# Zeros of modular functions and automorphisms

Let $$F_N$$ be the field of modular functions of level $$N$$ and with Fourier coefficients in $$\mathbf Q(\zeta_N)$$. We have $$F_N=\mathbf Q(j, f^{(r,s)}_N),$$ where the $$f^{(r,s)}_N$$ are the Fricke functions, that is, the $$x$$-coordinates of $$N$$-torsion points on a suitable elliptic curve defined over $$\mathbf Q(j)$$. We further have $$\operatorname{Gal}(F_N/\mathbf Q(j))\cong \operatorname{GL}_2(\mathbf Z/N\mathbf Z)/\lbrace \pm 1\rbrace.$$ If $$\alpha\in\operatorname{GL}_2(\mathbf Z/N\mathbf Z)$$ then we can write $$\alpha=\alpha_d\gamma$$, where $$\alpha_d=\begin{pmatrix}1 \\ &d\end{pmatrix}\gamma$$, $$\det \gamma =1$$, and $$\det \alpha =d$$. The corresponding automorphisms are given by $$f^\gamma=f\circ \gamma$$, and if $$f=\sum_na_nq^{n/N}$$ then $$f^{\alpha_d}=\sum_na_n^{\sigma_d}q^{n/N}$$, where $$\sigma_d\colon Q(\zeta_N)\rightarrow Q(\zeta_N)\colon \zeta_N\mapsto\zeta_N^d$$.

Let $$\tau_0\in \mathfrak H$$ be a CM point. Suppose that $$f\in F_N$$ and that $$f(\tau_0)=0$$. Let $$\alpha$$ be a $$2$$-by-$$2$$ matrix with integral coefficients such that $$\alpha\tau_0=\tau_0,\qquad (\det \alpha,N)=1.$$ Let $$\overline{\alpha}$$ be the image of $$\alpha$$ in $$\operatorname{GL}_2(\mathbf Z/N\mathbf Z)$$. How to prove directly that $$f^{\overline{\alpha}}(\tau_0)=0$$?