 Fricke involution

The Fricke matrix $W_N=\left(\begin{array}{cc}0& -1\\ N& 0\end{array}\right)$ normalizes ${\Gamma }_1(N)$ and ${\Gamma }_0(N)$ under the appropriate determinant-normalized slash action. It exchanges the cusps zero and infinity.

Fricke involution is a concept found in the context of modular forms and algebraic geometry, particularly in relation to the study of modular curves. It is a specific type of involution—meaning it is an operation that can be applied twice to return to the original state—defined on the upper half-plane or on modular forms.