I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things.
I was wondering if the following interpretation of why modular forms are important is correct a) technically and b) in terms of getting the right picture.
Possible Intuition for Importance of Modular Forms:
We want arithmetic data about elliptic curves. Given a congruence subgroup $$\Gamma\in\{\Gamma_1(N),\Gamma_0(N),\Gamma(N)\}$$
we let $X(\Gamma)$ denote $\Gamma/\mathfrak{h}^\ast$ (where $\mathfrak{h}^\ast$ is the upper half-plane $\mathfrak{h}$ union the cusps $\mathbb{Q}\cup\{\infty\}$). We know then that $X(\Gamma)$ is the compactification of a moduli space whose points classify elliptic curves with some torsion data. Because of this, we want to understand the geometry of $X(\Gamma)$ because hopefully this will translate, via the moduli space concept, back to arithmetic data about elliptic curves.But, given a compact Riemann surface, one ideologically gets a lot of the information concerning the surface by studying meromorphic sections of certain holomorphic line bundles over $X$. A very natural line bundle attached to $X(\Gamma)$ is $(T_{X(\Gamma)}^{\ast1,0})^{\otimes n}$ (which is the $n^{\text{th}}$-tensor power of its holomorphic cotangent bundle). Thus, a natural place to look for geometric information about $X(\Gamma)$ is in the meromorphic sections of this bundle, denoted $\Omega^{\otimes n}(X(\Gamma))$. More specifically one may focus on the holomorphic sections $H^0(X(\Gamma),(T_{X(\Gamma)}^{\ast1,0})^{\otimes n})$ of this line bundle.
That said, the natural equivalence map $\pi:\mathfrak{h}\to X(\Gamma)$ gives rise to the pullback map $\pi^\ast:\Omega^{\otimes n}(X(\Gamma))\to\Omega^{\otimes n}(\mathfrak{h})$. But, since $\mathfrak{h}$ has only one chart, we can naturally identify $\Omega^{\otimes n}(\mathfrak{h})$ with $\text{Mer}(\mathfrak{h})$. Thus, we have a linear embedding $\pi^\ast:\Omega^{\otimes n}(X(\Gamma))\to \text{Mer}(\mathfrak{h})$. Since $\text{Mer}(\mathfrak{h})$ is easier to deal with (at least its easier to "see") we would like to identify the objects of interest, $\Omega^{\otimes n}(X(\Gamma))$ and its subspace $H^0(X(\Gamma),(T_{X(\Gamma)}^{\ast 1,0})^{\otimes n})$, with their image under $\pi^\ast$.
That said, one can prove that the image under $\pi^\ast$ of $\Omega^{\otimes n}(X(\Gamma))$ is $\mathcal{A}_{2n}(\Gamma)$ (the automorphic forms of weight $2n$ with respect to $\Gamma$) and the image of $H^0(X(\Gamma),(T_{X(\Gamma)}^{\ast 1,0})^{\otimes n})$ is contained in $\mathcal{M}_{2n}(\Gamma)$ (the modular forms of weight $2n$ with respect to $\Gamma$).
Ok, assuming the above is correct, there are three questions that begged to be asked:
- Why do we care about all of $\mathcal{M}_{2n}(\Gamma)$? Why don't we care more specifically about the image of $H^0(X(\Gamma),(T_{X(\Gamma)}^{\ast 1,0})^{\otimes n})$ under $\pi^\ast$? Can we describe this image (e.g. it's the cups forms for $n=1$)?
- What do odd weight modular or automorphic forms tell us? If $-I\in\Gamma$ then there are no non-zero such objects, but in the cases when $-I\notin\Gamma$, do we gain anything by looking at odd weights?
- What does the geometry of $X(\Gamma)$ tell us about elliptic curves? For example, the genus of $X(\Gamma)$ tells us things about the objects we parameterize. That said, we don't need to study sections of line bundles to get this geometric data. Indeed, the genus for $X_0(1)$ can be deduced from the fact that the $j$-invariant has one simple pole, and thus must be induced a biholomorphism $j:X_0(1)\to\mathbb{P}^1$. From there, we can find the genus of $X(\Gamma)$ by using the natural projection $X(\Gamma)\to X_0(1)$ and the Riemann-Hurwitz formula. So, what geometric information about $X(\Gamma)$ is important that one needs automorphic/modular forms to get?
Thank you so, so much friends! I have been grappling with the "big picture" of modular forms of late, and this is the best I cam up with. I am very excited to hear your responses!
Best Answer
The theory of modular forms arose out of the study of elliptic integrals (as did the theory of elliptic curves, and much of modern algebraic geometry, and indeed much of modern mathematics). People understood that (complete) elliptic integrals (which we would think of as the number obtained by integrating a de Rham cohomology class, e.g. the one associated to the holomorphic differential on an elliptic curve, over a homology class on the curve) depended on an invariant (what we would think of as the $j$-invariant of the elliptic curve, although historically people used other invariants, often depending on some auxiliary level structure, such as $\lambda$, or $k$ (the square-root of $\lambda$)). This invariant was called the modulus (which is the origin of the adjective modular in this context).
