Why does the "automorphic factor" exist?

Question

In the theory of automorphic forms, functions we consider are not totally invariant (why? is it because otherwise there would be too few of them, like none except constants?), but "twisted" by a certain "automorphic factor": $$f(\gamma z) = j(\gamma,z)^kf(z)$$ or a fixed $k$ and $j(\gamma,z) = (cz+d)/|cz+d|$ (or is it sometimes just $cz+d$?). I do not get at all reasons for this factor to be this one, rather than... anything else. Are there theories of automorphic forms for more generic $j(\gamma,z)$? or has this one very specific reasons to be of this form?

Answer 1

There are several slightly different traditions in story-line and in the objects that first arise, for one thing. One well-known version is that "elliptic modular forms" are holomorphic functions on the complex upper half-plane meeting a cocycle condition $f(\gamma(z))=(cz+d)^{2k}\cdot f(z)$, for $\gamma$ in $SL_2(\mathbb Z)$ (etc.) with lower row $(c\,d)$, where $2k$ is the "weight". So these functions are not invariant... but the symmetric differential forms (sections of line bundles!) $f(z)\cdot dz^k$ are invariant. (So, yes, the $cz+d$ powers are special, in this sense...)

The other primary story is about Maass' waveforms, which are $SL_2(\mathbb Z)$-invariant, but merely required to be eigenfunctions for the invariant Laplacian on the upper half-plane.

These two ideas can be merged by converting functions $f$ on the upper half-plane to left $\Gamma=SL_2(\mathbb Z)$-invariant functions $F_f$ on the group $G=SL_2(\mathbb R)$ that acts transitively on it... For a waveform $f$ we can just let $F_f(g)=f(g(i))$. This creates a left $\Gamma$-invariant function on $G$ that is also right $K=SO(2,\mathbb R)$-invariant, and an eigenfunction for Casimir.

For a modular form $f$ of weight $2k$, let $F_f(g) = (cz+d)^{2k}f(g(i))$. A little computation (if I got the sign in the exponent right) verifies that this has become left $\Gamma$-invariant (no cocycle stuff any more), and is right $K$-equivariant by a character corresponding to the weight. EDIT: and is again an eigenfunction for Casimir, with eigenvalue determined by the weight $2k$.

There are other possible cocycles in general, for example for half-integral weight forms... which do not extend from the discrete subgroup to the ambient Lie group... requiring an extension (e.g., metaplectic group...).