Effective equidistribution of horocycle lifts

We give a rate of equidistribution of lifts of horocycles from the space $\mathrm{SL}(2,\mathbb Z)\backslash \mathrm{SL}(2, \mathbb R)$ to the space $\mathrm{ASL}(2,\mathbb Z)\backslash \mathrm{ASL}(2,\mathbb R)$, making effective a theorem of Elkies and McMullen. This result constitutes an effective version of Ratner's measure classification theorem for measures supported on general horocycle lifts. The method used relies on Weil's resolution of the Riemann hypothesis for function fields in one variable and generalizes the approach of Str\"ombergsson to the case of linear lifts and that of Browning and the author to rational quadratic lifts.