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

Takashi Taniguchi[1]

  • [1] University of Tokyo Graduate School of Mathematical Sciences 3–8–1 Komaba Meguro-Ku Tokyo 153-0041 (Japan)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 2, page 625-670
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Taniguchi, Takashi. "A mean value theorem for the square of class number times regulator of quadratic extensions." Annales de l’institut Fourier 58.2 (2008): 625-670. <http://eudml.org/doc/10327>.

@article{Taniguchi2008,
abstract = {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.},
affiliation = {University of Tokyo Graduate School of Mathematical Sciences 3–8–1 Komaba Meguro-Ku Tokyo 153-0041 (Japan)},
author = {Taniguchi, Takashi},
journal = {Annales de l’institut Fourier},
keywords = {Density theorem; prehomogeneous vector space; quaternion algebra; local zeta function; density theorem},
language = {eng},
number = {2},
pages = {625-670},
publisher = {Association des Annales de l’institut Fourier},
title = {A mean value theorem for the square of class number times regulator of quadratic extensions},
url = {http://eudml.org/doc/10327},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Taniguchi, Takashi
TI - A mean value theorem for the square of class number times regulator of quadratic extensions
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 2
SP - 625
EP - 670
AB - 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.
LA - eng
KW - Density theorem; prehomogeneous vector space; quaternion algebra; local zeta function; density theorem
UR - http://eudml.org/doc/10327
ER -

References

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  8. A. C. Kable, A. Yukie, The mean value of the product of class numbers of paired quadratic fields, III, J. Number Theory 99 (2003), 185-218 Zbl1039.11086MR1957252
  9. D. Mumford, J. Fogarty, Geometric invariant theory, Springer-Verlag (1982) Zbl0504.14008MR719371
  10. M. Peter, Momente der Klassenzahlen binärer quadratischer Formen mit ganzalgebraischen Koeffizienten, Acta Arithm. 70 (1995), 43-77 Zbl0817.11025MR1318761
  11. Waterloo Maple Software, Maple V, Waterloo Maple Inc., Waterloo, Ontario (1994) 
  12. T. Taniguchi, Distributions of discriminants of cubic algebras 
  13. T. Taniguchi, Distributions of discriminants of cubic algebras II 
  14. T. Taniguchi, On propotional constants of the mean value of class numbers of quadratic extensions, Trans. Amer. Math. Soc. 359 (2007), 5517-5524 Zbl1134.11041MR2327040
  15. T. Taniguchi, On the zeta functions of prehomogeneous vector spaces for a pair of simple algebras, Ann. Inst. Fourier 57 (2007), 1331-1358 Zbl1173.11050MR2339334
  16. M. F. Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics 800 (1980), Springer-Verlag, Berlin, Heidelberg, New York Zbl0422.12008MR580949
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  18. D. J. Wright, A. Yukie, Prehomogeneous vector spaces and field extensions, Invent. Math. 110 (1992), 283-314 Zbl0803.12004MR1185585

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