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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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An approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative
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by Jay Jorgenson, Lejla Smajlović and Holger Then HTML | PDF
Math. Comp. 91 (2022), 331-379 Request permission


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.
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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:
  • 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:
  • 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:
  • 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:
  • MathSciNet review: 4350542