| Abstrakt: |
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. [ABSTRACT FROM AUTHOR] |
