A computational technique for determining relative ... - Springer Link

A C o m p u t a t i o n a l Technique for D e t e r m i n i n g R e l a t i v e Class N u m b e r s of C M - F i e l d s St6phane Louboutin Univ.ersit6 de Caen, D6partement de Math~matiques, 14032 Caen Cedex, France A b s t r a c t . It was well known that it is easy to compute relative class numbers of abelian CM-fields by using generalized Bernoulli numbers (see [9]). Here, we provide a technique for computing the relative class number of any CM-field.

1

Statement

of the Results

P r o p o s i t i o n 1. Let n >_ 1 be an integer and ~ > 1 be real. Set n--1

1

olx)=Z x

k

k=0

and

A 2 fa+ioo K n ( A ) = -i -r ~ ~ _ ~

with fn(s) = Pn(s)A-2S

(1 ~

f.(s)ds

(1)

+

Then, it holds 0 < K n ( A ) < 2Pn(nA2/'~)e-'~A2/" < 2 n e x p ( - A 2 / n ) .

(2)

2. Let N be a totally imaginary number field of degree 2n which is a quadratic extension of a totally real number field N + of degree n, i.e. N is a CM-field. Let WN be the number of roots of unity in N, QN E {1, 2} be the Hasse unit index of N , and dN and dN+ be the absolute values of the discriminants of N and N +, respectively. Let XN/N+ be the quadratic character assocciated with the quadratic extension N / N + and let Ck be the coefficients of the Dirichlet seines (fiN/fiN+)(8) = L(8, XN/N+ ) = ~-~r -s (~(8) > 1). k>l Theorem

Set AN/N+ = ~/dN/TrndN+. We have

214

hN = QNWN

d~N+

(3)

and according to (2) this series (3) is absolutely convergent. Moreover, set

B(N) de_jAN/N+

/ ,~ ,~ ~/2 ~ n lOgAN/N+) 9

(4)

Then, i] X > 1 andn are given, then the limit o] lhN--hN( M)] as dN approaches infinity is equal to O, where hN(M ) is the approximation of the relative class number obtained by disregarding in the series oecuring in (3) the indices k > i _ B(N). For example, if N of degree m = 2n is the narrow Hilbert class field of a real quadratic number field L of discriminant dL, we have

d~/s log"~/4(dL/Ir 2).

B (N) =

The following Proposition 3 explains how we compute the numerical values of the function A ~-} K~(A) according to its series expansion : P r o p o s i t i o n 3. Take A > O. It holds

Kn(A) = 1 + lrn/2A + 2A2 Z Res,=_m(fn). m>0

(5)

This series is absolutely convergent and for any M > 0 we have 2A2 ~

I 7rn/2A2MT3 Res~=_m(f,~) < (M + 1)(M!/2) n"

(6)

m>M

Finally, the following Proposition 4 explains how to compute recursively the values of the residues Ress=_m(f~) occuring in (5) : P r o p o s i t i o n 4. We have t A2 m

-1

Z. Ress=-m(fn) = - ( - 1 ) n ' ~(m!)n -

2-t-ihi(m)((2m + 1)i + (2m + 2) i)

where the hi(m ) 's are computed recursivety from the hi(O) 's by using 1 note the misprint in the formula given in [2]

(7)

215

hi(m+l)=

hj(m)(m

E

1)i-/

j=-n

and

+

~ h~(O)sJ+ O ( 1 ) = P n ( s ) A

-2".

(s)

n--1

where bk = Cn+k_ 1 = ((k + n - 1)!/k!(n - 1)!) Thus, if n--1

r = ( , + 1) =

h,,' +

(9)

i=0

then hi-n(0) = ~

(-21~ i!

hl-j

(O