Whenever I encounter anything about modular parametrizations, I have a feeling it is something very unnatural: you have some kind of moduli space and all of a sudden it parametrizes an object represented by a point of some similar moduli space (sometimes even of itself?). Why on earth should this happen? How such a strange possibility could possibly occur to anybody?

And yet I know this is one of the most deep and important achievements in current mathematics. Can it be given some sort of heuristic/intuitive justification an outsider could understand?

Proper reaction to such questions probably is "why don't you read this or that paper"? However I would only accept such answer if the corresponding paper contains a good noob-friendly introduction since I certainly am a noob in this case.

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    $\begingroup$ I think this all started with Ramanujan (accidentally) and, then, coincidentally people (Mordell, Hecke and others) discovered that these power series expansion on Fourier series had a geometrical meaning. See, for instance, mathoverflow.net/questions/75709/… $\endgroup$ – user40276 Nov 11 '15 at 19:24
  • $\begingroup$ @user40276 Sorry even this (and also another question linked from there) seems to be too technical for me. Besides, I would prefer staying on the geometric side of the intuition if possible, without diving into analytic subtleties like $L$-functions, etc. I don't even understand what the modular parametrization has to do with power series expansions of Fourier series. $\endgroup$ – მამუკა ჯიბლაძე Nov 11 '15 at 19:33
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    $\begingroup$ The link between power series expansion of modular forms and moduli is given by Hecke operators. They can be seen as acting on a moduli space (of elliptic curves with additional structure on torsion points of a given order) or as acting on modular forms (because modular forms are just sections of a line bundle). Shimura constructed long time ago an abelian variety $A_f$ from a modular eigenfunction $f$ (an eingenfunction of the Hecke operator) of weight $k =2$ by decomposing the Jacobian of the moduli space $J(X_0 (N)) \cong \bigoplus_f A_f$ with the action of the Hecke algebra. $\endgroup$ – user40276 Nov 11 '15 at 19:57
  • $\begingroup$ Now, modularity asks for the opposite direction. $\endgroup$ – user40276 Nov 11 '15 at 19:57
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    $\begingroup$ The motivation for starting a formalization of this relation is coincidental as I said in my first comment. People were interested in understanding the Ramanujan work about the tau function, that is the coefficients of power series expansion of the discriminant (which coincides, by the way, with the discriminant of an elliptic curve after the q-expansion) and suddenly discovered that these coefficients codified the solutions mop $p$ of certain equations. Maybe these remarks will help. Modular forms can be seen as functions on lattices. … $\endgroup$ – user40276 Nov 12 '15 at 10:31

Take an elliptic curve. At each prime p you have some local information. It either has bad reduction, and there is a reduction type, or good reduction, and then you count the points and the number is $1+p-a_p$ with $a_p$ an integer between $-2 \sqrt{p}$ and $2 \sqrt{p}$. Also if the reduction is multiplicative, one should remember split or nonsplit reduction. One should also ignore the difference between reduction types that can be isogenous to each other. This information turns out to be the natural information about the elliptic curve to use in a whole lot of arithmetic situations.

Suppose I give you, for each prime, something that looks like this bit of information about the reduction of the elliptic curve - either a fiber type or a number $a_p$. How can you ever hope to tell if there is an elliptic curve that stitches all these disparate numbers together? Well, you can write down some necessary conditions, like that there are only finitely many primes of bad reduction. But there will still be only uncountably many "plausible" sequences, of which only countably many come from an elliptic curve. What makes them special?

Now take a modular form. This is an object that has primes of good reduction and bad reduction. The good reduction primes will just be the primes not dividing the level. At these primes, we have a Hecke eigenvalue, which is a real number between $-2 \sqrt{p}$ and $2\sqrt{p}$ - in fact it lies in the ring of integers of some number field, depending on the modular form. What's more, we can look at the action of the automorphism group of the modular curve on our modular form. If we look at the different representations that can appear, they turn out to be in perfect correspondence with the different reduction types of an elliptic curve (at least when the coefficient field is $\mathbb Q$).

So you have two different ways of stitching together this local information to get global objects - elliptic curves and modular forms whose coefficient field happens to be $\mathbb Q$. In both, you choose only a countable set of special sequences out of an uncountable set of possibilities. Could they be the same?

That would be quite impressive, because the ways they arise are so different! One comes from looking arithmetically at a natural class of algebraic geometry objects, and the other from doing analysis on some special symmetric spaces.

It would be particularly impressive because some facts (like the bound on the $a_p$) are much easier to prove on the elliptic curve side, while others (like a bound on the average of $a_p/\sqrt{p}$) are much easier to prove on the modular curve side.

Of course this sounds totally ridiculous. Those two things should be totally unrelated! Except that you can get, by a not-too-difficult construction, elliptic curves from modular forms...

So you have two ways to solve this incredibly mysterious problem of patching together local data at each prime to get a single global picture. They behave the same, and one is a subset of the other. Might the subset in fact be the whole - could these two pictures be the same? Once you came up with the question, and understood its importance, you would start searching for evidence, counterexamples, etc., and like the mathematical community in the mid-late 20th century, be convinced that they are.

Now this correspondence between elliptic curves and modular forms does not mention the actual map from the modular curve to the elliptic curve. Constructing it, giving the equality of $a_p$s, is nontrivial (the Tate conjecture for morphisms of abelian varieties) and doesn't really generalize. That is one reason you shouldn't think about this too geometrically. This is an arithmetic statement, not a geometric one. Note that it is only true for elliptic curves over $\mathbb Q$.

Also note that the fact that the modular curves are moduli spaces of elliptic curves is a coincidence, akin to the small number coincidences from Lie groups, like the fact that $PGL_2=SO_3$.

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  • $\begingroup$ This is a very good answer, thank you! The key seems to be in the third paragraph, may I ask you to just slightly expand it? I mean, is there any sense in the combination of words "reduction of a modular form at a prime"? $\endgroup$ – მამუკა ჯიბლაძე Nov 12 '15 at 6:54
  • $\begingroup$ As for your last remark - this is in fact a good example to argue that there are never any pure coincidences. After all, your equality of Lie groups can be explained by $\mathbb C\mathbf P^1=S^2$. You can call also this a coincidence but... $\endgroup$ – მამუკა ჯიბლაძე Nov 12 '15 at 6:57
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    $\begingroup$ @მამუკაჯიბლაძე There is. From a modular form and a prime $p$ you obtain an irreducible representation of $GL_2(\mathbb Q_p)$. This is from viewing the modular form as a differential form on $\lim_{n \to \infty} X(p^n)$, which has an action of $GL_2(\mathbb Q_p)$, and taking the representation it generates, which turns out to be irreducible. This is the "reduction mod $p$". $\endgroup$ – Will Sawin Nov 12 '15 at 13:54
  • $\begingroup$ So do you imply that ultimately to understand all this one has to translate modular forms into the language of automorphic representations? $\endgroup$ – მამუკა ჯიბლაძე Nov 13 '15 at 8:53
  • $\begingroup$ @მამუკაჯიბლაძე If you want to understand everything about how the local theory relates to the global theory, I think yes. You can express everything with classical modular forms, but some things will be less elegant. $\endgroup$ – Will Sawin Nov 14 '15 at 18:25

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