Computation of relative class numbers of CM-fields by using Hecke 𝐿-functions

Louboutin, Stéphane

doi:10.1090/S0025-5718-99-01096-0

Computation of relative class numbers of CM-fields by using Hecke $L$-functions
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Math. Comp. 69 (2000), 371-393 Request permission

Abstract:

We develop an efficient technique for computing values at $s=1$ of Hecke $L$-functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields $\mathbf { N}$ which are abelian extensions of some totally real subfield $\mathbf { L}$. We note that the smaller the degree of $\mathbf { L}$ the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing $\mathbf { L} =\mathbf { N}^+$ (the maximal totally real subfield of $\mathbf { N}$) we can choose $\mathbf { L}$ real quadratic. We finally give examples of computations of relative class numbers of several dihedral CM-fields of large degrees and of several quaternion octic CM-fields with large discriminants.

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