simpler way to define modular forms

Question

i'm trying to define modular forms for the full modular group for a general audience and want to make it as simple as possible. please tell me if this is correct:

definition of a modular form $f$:

1. holomorphic function $f$ from upper half plane to $\mathbb{C}$.
2. $f(z)$ bounded as Im($z$) tends to infinity.
3. $f$ satisfies the usual transformation law under $SL_2(Z)$

All the definitions I've seen say that, instead of 2., f should be holomorphic at infinity. I know that this is more beautiful and better for generalizations of the definition, for example to congruence subgroups... but then I have to explain what it means to be holomorphic at infinity, by using a power series representation. That's an extra step which I think is implicit in my definition because.....

By 3., $f(z)$ is periodic and so by complex analysis it must have a fourier series expansion $f(z)=\sum_{n=-\infty}^{\infty} a(n) e^{2\pi i nz}$. By 2. and complex analysis, we must have a(n)=0 for $n<0$. Then the resulting fourier series shows that $f$ is holomorphic at infinity and we arrive at the same thing.

Is all this right or am I missing something?

Answer 1

(Edited/corrected/amplified) "Bounded at infinity" is a condition that includes not only cuspforms but also Eisenstein series. If by "holomorphic at infinity" one means exactly holomorphy of $f(z)$ on the quotient by translations $z\rightarrow z+1$, then this is the same as "bounded at infinity".

On another hand, if one is trying to make a (holomorphic weight $2k$) modular form $f(z)$ descend to a Riemann surface $\Gamma\backslash \mathfrak H$, it can only do so as a "symmetric differential form $f(z)\,dz^k$, that is, a section of some line bundle. Then with $q=e^{2\pi iz}$ locally, that $dz^k$ shifts the apparent Fourier expansion...

Why not just give the condition at infinity in terms of the Fourier expansion, anyway, rather than wrangle about implications?

Edit-further: while I myself do not at all think of Eisenstein series as "having poles at cusps", one can easily find such remarks in well-established literature, and the (at least two) viewpoints probably should be reconciled, whatever one thinks of the reasonableness of one or the other. Depending on "where one is going" in talking to a "general audience", the issue might not merit attention.

And one more edit! :) To be clear: for "level one" holomorphic elliptic modular forms, the three conditions in the question do specify cuspforms and Eisenstein series correctly. That is literally so. On another hand, as already noted in the question, this style of description develops problems in any more general circumstance, partly due to ambiguities in the language, and/or conflicts in usage, but also to wanting to make "holomorphy" independent of choice of coordinates (which is, in general, a good impulse). For example, cuspforms do go to zero "at infinity" in the sense of approaching a rational number inside a fixed image of fundamental domain, and, in fact, in the stronger sense of going to zero uniformly approaching the real line. But Eisenstein series blow up as the imaginary part goes to $0$, and so on. That is, this and other "dangers" easily entrap the unwary, which may include people looking at these things in an elementary way, despite the appeal of not setting up machinery. It's not just that "more complicated" set-ups are more aesthetically pleasing or stylistically "cooler", but essentially necessary to avoid troubles. So I recommend being aware of the delicacy of the literally-correct assertions 1,2,3 in the question, despite the seeming innocence of the situation.

Answer 2

This is not exactly an answer to the question as asked, but I was recently confronted to the same problem: how to define, for a general audience, a modular form for the full modular group in a way which is both correct and quick? Here is a solution I used:

A modular form of weight $k$ (and level 1) is a formal series $f(q)=\sum_{n \geq 0} a_n q^n$, converging on the open unit disc, such that, in terms of the variable $z$ defined by $q=e^{2 i \pi z}$, $f$ satisfies $f(-1/z)=z^k f(z)$.

It is short and allows you to talk without further steps of the $q$-expansion of $f$, and avoid even mentioning the full modular group $Sl_2(\mathbb Z)$ or using any explicit element in it. This also avoids having to justify the existence of a $q$-expansion from $f(z+1)=f(z)$, for which I have always felt unsatisfied by the common three-word justification "by Fourier theory".

Of course, depending of what you want to do next, this may be either an advantage or a shortcoming. If you need to discuss other congruences subgroups later, this is not the definition you want. But if what you need about modular forms is mainly their $q$-expansions, as in many arithmetical applications, this definition may be useful.