Class Number Formula Research Articles

A study of Inkeri's class number formula

Let p be an odd prime and Q ( ζ p ) be the cyclotomic field defined by a primitive pth root of unity ζ p . It is well-known that the relative class number h p - of Q ( ζ p ) can be expressed by means of the determinant of Inkeri's matrix. The main purpose of this paper is to study Inkeri's class number formula from the viewpoints of bases of Stickelberger ideals in a certain group ring and some special polynomial values.

Class number formulae in the form of a product of determinants in function fields

AbstractIn this paper, we generalize the Kučera's group-determinant formulae to obtain the real and relative class number formulae of any subfield of cyclotomic function fields with arbitrary conductor in the form of a product of determinants. From these formulae, we generalize the relative class number formula of Rosen and Bae-Kang and obtain analogous results of Tsumura and Hirabayashi for an intermediate field in the tower of cyclotomic function fields with prime power conductor.

Mean values of biquadratic zeta functions

Let h(d) be the number of primitive inequivalent binary quadratic forms of discriminant d. In Disquisitiones §302 and 304 [8], Gauss gives conjectures for the average value of h(d) if d 0, where in the latter case, d is derived from the fundamental solution to Pell’s equation. Gauss’s conjectures were first proved by Lipschitz [12] (for negative discriminants) and Siegel (for positive discriminants) [14]. When d is a fundamental discriminant and χd the primitive real Dirichlet character associated to the quadratic extension Q[ √ d], Dirichlet’s class number formula relates the value of the L-series of χd at s = 1 to the class number h(d). Thus Gauss’s conjectured asymptotics are equivalent to conjectures for sums of the type ∑

On class number formula for the real quadratic fields

Let $k > 1$ be the fundamental discriminant, and let $\chi(n)$, $\varepsilon$ and $h$ be the real primitive character modulo $k$, the fundamental unit and the class number of the real quadratic field $\mathbf{Q}(\sqrt{k} )$, respectively. And let $[x]$ denote the greatest integer not greater than $x$. In [3], M.-G. Leu showed $h = \big[ \sqrt{k} / (2\log{\varepsilon}) \sum_{n=1}^k \chi(n) / n \big] + 1$ for all $k$, and $h = \big[ \sqrt{k} / (2\log{\varepsilon}) \sum_{n=1}^{[k/2]} \chi(n) / n \big]$ in the case $k \neq m^2 + 4$ with $m \in \mathbf{Z}$. In this paper we will show $h = \big[ \sqrt{k} / (2\log{\varepsilon}) \sum_{n=1}^{[k/2]} \chi(n) / n \big]$ for all fundamental discriminants $k > 1$.

On the Iwasawa Invariants of $\mathbb Z_2$-Extensions of Certain Real Quadratic Fields

For a real quadratic field $k$, we denote by $\lambda_2(k)$, $\mu_2(k)$ and $\nu_2(k)$ the Iwasawa $\lambda$-, $\mu$- and $\nu$-invariants of the cyclotomic $\mathbb Z_2$-extension of $k$, respectively. In this paper, we give certain families of real quadratic fields $k$ such that $\lambda_2(k)=\mu_2(k)=0$ and $\nu_2(k)=2$, by using Kuroda's class number formula.

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.

The arithmetic of the values of modular functions and the divisors of modular forms

We investigate the arithmetic and combinatorial significance of the values of the polynomials jn(x) defined by the q-expansion \[\sum_{n=0}^{\infty}j_n(x)q^n:=\frac{E_4(z)^2E_6(z)}{\Delta(z)}\cdot\frac{1}{j(z)-x}.\] They allow us to provide an explicit description of the action of the Ramanujan Theta-operator on modular forms. There are a substantial number of consequences for this result. We obtain recursive formulas for coefficients of modular forms, formulas for the infinite product exponents of modular forms, and new p-adic class number formulas.

AN ANALOGUE OF CIRCULAR UNITS FOR PRODUCTS OF ELLIPTIC CURVES

AbstractWe construct certain elements in the motivic cohomology group $H^3_{\mathcal{M}}(E\times E',\mathbb{Q}(2))$, where $E$ and $E'$ are elliptic curves over $\mathbb{Q}$. When $E$ is not isogenous to $E'$ these elements are analogous to circular units in real quadratic fields, as they come from modular parametrizations of the elliptic curves. We then find an analogue of the class-number formula for real quadratic fields, which specializes to the usual quadratic class-number formula when $E$ and $E'$ are quadratic twists.AMS 2000 Mathematics subject classification: Primary 11F67; 14G35. Secondary 11F11; 11E45; 14G10

On the values of equivariant zeta functions of curves over finite fields

Let K/k be a finite Galois extension of global function fields of characteristic p . Let C_K denote the smooth projective curve that has function field K and set G:= \operatorname{Gal}(K/k) . We conjecture a formula for the leading term in the Taylor expansion at zero of the G -equivariant truncated Artin L -functions of K/k in terms of the Weil-étale cohomology of \mathbb G_m on the corresponding open subschemes of C_K . We then prove the \ell -primary component of this conjecture for all primes \ell for which either \ell \not= p or the relative algebraic K -group K_0(\mathbb z _\ell [G],\mathbb Q_\ell) is torsion-free. In the remainder of the manuscript we show that this result has the following consequences for K/k : if p \nmid |G| , then refined versions of all of Chinburg's ' \Omega -Conjectures' in Galois module theory are valid; if the torsion subgroup of K^\times is a cohomologically-trivial G -module, then Gross's conjectural 'refined class number formula' is valid; if K/k satisfies a certain natural class-field theoretical condition, then Tate's recent refinement of Gross's conjecture is valid.

