# Do general automorphic factors arise in some canonical way?

Disclaimer: none of what I'm about to say I understand particularly well; corrections and clarifications are not only welcome but will be accepted with great gratitude.

The wikipedia article on Factors of Automorphy gives the closest to an intuitive definition of automorphic form that I have encountered. Paraphrasing:

Given a group $G$ acting on a complex-analytic manifold $X$, a holomorphic function $f\colon X\to\mathbb C$ is called automorphic if the natural action of $G$ on $f$ preserves the divisor of $f$.

From this definition, my understanding is that one can associate to every automorphic form $f$ and element $g\in G$ a function $j_g\colon X\to\mathbb C$ given by $f(gz)=j_g(z)f(z)$.

It is immediately clear that $j_g$ is holomorphic and nowhere zero. Additionally (based on what little I've gleaned from J.S. Milne's notes on Modular Functions and Modular Forms), we have that $j_g$ satisfies the cocyle condition since $$j_{gh}(z)=\frac{f(ghz)}{f(z)}=\frac{f(ghz)}{f(hz)}\frac{f(hz)}{f(z)}=j_g(hz)j_h(z)$$

This motivates that we define a factor of automorphy to be a function $j\colon G\times X\to\mathbb C$ such that the restrictions $j_g\colon X\to\mathbb C$ are holomorphic and nowhere zero, and such that $j$ satisfies the cocyle condition.

Then we can think of functions being automorphic relative to specific factors of automorphy. In some sense if two functions have the same factors of automorphy, then they are homogeneous with respect to the group action in the same way, hence their quotients ought to well-define meromorphic functions on the quotient $G\backslash X$.

An example I like to illustrate in my head comes from considering how the multiplicative group $\mathbb C^\times$ acts on $\mathbb C\times\mathbb C\setminus\{0\}$. The quotient is the Riemann sphere, and one possible factor of automorphy is given by $\mathbb C\ni z_0\to z_0$. Recalling that holomoprhic functions have power series expansions, it is evident that the condition $f(z_0z_1,z_0z_2)=z_0f(z_1,z_2)$ has solutions only when $f$ is a homogeneous polynomial of degree $1$. If we further observe that products of automorphic factors are automorphic factors, we see that all homogeneous polynomials of degree $k$ are automorphic with respect to $\mathbb C^\times$ and factor of automorphy $z_0\to z_0^k$.

In the case where $X$ is an open subset of $\mathbb C$, Milne shows that we have in fact a canonical factor of automorphy for any (discrete) group action on $\mathbb C$ given by $j_g(z)=(dg)_z$ where $(dg)_z$ is the differential of the action of $g\in G$ on $X$ evaluated at the point $z$ in $X$. In particular, when $G=SL(2,\mathbb Z)$ acting on $\mathbb H$ by fractional linear transformations, we have that the canonical factor of automorphy is $(cz+d)^{-2}$ for $g=\left[\begin{matrix}a&b\\c&d\end{matrix}\right]$, and the automorphic forms are then precisely the (weak? classical?) modular forms.

Now, the wikipedia article on automorphic factors, defines an automorphic factor to be one such that $|j_g|=|cz+d|^k$ for $g=\left[\begin{matrix}a&b\\c&d\\\end{matrix}\right]$, which are needed to talk about the Dedekind eta function, apparently.

My question then has three (small) parts:

1. Is my analogy with functions on the Riemann sphere correct?
2. Are the automorphic factors (as defined in the wikipedia article) canonical in any way, or does only the factor $(cz+d)^k$ arise from a canonical construction?
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Yes, your comparison with "homogenous" functions on the Riemann sphere (that is, complex projective one-space), is good, insofar as the objects are not really functions on the thing, but "global sections of a vector bundle". Automorphic forms on the upper half-plane $H$ are (mostly) not $\Gamma$-invariant, so do not "live" on the quotient space $\Gamma\backslash H$.
On another hand, one of the modernizations of the theory of automorphic forms converts them to genuinely left $\Gamma$-invariant functions on a group, such as $SL(2,R)$, acting transitively on $H$, for example: for $f(\gamma z)=j(gamma,z)f(g)$, then $F(g)=j(g,i)^{-1}f(gi)$ is the conversion. It is an exercise to check that $F$ is left $\Gamma$-invariant. The automorphy factor/cocycle is converted into right equivariance under the maximal compact subgroup $SO(2)$ fixing the point $i\in H$. (One warning: for "half-integral weight", the automorphy factor is such that the automorphic forms lift to _metaplectic_groups_, rather than $SL(2,R)$.)
The latter context correctly insinuates that, apart from normalization and fooling-around, the collection of possible automorphy factors is actually indexed by the (irreducible) representations of the isotropy subgroup $SO(2)$ of the base point $i\in H$ (and similarly for other groups and spaces). Since this group is a circle, its repns are indexed by integers $k$, with $\pmatrix{\cos\theta & \sin \theta \cr -\sin \theta & \cos \theta}\rightarrow e^{ik\theta}$. In particular, there are obvious relations among these. Note that $cz+d$ at $z=i$ and with $c,d$ the lower entries of such a matrix becomes $-i\sin\theta+\cos\theta=e^{-i\theta}$.
(The $|cz+d|$ arises by tweaking, using $\Im(gz)=\Im(z)/|cz+d|^2$.)
So, yes, it is certainly reasonable to think about automorphic forms as "sections of a line bundle". Yes, the expressions $(cz+d)^k$ are special and canonical. Yes, even in more general circumstances (Siegel modular forms, Hilbert modular forms, ...) the line-bundle viewpoint is helpful.