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Orbital $L$-functions for the Space of Binary Cubic Forms

Published online by Cambridge University Press:  20 November 2018

Takashi Taniguchi  and
Frank Thorne
Takashi Taniguchi
Affiliation:
Department of Mathematics, Graduate School of Science, Kobe University, Kobe 657-8501, Japan Department of Mathematics, Princeton University, Princeton, NJ 08540, USA, e-mail: tani@math.kobe-u.ac.jp
Frank Thorne
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA, e-mail: thorne@math.sc.edu
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Abstract

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We introduce the notion of orbital $L$-functions for the space of binary cubic forms and investigate their analytic properties. We study their functional equations and residue formulas in some detail. Aside from their intrinsic interest, the results from this paper are used to prove the existence of secondary terms in counting functions for cubic fields. This is worked out in a companion paper.

Keywords

11M4111E76binary cubic formsprehomogeneous vector spacesShintani zeta functionsL-functionscubic rings and fields
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 6 , 01 December 2013 , pp. 1320 - 1383
Copyright
Copyright © Canadian Mathematical Society 2013

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