Units and class groups in cyclotomic function fields

A unit index-class number formula is proven for “cyclotomic function fields” in analogy with similar results in cyclotomic number fields. The main theorem generalizes to arbitrary modules work done earlier for prime power modules.     

The parity of the class number of the cyclotomic fields of prime conductor

Using a duality result for cyclotomic units proved by G.Gras, we derive a relation between the vanishing of some -components of the ideal class groups of abelian fields of prime conductor (Theorem 1). As a consequence, we obtain a criterion for the parity of the class number of any abelian number field of prime conductor.

     

On the first factor of the class number of prime cyclotomic fields

Let H(l) be the first factor of the class number of the field $\text{Q}$(exp 2πi/l), l a prime. The best-known upper and lower bounds on H(l) are improved for small l. The methods would also improve the best-known bounds for large l. It is shown that H(l) is the absolute value of the determinant of an easily written down matrix whose only entries are 0 and 1. The upper bounds obtained on H(l) significantly improve the Hadamard bound on the determinant of this matrix. Results of Lehmer on the factors of H(l) are explained via class field theory.     

Class number parity for cyclotomic fields

We give a simple criterion for the parity of the class number of the cyclotomic field.

     

Class numbers of cyclotomic function fields

Let be a prime power and let be the finite field with elements. For each polynomial in , one could use the Carlitz module to construct an abelian extension of , called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of , similar to the role played by cyclotomic number fields for abelian extensions of . We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in . Two types of properties are obtained for the -parts of the class numbers of the fields in this tower, for a fixed prime number . One gives congruence relations between the -parts of these class numbers. The other gives lower bound for the -parts of these class numbers.

     

On the class numbers of cyclotomic fields

Let g_n denote the first factor of the class number of the nth cyclotomic field. It is proved that if n runs through a sequence of prime powers p^r tending to infinity, then

$\text{log}\text{g}{}_{\mathrm{n}}\text{～}\text{1}\text{4}\text{[1 ? (}\text{1}\text{p}\text{)]n}\text{log}\text{n}$

.     

On the class numbers of cyclotomic fields

The finiteness of the number of cyclotomic fields whose relative class numbers have bounded odd parts will be verified and then all the cyclotomic fields with relative class numbers non-trivial 2-powers will be determined.This research was supported in part by Grant-in-Aid for Science (No. 01740051), Ministry of Education, Science, and Culture of Japan     

Algebraic function fields with small class number

It is proved that there is no congruence function field of genus 4 over GF(2) which has no prime of degree less than 4 and precisely one prime of degree 4. This shows the nonexistence of function fields of genus 4 with class number one and gives an example of an isogeny class of abelian varieties which contains no jacobian. It is shown that, up to isomorphism, there are two congruence function fields of genus 3 with class number one. It follows that there are seven nonisomorphic function fields of genus different from zero with class number one. Congruence function fields with class number 2 are fully classified. Finally, it is proved that there are eight imaginary quadratic function fields $\text{F}\text{K(x)}$ for which the integral closure of K[x] in F has class number 2.     

Arithmetic actions on cyclotomic function fields

We derive the group structure for cyclotomic function fields obtained by applying the Carlitz action for extensions of an initial constant field. The tame and wild structures are isolated to describe the Galois action on differentials. We show that the associated invariant rings are not polynomial.     

On the class number of p-th cyclotomic field

Let h^[-(p)h^-(p) be the relative class number of the p-th cyclotomic field. We show that logh^-(p) = [(p+3)/4] logp - [(p)/2] log2p+ log(1-b) + O(log₂² p)\log h^-(p) = {{p+3} \over {4}} \log p - {{p} \over {2}} \log 2\pi + \log (1-\beta ) + O(\log _2^2 p), where b\beta denotes a Siegel zero, if such a zero exists and p o -1 mod 4p\equiv -1\pmod {4}. Otherwise this term does not appear.     

On the subfields of cyclotomic function fields

Let K = F(T) be the rational function field over a finite field of q elements. For any polynomial f(T) ∈ F [T] with positive degree, denote by Λ _f the torsion points of the Carlitz module for the polynomial ring F[T]. In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield M of the cyclotomic function field K(Λ _P ) of degree k over F(T), where P ∈ F[T] is an irreducible polynomial of positive degree and k > 1 is a positive divisor of q ? 1. A formula for the analytic class number for the maximal real subfield M ⁺ of M is also presented. Futhermore, a relative class number formula for ideal class group of M will be given in terms of Artin L-function in this paper.     

On L-functions of cyclotomic function fields

We study two criterions of cyclicity for divisor class groups of function fields, the first one involves Artin L-functions and the second one involves “affine” class groups. We show that, in general, these two criterions are not linked.     

The first factor of the class number of the p-th cyclotomic field

Kummer’s conjecture states that the relative class number of the p-th cyclotomic field follows a strict asymptotic law. Granville has shown it unlikely to be true—it cannot be true if we assume the truth of two other widely believed conjectures. We establish a new bound for the error term in Kummer’s conjecture, and more precisely we prove that ${\log(h_p^-)=\frac{p+3}{4} \log p +\frac{p}{2}\log(2\pi)+\log(1-\beta)+O(\log_2 p)}$ , where β is a possible Siegel zero of an ${L(s,\chi), \chi}$ odd.