Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

An approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative
HTML articles powered by AMS MathViewer

by Jay Jorgenson, Lejla Smajlović and Holger Then HTML | PDF
Math. Comp. 91 (2022), 331-379 Request permission

Abstract:

Let $N$ be one of the $38$ distinct square-free integers such that the arithmetic group $\Gamma _0(N)^+$ has genus one. We constructed canonical generators $x_N$ and $y_N$ for the associated function field (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 25 (2016), pp. 295–319]). In this article we study the Schwarzian derivative of $x_N$, which we express as a polynomial in $y_N$ with coefficients that are rational functions in $x_N$. As a corollary, we prove that for any point $e$ in the upper half-plane which is fixed by an element of $\Gamma _0(N)^+$, one can explicitly evaluate $x_N(e)$ and $y_N(e)$. As it turns out, each value $x_N(e)$ and $y_N(e)$ is an algebraic integer which we are able to understand in the context of explicit class field theory. When combined with our previous article (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 29 (2020), pp. 1–27]), we now have a complete investigation of $x_N(\tau )$ and $y_N(\tau )$ at any CM point $\tau$, including elliptic points, for any genus one group $\Gamma _0(N)^+$. Furthermore, the present article when combined with the two aforementioned papers leads to a procedure which we expect to yield generators of class fields, and certain subfields, using the Schwarzian derivative and which does not use either modular polynomials or Shimura reciprocity.
References
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC (2020): 11F11, 11R37, 11G05
  • Retrieve articles in all journals with MSC (2020): 11F11, 11R37, 11G05
Additional Information
  • Jay Jorgenson
  • Affiliation: Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, New York 10031
  • MR Author ID: 292611
  • Email: jjorgenson@mindspring.com
  • Lejla Smajlović
  • Affiliation: Department of Mathematics, University of Sarajevo, Zmaja od Bosne 35, 71 000 Sarajevo, Bosnia and Herzegovina
  • ORCID: 0000-0002-2709-5535
  • Email: lejlas@pmf.unsa.ba
  • Holger Then
  • Affiliation: Freie Waldorfschule und Waldorfkindergärten Ausburg e.V., Dr.-Schmelzing-Straße 52, 86169, Germany
  • MR Author ID: 742378
  • ORCID: 0000-0002-0368-639X
  • Email: holger.then@gmx.de
  • Received by editor(s): July 30, 2019
  • Received by editor(s) in revised form: July 29, 2020, and November 3, 2020
  • Published electronically: October 21, 2021
  • Additional Notes: The first author acknowledges grant support from PSC-CUNY Awards, which were jointly funded by The Professional Staff Congress and The City University of New York
  • © Copyright 2021 American Mathematical Society
  • Journal: Math. Comp. 91 (2022), 331-379
  • MSC (2020): Primary 11F11, 11R37; Secondary 11G05
  • DOI: https://doi.org/10.1090/mcom/3619
  • MathSciNet review: 4350542