Some explicit upper bounds for residues of zeta functions of number fields taking into account the behavior of the prime 2

We recall the known explicit upper bounds for the residue at s = 1 of the Dedekind zeta function of a number field K. Then, we improve upon these previously known upper bounds by taking into account the behavior of the prime 2 in K. We finally give several examples showing how such improvements yield better bounds on the absolute values of the discriminants of CM-fields of a given relative class number. In particular, we will obtain a 4,000-fold improvement on our previous bound for the absolute values of the discriminants of the non-normal sextic CM-fields with relative class number one.     

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).     

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.     

The ideal class groups of dihedral extensions over imaginary quadratic fields and the special values of the Artin L-function

We study the relation between the minus part of the p-class subgroup of a dihedral extension over an imaginary quadratic field and the special value of the Artin L-function at 0.     

The class-number one problem for some real cubic number fields with negative discriminants

We prove that there are effectively only finitely many real cubic number fields of a given class number with negative discriminants and ring of algebraic integers generated by an algebraic unit. As an example, we then determine all these cubic number fields of class number one. There are 42 of them. As a byproduct of our approach, we obtain a new proof of Nagell's result according to which a real cubic unit ?>1 of negative discriminant is generally the fundamental unit of the cubic order Z[?].     

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.     

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.     

Congruences involving Bernoulli polynomials

Let {B_n(x)} be the Bernoulli polynomials. In the paper we establish some congruences for , where p is an odd prime and x is a rational p-integer. Such congruences are concerned with the properties of p-regular functions, the congruences for and the sum , where h(d) is the class number of the quadratic field of discriminant d and p-regular functions are those functions f such that are rational p-integers and for n=1,2,3,… . We also establish many congruences for Euler numbers.     

A generalization of Newman’s formula

We give a determinant formula for the relative class number of an imaginary abelian number field, which is a generalization of Newman’s formula of 1970 and of Skula’s of 1981. By our formula we can determine the signs of these determinants, which these authors did not give. Received: 17 May 2007     

A study on functional independence of the Iwasawa power series

We study a class of functional independences that the Iwasawa power series satisfy for both zero and non-zero characteristics. As results, we prove a generalization of Anglès and Ranieri [B. Anglès, G. Ranieri, On the linear independence of p$p$-adic L$L$-functions modulo p$p$, Ann. Inst. Fourier (Grenoble) 60 (5) (2010) 1831–1855] and transcendence of the Iwasawa power series over the rational functions for non-zero characteristics. We also verify that the power series is not a solution of any non-trivial linear differential equation with the coefficients of rational functions over p$p$-adic numbers.     

Class number and class group problems for some non-normal totally real cubic number fields

Let \lcub;K _m } _{m ≥ 4} be the family of non-normal totally real cubic number fields associated with the Q-irreducible cubic polynomials P _m (x) =x ³−mx ²−(m+1)x− 1, m≥ 4. We determine all these K _m 's with class numbers h _m ≤ 3: there are 14 such K _m 's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents e _m of the ideal class groups of these K _m go to infinity with m and we determine all these K _m 's with ideal class groups of exponents e _m ≤ 3: there are 6 suchK _m with ideal class groups of exponent 2, and 6 such K _m with ideal class groups of exponent 3. Received: 16 November 2000 / Revised version: 16 May 2001     

Universal deformation rings need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose (unrestricted) universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. Finally, we discuss bounds on the singularities of universal deformation rings of representations of finite groups in terms of the nilpotency of the associated defect groups. The first author was supported in part by NSF Grant DMS01-39737 and NSA Grant H98230-06-1-0021. The second author was supported in part by NSF Grants DMS00-70433 and DMS05-00106.     

A finiteness theorem for imaginary abelian number fields

Lately, I. Miyada proved that there are only finitely many imaginary abelian number fields with Galois groups of exponents ≤2 with one class in each genus. He also proved that under the assumption of the Riemann hypothesis there are exactly 301 such number fields. Here, we prove the following finiteness theorem: there are only finitely many imaginary abelian number fields with one class in each genus. We note that our proof would make it possible to find an explict upper bound on the discriminants of these number fields which are neither quadratic nor biquadratic bicyclic. However, we do not go into any explicit determination.     

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.     

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.     

A note on the 3-rank of quadratic fields

The difference between the 3-rank of the ideal class group of an imaginary quadratic field and that of the associated real quadratic field is equal to 0 or 1. In this note, we give an infinite family of examples in each case.Received: 9 September 2002     

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.     

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.     

A density result for some imaginary quadratic fields with infinite Hilbert 2-class field towers

This paper shows that a positive proportion of the imaginary quadratic fields with 2-class rank equal to 3 have 4-class rank equal to zero and infinite Hilbert 2-class field towers. Received: 14 January 2003     

A family of quintic cyclic fields with even class number parameterized by rational points on an elliptic curve

We give a family of quintic cyclic fields with even class number parametrized by rational points on an elliptic curve associated with Emma Lehmer's quintic polynomial. Further, we use the arithmetic of elliptic curves and the Chebotarev density theorem to show that there are infinitely many such fields.