I've been trying to implement Schoof's Algorithm, and I understand it except for one part.
Near the bottom of page 7 of this paper is where my issue is: http://www-users.math.umn.edu/~musiker/schoof.pdf
When we check $x' - x_\overline{t}^q \equiv 0$ (mod $\psi_l$), how does this make sense? The left side is a rational function, while the modulo is a polynomial. Are we only checking the numerator? If not, how do you take rational functions modulo a polynomial? Thank you!