People knew that if you replaced an elliptic curve by an $N$-isogenous one, then the elliptic integral would be multiplied by $N$ (in terms of $\mathbb C/\Lambda$, the elliptic integral is just one of the basis elements for $\Lambda$, and multiplying this by $N$, while keeping the other one fixed, gives a new elliptic curve related to the original one by an $N$-isogeny). They asked themselves how they could describe the modulus for this $N$-isogenous elliptic curve (or integral) in terms of the original one. This led them to find explicit equations for the modular curves $X_0(N)$ (for small values of $N$).
With these kinds of investigations (and remember, these were brilliant people --- Jacobi, Kronecker, Klein, just to mention some spanning a good part of the 19th century), it was natural that they were led to modular forms as well as modular functions (as one example, the Taylor coefficients of elliptic functions give modular forms; as another, the coordinates --- say with respect to Weierstrass elliptic functions --- of $N$-torsion points give level $N$ modular forms).
So all these investigations grew out of the study of elliptic integrals, but became intimately connected with the invention of algebraic topology, the development of complex analysis (by Riemann, and then Schwarz, and then the uniformization theorem), the development of hyperbolic geometry; basically all the fundamental mathematics of the 19th century that then drove much of the developments of 20th century mathematics.
The connections with arithmetic were also observed early on. Jacobi already introduced theta series and saw the relationship with counting representations by quadratic forms (e.g. he proved that the number of ways of writing $n \geq 0$ as a sum of four squares is equal to $\sum_{d | n, 4 \not\mid d} d$, using weight $2$ modular forms on $\Gamma_0(4)$).
But Kronecker (and maybe Abel, Eisenstein and even Gauss before him) also knew that modular forms, when evaluated at CM elliptic curves (i.e. at quadratic imaginary values of $\tau$) gave algebraic number values in some contexts. Gauss was led to this by the analogy with cyclotomy: $N$-torsion on an elliptic curve was analogous to $N$th roots of $1$ on the unit circle, and the analogy is tighter when the elliptic curve has CM, because then the $N$-torsion points become a cyclic module over the ring of CMs, just as the $N$th roots of $1$ are a cyclic module over $\mathbb Z$ (i.e. a cyclic group).
Kronecker (and again, maybe people before him) realized that CM elliptic curves corresponded to lattices $\Lambda \subset \mathbb C$ that belong to ideal classes in quadratic imaginary fields, and so saw a relationship between CM elliptic curves and class field theory for quadratic imaginary fields (Kronecker's Jugendtraum). This also related to the previous work on evaluating modular forms at CM points.
All this is just to say that even in the 19th century the subject was very deep, and already very connected to number theory, as well as everything else.
Ramanujan knew the theory very well, and discovered new phenomena (e.g. his conjectures on the behavious of $\tau(n)$, defined by $\Delta = q\prod_{n=1}^{\infty} (1- q^n)^{24} = \sum_{n=1}^{\infty} \tau(n) q^n$). Mordell proved Ramanujan's conjecture on the multiplicative nature of $\tau$, and Hecke introduced his operators to systematize Mordell's method of proof.
At this point, the subject moved in a more representation-theoretic and analytic direction, with the generalization to automorphic forms. With the discovery in the 50s, 60s, and 70s of the modularity conjecure for elliptic curves over $\mathbb Q$, and related ideas, the arithmetic theory of modular forms became a central topic again. See this answer on MO for more on that.
Mazur's theorem on torsion points on elliptic curves over $\mathbb Q$ is one of the deepest results that comes from thinking of $X_0(N)$ and $X_1(N)$ directly in modular terms. But already the proofs are more automorphic in nature, and are focussed on the relationships between modular forms, particularly Hecke eigenforms, and Galois representations. That's where the modern focus primarily lies. You can see some of the other answers linked from my webpage (here) for more on that.
Let me close this long discussion by just saying that the passage to Galois representations as a focus is a natural development from Kronecker's Jugendtraum, but reflects a shifting of attention from abelian class field theory for quadratic imaginary fields to non-abelian (more precisely, $\mathrm{GL}_2$) class field theory for $\mathbb Q$. (Note that the former embeds in the latter, since the indcution of a Galois character of a quadratic extension gives a two-dimensional rep. of $G_{\mathbb Q}$.)
Finally, let me mention that the main theme of Mazur's article is congruences between cuspforms and Eisenstein series (this is what the Eisenstein ideal measures), and so it's hard to have one without the other. (In some sense, Eisenstein series are like the trivial Dirichlet character mod $N$, while cuspforms are like the non-trivial characters. Which is more important depends on what you are doing; in many problems you need to consider both.)