New modular forms from given ones (awaiting review)

In addition to the different situations we have seen above, which involve modular functions and forms, there are also many ways to create new modular forms from given ones.

Vector space or graded ring structure

The most trivial way to construct a modular form from given ones is to use the vector space structure: if $f$ and $g$ have the same weight and multiplier system on some group $G$, then of course so does $\lambda f+\mu g$, for any constants $\lambda$ and $\mu$. Another way is to use the fact that the set of all modular forms with trivial multiplier for a given group has the structure of a graded ring. Even more is true: if $f$ and $g$ are modular for the same group $G$, then so is the product $fg$, whose weight will be the sum of the weights and multiplier system, the product of the multiplier systems.

Differentiation

If we differentiate the modular identity $f(\gamma z)=v(\gamma)(cz+d)^kf(z)$ and use $(\gamma z)'=(cz+d)^{-2}$, we obtain \[f'(\gamma z)=v(\gamma)((cz+d)^{k+2}f'(z)+k(cz+d)^{k+1}f(z))\;.\] Thus $f'$ is almost a modular form (it is in fact a quasi-modular form) of weight $k+2$, and it is exactly modular if $k=0$. There are many ways to "repair" this defect of modularity: two of them involve modifying the differentiation operator by using the auxiliary functions $1/y=1/\Im(z)$ or the quasi-Eisenstein series $E_2$. Another is to take suitable combinations which remove the extra terms that prevent modularity: for instance, if $f$ is of weight $k$ and $g$ of weight $\ell$, then $f^\ell/g^k$ is of weight $0$, so its derivative is really modular of weight $2$, and expanding shows that $\ell f'g-kfg'$ is modular of weight $k+\ell+2$. This is a special case of a series of bilinear operators called the Rankin‒Cohen operators.

Changing the group

If we accept to modify the group on which a function is modular, there are many other ways to create new modular forms. For instance, if $f$ is modular on some group, then $f(mz)$ will also be modular of the same weight on some other group for any $m\in\Q^{\times}$. A similar construction implies that if $f(z)=\sum_{n\ge n_0}a_nq^{n/N}$ is modular, then so is $\sum_{n\equiv r\pmod{M}}a_nq^{n/N}$ and $\sum_{n\ge n_0}\psi(n)a_nq^{n/N}$ for any periodic function $\psi$. This last construction is called twisting by $\psi$.

Enlarging the group

Even more interesting is the possibility to enlarge the group on which a function is modular: if $f$ is modular on some subgroup $H$ of finite index of some other group $G$, say with trivial multiplier system, and if $(\gamma_i)$ is a system of left coset representatives of $H\backslash G$, so that $G=\bigsqcup_iH\gamma_i$, then it is clear that $\sum_i$ $f|_k\gamma_i$ will be modular on $G$. This is a special case of the averaging procedure mentioned at the very beginning. An important example combining the above two methods is the construction of Hecke operators: let $p$ be a prime. If, for instance, $f$ is modular of weight $k$ on the full modular group $\Gamma$, the functions $f((z+j)/p)$ and $f(pz)$ will be modular only on the subgroup $\Gamma_0(p)$ of $\Gamma$, but it is immediate to show that the linear combination $g=\sum_{0\le j\le p-1}f((z+j)/p)+p^kf(pz)$ is again modular on the full modular group $\Gamma$, and we can define the Hecke operator $T(p)$ by $T(p)(f)=g/p$.