The mean value of the product of class numbers of paired quadratic fields. I

Abstract

Let k be a number field and k a fixed quadratic extension of k. In this paper and its companion, we find the mean value of the product of class numbers and regulators of two quadratic extensions F, F * = k contained in the biquadratic extensions of k containing k.

On the zeta functions of prehomogeneous vector spaces for a pair of simple algebras

“…We prove Theorem 3.11 by choosing D such that M D = {v 1 , v 2 }. Combined with the result of Kable-Yukie [7] ANNALES DE L'INSTITUT FOURIER we also obtain the limit of correlation coefficients of h F R F for certain families of quadratic extensions of k. For more details including this result, see [15,Introduction].…”

“…Also recent work on the space of pairs of binary Hermitian forms [7] by Kable and Yukie, combined with Yukie's global theory [20], gave some new density theorems. For the statement of the density theorem, see the introduction of [7]. Note that this case is another k-form of the D 4 type.…”

A mean value theorem for the square of class number times regulator of quadratic extensions

“…We consider the localizations of (G, V ) at each place of k. We note that the localizations of (G, V ) are equivalent to (G , V ) all but finite number of places at which B ramifies. An outer form of the representation (G , V ) is studied by Kable and Yukie [6,7,8] and some of their results are useful for us. After we prove Theorem 10.12, we study the correlation coefficient in the final section combined with the results of [6].…”

“…An outer form of the representation (G , V ) is studied by Kable and Yukie [6,7,8] and some of their results are useful for us. After we prove Theorem 10.12, we study the correlation coefficient in the final section combined with the results of [6].…”

“…In Section 6, we define the local zeta function and the local density. Also we quote from [6] an estimate of the standard local zeta function, which we need in order to apply the filtering process in the proof of the mean value theorem in Section 10. In Sections 7, 8 and 9, we compute the local densities.…”