Reflection theorem for divisor class groups of relative quadratic function fields

We present the reflection theorem for divisor class groups of relative quadratic function fields. Let K be a global function field with constant field Fq. Let L₁ be a quadratic geometric extension of K and let L₂ be its twist by the quadratic constant field extension of K. We show that for every odd integer m that divides q+1 the divisor class groups of L₁ and L₂ have the same m-rank.     

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     

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.     

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.     

The distribution of class groups of function fields

For any sufficiently general family of curves over a finite field F_q and any elementary abelian ?-group H with ? relatively prime to q, we give an explicit formula for the proportion of curves C for which Jac(C)[?](F_q)≅H. In doing so, we prove a conjecture of Friedman and Washington.     

Quadratic equations over finite fields and class numbers of real quadratic fields

Letp be an odd prime and the finite field withp elements. In the present paper we shall investigate the number of points of certain quadratic hypersurfaces in the vector space and derive explicit formulas for them. In addition, we shall show that the class number of the real quadratic field (wherep1 (mod 4)) over the field of rational numbers can be expressed by means of these formulas.     

The Scholz theorem in function fields

The Scholz theorem in function fields states that the l-rank difference between the class groups of an imaginary quadratic function field and its associated real quadratic function field is either 0 or 1 for some prime l. Furthermore, Leopoldt's Spiegelungssatz (= the Reflection theorem) in function fields yields a comparison between the m-rank of some subgroup of the class group of an imaginary cyclic function field L₁ and the m-rank of some subgroup of the class group of its associated real cyclic function field L₂ for some prime number m; then their m-ranks also equal or differ by 1. In this paper we find an explicit necessary condition for their m-ranks (respectively l-ranks) to be the same in the case of cyclic function fields (respectively quadratic function fields). In particular, in the case of quadratic function fields, if l does not divide the regulator of L₂, then their l-ranks are the same, equivalently if their l-ranks differ by 1, then l divides the regulator of L₂.     

Higher rank subgroups in the class groups of imaginary function fields

Let F be a finite field and T a transcendental element over F. In this paper, we construct, for integers m and n relatively prime to the characteristic of F(T), infinitely many imaginary function fields K of degree m over F(T) whose class groups contain subgroups isomorphic to (Z/nZ)^m. This increases the previous rank of m−1 found by the authors in [Y. Lee, A. Pacelli, Class groups of imaginary function fields: The inert case, Proc. Amer. Math. Soc. 133 (2005) 2883-2889].     

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.     

The class number one problem for some totally complex quartic number fields

We prove that there are 95 non-isomorphic totally complex quartic fields whose rings of algebraic integers are generated by an algebraic unit and whose class numbers are equal to 1. Moreover, we prove Louboutin's Conjecture according to which a totally complex quartic unit εu generally generates the unit group of the quartic order Z[εu].     

On the factorization of the relative class number in terms of Frobenius divisions

The relative class number of an imaginary abelian number fieldK is—up to trivial factors—the product of the first Bernoulli numbersB _x belonging to the odd characters ofK. This product splits into rational factorsF _Z = {B ; Z}, whereZ runs through the Frobenius divisions of odd characters. It is shown that each numberF _z is—up to a certain prime power—the index of two explicitly given subgroups of (K, +). These subgroups are cyclic Galois modules, whose generators arise from roots of unity and cotangent numbers, resp. Our result is an analogue of a result concerningh ⁺ which was given by Leopoldt many years ago.To the memory of my friend Kurt Dietrich     

Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields

We show that, for all characteristic p global fields k and natural numbers n coprime to the order of the non-p-part of the Picard group Pic⁰(k) of k, there exists an abelian extension L/k whose local degree at every prime of k is equal to n. This answers in the affirmative in this context a question recently posed by Kisilevsky and Sonn. As a consequence, we show that, for all n and k as above, the n-torsion subgroup Br_n(k) of the Brauer group Br(k) of k is algebraic, answering a question of Aldjaeff and Sonn in this context.     

Class number indivisibility for quadratic function fields

Let M?5. For any odd prime power q and any prime ??q, we show that there are at least pairwise coprime D∈Fq[T] which are square-free and of odd degree ?M, such that ? does not divide the class number of the complex quadratic functions fields .     

On some questions related to the Gauss conjecture for function fields

We show that, for any finite field Fq, there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial.     

Ono invariants of imaginary quadratic fields with class number three

Let Ed(x) denote the “Euler polynomial” x²+x+(1−d)/4 if and x²−d if . Set Ω(n) to be the number of prime factors (counting multiplicity) of the positive integer n. The Ono invariantOnod of is defined to be except when d=−1,−3 in which case Onod is defined to be 1. Finally, let hd=hk denote the class number of K. In 2002 J. Cohen and J. Sonn conjectured that hd=3⇔Onod=3 and is a prime. They verified that the conjecture is true for p<1.5×¹⁰⁷. Moreover, they proved that the conjecture holds for p>¹⁰¹⁷ assuming the extended Riemann Hypothesis. In this paper, we show that the conjecture holds for p?2.5×¹⁰¹³ by the aid of computer. And using a result of Bach, we also proved that the conjecture holds for p>2.5×¹⁰¹³ assuming the extended Riemann Hypothesis. In conclusion, we proved the conjecture is true assuming the extended Riemann Hypothesis.     

Cohen-Lenstra heuristics and the Spiegelungssatz; function fields

In this paper, we study the compatibility of Cohen-Lenstra heuristics with Leopoldt's Spiegelungssatz (= the reflection theorem) in the case of cyclic function fields. First, we prove a group-theoretical version of the Spiegelungssatz for cyclic function fields. Then we show the internal consistency of the function field analogue of Cohen-Lenstra heuristics in the case of cyclic function fields. It thus supports the validity of Cohen-Lenstra heuristics in function fields.     

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.     

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     

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     

On the refined class number formula of Gross

We verify Gross's refined class number formula for abelian extensions of global function fields of prime exponent.