A class number formula for higher derivatives of abelian L-functions

Gross and Rubin have made conjectures about special values of equivariant L -functions associated to abelian extensions of global fields. We describe a common refinement, due to Burns, and give evidence in favour of this conjecture for quadratic extensions and cyclotomic fields. We also note that the statement provides a new interpretation of further conjectures of Darmon and Gross.

Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters

The Tamagawa number conjecture proposed by S. Bloch and K. Kato describes the "special values" of L-functions in terms of cohomological data. The main conjecture of Iwasawa theory describes a p-adic L-function in terms of the structure of modules for the Iwasawa algebra. We give a complete proof of both conjectures (up to the prime 2) for L-functions attached to Dirichlet characters. We use the insight of Kato and B. Perrin-Riou that these two conjectures can be seen as incarnations of the same mathematical content. In particular, they imply each other. By a bootstrapping process using the theory of Euler systems and explicit reciprocity laws, both conjectures are reduced to the analytic class number formula. Technical problems with primes dividing the order of the character are avoided by using the correct cohomological formulation of the main conjecture.

Computation of Stark-Tamagawa units

Let K be a totally real number field and let l denote an odd prime number. We design an algorithm which computes strong numerical evidence for the validity of the Equivariant Tamagawa Number Conjecture for the Q[G]-equivariant motive h0(Spec(L)), where L/K is a cyclic extension of degree l and group G. This conjecture is a very deep refinement of the classical analytic class number formula. In the course of the algorithm, we compute a set of special units which must be considered as a generalization of the (conjecturally existing) Stark units associated to first order vanishing Dirichlet L-functions.

Zeros of families of automorphic L-functions close to 1

For many L-functions of arithmetic interest, the values on or close to the edge of the region of absolute convergence are of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to non-vanishing of ζ(s) for Re(s) = 1). Other examples are the Dirichlet L-functions (e.g., because of the Dirichlet class-number formula) and the symmetric square L-functions of classical automorphic forms. For analytic purposes, in the absence of the Generalized Riemann Hypothesis, it is very useful to have an upper-bound, on average, for the number of zeros of the L-functions which are very close to 1. We prove a very general statement of this type for forms on GL(n)/Q for any n ≥ 1, comparable to the log-free density theorems for Dirichlet characters first proved by Linnik.

A relative class number formula for an imaginary abelian number field by means of Dedekind sum

We give a relative class number formula for an imaginary abelian number field by means of some ``Dedekind sum''. This is a generalization of Carlitz and Olson's formula presented in 1955.

Inkeri‚s determinant for an imaginary abelian number field

We give a relative class number formula for an imaginary abelian number field by means of a determinant, which yields a generalization of Inkeri‚s determinant.

Higher relative class number formulae

Let E be a totally complex abelian number field with maximal real subfield F, and let \(\chi\) denote the non-trivial character of \(Gal (E/F)\). Similar to the classical case n=1 the value of the Artin L-function \(L(E/F,\chi,1-n)\) at \(1-n\) for odd \(n \geq 3\) is given by a relative class number formula of the form

On a class number formula for real quadratic number fields

For an even Dirichlet character ψ, we obtain a formula for L (1, ψ) in terms of a sum of Dirichlet L-Series evaluated at s = 2 and s = 3 and a rapidly convergent numerical series involving the central binomial coefficients. We then derive a class number formula for real quadratic number fields by taking L (s, ψ) to be the quadratic L-series associated with these fields.

A generalization of maillet and demyanenko determinants for the Cyclotomic Zp-Extension

LetK be an imaginary abelian number field. By means of a generalization of Maillet and Demyanenko determinants we give a relative class number formula for an intermediate field of the cyclotomic ℤp-extension ofK. The degree of the generalized determinant is a half of the degree ofK over ℚ.

Class number of (v, n, M)-extensions

An analogue of cyclotomic number fields for function fields over the finite field q, was investigated by L. Carlitz in 1935 and has been studied recently by D. Hayes, M. Rosen, S. Galovich and others. For each nonzero polynomial M in q [T], we denote by k (ΛM) the cyclotomic function field associated with M, where k = q(T). Replacing T by 1/T in k and considering the cyclotomic function field Fv that corresponds to (1/T)v+1 gets us an extension of k, denoted by Lv, which is the fixed field of Fv modulo . We define a (v, n, M)-extension to be the composite N = knk (Λm) Lv where kn is the constant field of degree n over k. In this paper we give analytic class number formulas for (v, n, M)-extensions when M has a nonzero constant term.

Kaplansky′s ternary quadratic form

This paper proves that if N is a nonnegative eligible integer, coprime to 7, which is not of the form x2 + y2 + 7z2, then N is square‐free. The proof is modelled on that of a similar theorem by Ono and Soundararajan, in which relations between the number of representations of an integer np2 by two quadratic forms in the same genus, the pth coefficient of an L‐function of a suitable elliptic curve, and the class number formula prove the theorem for large primes, leaving 3 cases which are easily numerically verified.