Elliptic functions and equations of modular curves
  Let $p\ge 5$ be a prime. We show that the space of weight one Eisenstein series defines an embedding into $\PP^{(p-3)/2}$ of the modular curve $X_1(p)$ for the congruence group $\Gamma_1(p)$ that is scheme-theoretically cut out by explicit quadratic equations. 
Lev Borisov, Paul Gunnells, Sorin Popescu