 Lattice model of a modular form

Let $\mathcal{L}$ be the set of complex lattices. A weight-$k$ lattice function is a function satisfyingSet ${\Lambda }_{\tau }=\mathbb{Z}\tau +\mathbb{Z}$ and $f(\tau )=F({\Lambda }_{\tau })$. For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in S{L}_2(\mathbb{Z})$, a change of oriented basis gives ${\Lambda }_{\gamma \tau }=(c\tau +d{)}^{-1}{\Lambda }_{\tau }$, henceThus the resulting function has the weight-$k$ transformation law; ordinary invariance under the modular group holds only when $k=0$.

Conversely, a function with this transformation law defines $F({\omega }_2{\Lambda }_{\tau })={\omega }_2^{-k}f(\tau )$ independently of the chosen oriented basis. To obtain actual modular forms, one must additionally require holomorphy on the complex upper half-plane and at the modular cusp; homogeneity alone imposes no analytic regularity. For even $k>2$, the lattice Eisenstein sum is an example, and its scaling law explains its weight.^[ref]