Abstract
In this chapter, as an application of quadratic forms and quadratic fields, we give an explicit formula of some simple zeta functions, related to some so-called prehomogeneous vector spaces. We also prove a class number formula of imaginary quadratic fields. Before that, we review the theory of multiplicative structure of ideals of quadratic field without proof.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Heinrich Martin Weber (born on May 5, 1842 in Heidelberg, Germany—died on May 17, 1913 in Strasbourg, Germany (now France)).
- 2.
Siméon Denis Poisson (born on June 21, 1781 in Pithiviers, France—died on April 25, 1840 in Sceaux, France).
- 3.
Jean Baptiste Joseph Fourier (born on March 21, 1768 in Auxerre, France—died on May 16, 1830 in Paris, France).
- 4.
August Ferdinand Möbius (born on November 17, 1790 in Schulpforta, Saxony (now Germany)—died on September 26, 1868 in Leipzig, Germany).
References
Andrianov, A.N.: Quadratic Forms and Hecke Operators. Grundlehren der mathematischen Wissenshaften, vol. 286. Springer (1987)
Dirichlet, P.G.L.: Lectures on Number Theory, with Supplements by R. Dedekind. History of Mathematics Sources, vol. 16. Amer. Math. Soc. (1999)
Ibukiyama, T., Saito, H.: On zeta functions associated to symmetric matrices and an explicit conjecture on dimensions of Siegel modular forms of general degree. Int. Math. Res. Notices 8, 161–169 (1992)
Ibukiyama, T., Saito, H.: On a L-functions of ternary zero forms and exponential sums of Lee and Weintraub. J. Number Theory 48-2, 252–257 (1994)
Ibukiyama, T., Saito, H.: On zeta functions associated to symmetric matrices I. Am. J. Math. 117-5, 1097–1155 (1995); II: Nagoya Math. J. 208, 265–316 (2012); III: Nagoya Math. J. 146, 149–183 (1997)
Ibukiyama, T., Saito, H.: On “easy” zeta functions (trans. by Don Zagier). Sugaku Exposition, 14(2), 191–204 (2001). Originally in Sugaku, 50–1, 1–11 (1998)
Kaneko, M.: A generalization of the Chowla-Selberg formula and the zeta functions of quadratic orders. Proc. Jpn. Acad. 66(A)-7, 201–203 (1990)
Lang, S.: Elliptic Functions, 2nd edn. Graduate Texts in Mathematics, vol. 112. Springer (1987). (Original edition, Addison-Wesley Publishing, 1973)
Serre, J.-P.: Cours d’arithmétique, Presses Universitaires de France, 1970. English translation: A course in arithmetic, Graduate Text in Mathematics, vol. 7. Springer (1973)
Weber, H.: Lehrbuch der Algebra, vol. III, Chelsea Publication, New York. (First edition, Friedrich Vieweg und Sohn, Braunschweig, 1908)
Weil, A.: Basic Number Theory. Springer, New York (1973)
Zagier, D.: Modular forms whose Fourier coefficients involve zeta functions of quadratic fields, in Modular functions of one variable VI. Lect. Notes in Math., vol. 627, pp. 105–169. Springer (1977)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Japan
About this chapter
Cite this chapter
Ibukiyama, T., Kaneko, M. (2014). Class Number Formula and an Easy Zeta Function of the Space of Quadratic Forms. In: Bernoulli Numbers and Zeta Functions. Springer Monographs in Mathematics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54919-2_10
Download citation
DOI: https://doi.org/10.1007/978-4-431-54919-2_10
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-54918-5
Online ISBN: 978-4-431-54919-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)