Toric varieties and modular forms

Let $N\subset {\text{RR}}^r$ be a lattice, and let $\mathrm{deg}\text{\hspace{0.17em}⁣}:N\to \text{CC}$ be a piecewise-linear function that is linear on the cones of a complete rational polyhedral fan. Under certain conditions on $\mathrm{deg}$, the data $(N,\mathrm{deg})$ determines a function $f\text{\hspace{0.17em}⁣}:\text{HHH}\to \text{CC}$ that is a holomorphic modular form of weight $r$ for the congruence subgroup ${\mathrm{\Gamma }}_1(l)$. Moreover, by considering all possible pairs $(N,\mathrm{deg})$, we obtain a natural subring $\text{TTT}(l)$ of modular forms with respect to ${\mathrm{\Gamma }}_1(l)$. We construct an explicit set of generators for $\text{TTT}(l)$, and show that ${\TTT} (l)$isstableundertheactionoftheHeckeoperators\mathrm{.}Finally,werelate${\TTT} (l)$totheHirzebruchellipticgenerathataremodularwithrespectto$\Gamma_{1} (l) $.