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

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.