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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



$L$-functions and class numbers of imaginary quadratic fields and of quadratic extensions of an imaginary quadratic field

Author: Stéphane Louboutin
Journal: Math. Comp. 59 (1992), 213-230
MSC: Primary 11R29; Secondary 11R16
MathSciNet review: 1134735
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Abstract: Starting from the analytic class number formula involving its L-function, we first give an expression for the class number of an imaginary quadratic field which, in the case of large discriminants, provides us with a much more powerful numerical technique than that of counting the number of reduced definite positive binary quadratic forms, as has been used by Buell in order to compute his class number tables. Then, using class field theory, we will construct a periodic character $\chi$, defined on the ring of integers of a field K that is a quadratic extension of a principal imaginary quadratic field k, such that the zeta function of K is the product of the zeta function of k and of the L-function $L(s,\chi )$. We will then determine an integral representation of this L-function that enables us to calculate the class number of K numerically, as soon as its regulator is known. It will also provide us with an upper bound for these class numbers, showing that Hua’s bound for the class numbers of imaginary and real quadratic fields is not the best that one could expect. We give statistical results concerning the class numbers of the first 50000 quadratic extensions of ${\mathbf {Q}}(i)$ with prime relative discriminant (and with K/Q a non-Galois quartic extension). Our analytic calculation improves the algebraic calculation used by Lakein in the same way as the analytic calculation of the class numbers of real quadratic fields made by Williams and Broere improved the algebraic calculation consisting in counting the number of cycles of reduced ideals. Finally, we give upper bounds for class numbers of K that is a quadratic extension of an imaginary quadratic field k which is no longer assumed to be of class number one.

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Article copyright: © Copyright 1992 American Mathematical Society