## An approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative

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Jay Jorgenson, Lejla Smajlović and Holger Then
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## 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

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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: 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