I find it somewhat discouraging that I took about 8 courses of graduate mathematics and still don't know what modular forms are.
It's just a analytic function on the moduli space of elliptic curves.
The collection of equivalence classes of elliptic curve (torii of the form C/lattice) has the structure of a complex space (it's not a complex manifold, but rather a complex moduli stack). Modular forms are just analytic functions on it. That's all.
This dumb article doesn't help matter by presenting a brazen lie in the headline. Fifth fundamental operation, my butthurt ass.
First, they can be differential forms, not only functions. Second, there's an important note that we don't look only at things over C. For example, specifically in the context of Fermat's Last Theorem, we need Hida's theory of p-adic families of modular forms. Much of the arithmetic of modular forms comes from the modular curves being algebraic and (almost) defined over the integers.
If modular forms are (global?) sections of the structural sheaf of the moduli space of elliptic curves, the differential forms view will just be the standard construction of sheaf of 1-differentials. Similarly, since elliptic curves are easily defined over arithmetic fields, arithmetic modular forms will just be same thing, but over C_p or something like that.
I actually might be totally off in the above, but I doubt I am: that’s the power of Grothendieck approach, where everything just falls into its natural place in the framework.
There is rich structure in this area of maths that goes well beyond just sections of some sheaf, or at least this is what Serre, Deligne, Langlands, Mazur, Katz, Hida, Taylor, Wiles and many others seem to think.
Now make M to be a Mobius transformation, so that f((ax + b)/(cx + d)) = (cx + d)^k f(x), where the coefficients are in SL(2,Z) ie. integers with ad - bc = 1. Mobius transformations like this are common.
That's more or less it as far as I'm aware. There's some growth rate condition too, I don't think it's as important to an intuitive understanding as the transformation law.
Disclaimer; my training was in mathematical physics not mathematics. To me, what l wrote here was enough for me to feel like I understand what they are, at least to some basic level.
let f(ax, ay) = a^k f(x,y) be a homogeneous function in the plane x,y for a scalar a, then f can also be written as:
f(x,y) = f(y*x/y, y) = y^k f(x/y,1)
and the symmetry by a matrix transformation in the plane
x-> ax + by and y-> cx+dy then transforms the function f as:
f(ax+by, cx+dy) = (cx+dy)^k f((ax+by)/(cx+dy),1)
now introduce z=x/y and f(x/y,1)=F(z) then
F((az+b)/(cz+d)) = (cz+d)^k F(z)
Projective spaces are cool ;-)
just my 2 ct
I forgot to add to my post that, SL(2) is the symmetry group of the Riemann sphere. So since you often use this as your space in complex analysis, this symmetry transformation is everywhere. So that's one reason why you might expect to see modular forms in lots of places. (At least, that's what I understood, maybe a mathematician would tell you differently.)