A mean value theorem for the square of class number times regulator of quadratic extensions  [ Un théorème de la moyenne pour le carré du nombre de classe multiplié par le régulateur d’extensions quadratiques ]
Annales de l'Institut Fourier, Tome 58 (2008) no. 2, p. 625-670
Soit k un corps de nombres. Dans cet article, nous donnons une formule pour la valeur moyenne du carré du nombre de classe multiplié par le régulateur pour certaines familles d’extensions quadratiques de k caractérisées par un nombre fini de conditions locales. Notre approche utilise la théorie de la fonction zêta associée à l’espace de paires d’algèbres de quaternions. Nous prouvons aussi une formule asymptotique pour le coefficient de corrélation du nombre de classe multiplié par le régulateur de certaines familles d’extensions quadratiques.
Let k be a number field. In this paper, we give a formula for the mean value of the square of class number times regulator for certain families of quadratic extensions of k characterized by finitely many local conditions. We approach this by using the theory of the zeta function associated with the space of pairs of quaternion algebras. We also prove an asymptotic formula of the correlation coefficient for class number times regulator of certain families of quadratic extensions.
DOI : https://doi.org/10.5802/aif.2363
Classification:  11M41
Mots clés: théorème densité, espace vectoriel préhomogène, fonction zêta local, algèbres de quaternions
@article{AIF_2008__58_2_625_0,
     author = {Taniguchi, Takashi},
     title = {A mean value theorem for the square of class number times regulator of quadratic extensions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {2},
     year = {2008},
     pages = {625-670},
     doi = {10.5802/aif.2363},
     mrnumber = {2410385},
     zbl = {pre05298315},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_2_625_0}
}
Taniguchi, Takashi. A mean value theorem for the square of class number times regulator of quadratic extensions. Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 625-670. doi : 10.5802/aif.2363. http://www.numdam.org/item/AIF_2008__58_2_625_0/

[1] Datskovsky, B. A mean value theorem for class numbers of quadratic extensions, Contemporary Mathematics, Tome 143 (1993), pp. 179-242 | MR 1210518 | Zbl 0791.11058

[2] Datskovsky, B.; Wright, D. J. The adelic zeta function associated with the space of binary cubic forms II: Local theory, J. Reine Angew. Math., Tome 367 (1986), pp. 27-75 | Article | MR 839123 | Zbl 0575.10016

[3] Datskovsky, B.; Wright, D. J. Density of discriminants of cubic extensions, J. Reine Angew. Math., Tome 386 (1988), pp. 116-138 | Article | MR 936994 | Zbl 0632.12007

[4] Granville, A.; Soundararajan, K. The distributions of values of L(1,χ d ), Geom. Funct. Anal., Tome 13 (2003), pp. 992-1028 | Article | MR 2024414 | Zbl 1044.11080

[5] Kable, A. C.; Wright, D. J. Uniform distribution of the Steinitz invariants of quadratic and cubic extensions, Compos. Math., Tome 142 (2006), pp. 84-100 | Article | MR 2196763 | Zbl 05017580

[6] Kable, A. C.; Yukie, A. The mean value of the product of class numbers of paired quadratic fields, I, Tohoku Math. J., Tome 54 (2002), pp. 513-565 | Article | MR 1936267 | Zbl 1020.11079

[7] Kable, A. C.; Yukie, A. The mean value of the product of class numbers of paired quadratic fields, II, J. Math. Soc. Japan, Tome 55 (2003), pp. 739-764 | Article | MR 1978221 | Zbl 1039.11087

[8] Kable, A. C.; Yukie, A. The mean value of the product of class numbers of paired quadratic fields, III, J. Number Theory, Tome 99 (2003), pp. 185-218 | Article | MR 1957252 | Zbl 1039.11086

[9] Mumford, D.; Fogarty, J. Geometric invariant theory, Springer-Verlag (1982) (Berlin, Heidelberg, New York, 2nd edition) | MR 719371 | Zbl 0504.14008

[10] Peter, M. Momente der Klassenzahlen binärer quadratischer Formen mit ganzalgebraischen Koeffizienten, Acta Arithm., Tome 70 (1995), pp. 43-77 | MR 1318761 | Zbl 0817.11025

[11] Software, Waterloo Maple Maple V, Waterloo Maple Inc., Waterloo, Ontario (1994)

[12] Taniguchi, T. Distributions of discriminants of cubic algebras (Preprint 2006, math.NT/0606109)

[13] Taniguchi, T. Distributions of discriminants of cubic algebras II (Preprint 2006, math.NT/0608658)

[14] Taniguchi, T. On propotional constants of the mean value of class numbers of quadratic extensions, Trans. Amer. Math. Soc., Tome 359 (2007), pp. 5517-5524 | Article | MR 2327040 | Zbl 1134.11041

[15] Taniguchi, T. On the zeta functions of prehomogeneous vector spaces for a pair of simple algebras, Ann. Inst. Fourier, Tome 57 (2007), pp. 1331-1358 | Article | Numdam | MR 2339334

[16] Vignéras, M. F. Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, Tome 800 (1980) | MR 580949 | Zbl 0422.12008

[17] Weil, A. Basic number theory, Springer-Verlag (1974) (Berlin, Heidelberg, New York) | Zbl 0823.11001

[18] Wright, D. J.; Yukie, A. Prehomogeneous vector spaces and field extensions, Invent. Math., Tome 110 (1992), pp. 283-314 | Article | MR 1185585 | Zbl 0803.12004