a consequence, we obtain that for all large positive real numbers x, the number of both real and complex quadratic fields whose class group has 3-rank at least 2 and absolute value of the discriminant ?x is >cx1/3, where c is some positive constant.     

Densities for some real quadratic fields with infinite Hilbert 2-class field towers

Let K be a real quadratic field with 2-class rank equal to 4 or 5 and 4-class rank equal to 3. This paper computes density information for such fields to have infinite Hilbert 2-class field towers.     

On the parity of the class number of multiquadratic number fields

This paper investigates the 2-class group of real multiquadratic number fields. Let p₁,p₂,…,pn be distinct primes and . We draw a list of all fields K whose 2-class group is trivial.     

Caliber number of real quadratic fields

We obtain lower bound of caliber number of real quadratic field using splitting primes in K. We find all real quadratic fields of caliber number 1 and find all real quadratic fields of caliber number 2 if d is not 5 modulo 8. In both cases, we don't rely on the assumption on ζK(1/2).