Intuition for the Importance of Modular Forms

Question

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:

1. 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 $n=1$)?
2. 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?
3. 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!

Answer 1

I don't have very specific answers to your questions (some of these might be better answered by someone with more background in complex geometry), but I think that I can address some aspects of the importance of modular forms for number theory.

To understand, I think that a little historical perspective is always good to have. This is not exactly a direct answer but please bear with me.

In his foundational paper on algebraic number theory, Riemann expressed the "completed zeta function" $\Lambda(s) = \Gamma(s/2) \pi^{-s/2}\zeta(s)$ as the Mellin transform of $(\theta(\tau)-1)/2$ , where $\theta(\tau) = \sum_{n \in \mathbf Z} q^{n^2}$ ($q=e^{2\pi i \tau}$) is the Riemann theta function. By the rapid convergence of this series for $\text{Im} \tau > 0$, the theta function is holomorphic in the upper-half plane, and it follows from the Poisson summation formula that $\theta(-1/\tau) = (-iq)^{1/2}\theta(q)$. By using this functional equation, Riemann proved that $\Lambda(s)$ extends to a holomorphic function on $\mathbf C$ and satisfies the functional equation $\Lambda(s)=\Lambda(1-s)$.

Now $\theta$ is obviously periodic of period $1$, so we see that it transforms nicely under the subgroup of $\text{SL}_2(\mathbf Z)$ generated by $\tau \mapsto \tau+1$ and $\tau \mapsto -1/\tau$. As you probably know, this subgroup maps isomorphically onto $\text{PSL}_2(\mathbf Z)$. Thus we see that $\theta$ is a modular form of weight $1/2$ for the full modular group, with the extra "character" $\chi\left(\begin{matrix} a & b\\ c & d\end{matrix}\right) = (-i)^{c/2}$.

In the $19^{th}$ century, following the pioneering work of Euler and Fagano on the transformation properties of elliptic integrals, mathematicians, led by Weierstrass and Jacobi, studied elliptic curves over $\mathbf C$. It was well understood then that these objects could be thought of either as smooth projective cubics over $\mathbf C$ or as complex tori of dimension $1$. During this period, the first modular forms of integral weight were discovered in the "invariants" of elliptic curves over $\mathbf C$.

Poincaré was the first to consider seriously elliptic curves over $\mathbf Q$. Poincaré conjectured that if $E/\mathbf Q$ is an elliptic curve, then $E(\mathbf Q)$ is a finitely generated abelian group. Some examples of this had already been supplied unknowingly by Fermat. It was proven some years later by Mordell and eventually for elliptic curves over any number field by Weil.

Weil, in 1949, formulated his famous conjectures about the zeta functions of smooth algebraic varieties over finite fields, and proved them in the case of curves. This led Hasse to define the zeta function of a smooth algebraic variety over $\mathbf Q$, and in particular of an elliptic curve. For an elliptic curve $E/\mathbf Q$, he defined $L(E, s)$ in the following way: by reducing $\mod p$ an integral model of $E$ for each prime $p$ not dividing $\Delta(E)$, he obtained an elliptic curve over $\mathbf F_p$ for each such $p$; he defined $L_0(E, s)$ as the product of the local zeta functions evaluated at $p^{-s}$. He conjectured the convergence of this product for $\text{Re }s > 3/2$, and conjectured a functional equation for it. For elliptic curves with complex multiplication, he was able to prove this conjecture essentially by class field theory over an imaginary quadratic field. This proof, like Riemann's, once again involved modular forms of half-integral weight.

(Hasse was well aware, however, that his $L$-function was missing factors for the primes dividing $\Delta$. It is only with the work of Grothendieck and his school that the missing factors could be accounted for, by interpreting $L(E,s)$ as the Artin $L$-function of the $\mathcal l$-adic cohomology of $E$.)

In the fifties and sixties, Shimura, Taniyama and Weil conjectured that the $L$-function of any elliptic curve over $\mathbf Q$ should come from a modular form. More precisely, they conjectured that given an elliptic curve of conductor $N$, there exists a Hecke newform $f$ of weight $2$ and level $N$ such that $L(E, s) = L(f, s)$. If that were true, then the analytic continuation and functional equation of $L(E,s)$ would follow directly from that of $f$, in the same spirit as for Riemann's proof. This is the celebrated theorem of Wiles, Breuil, Conrad, Diamond and Taylor, of which Fermat's last theorem is a consequence.

Eichler and Shimura provided a construction going in the other direction - namely, given a Hecke eigenform of weight $2$ and level $N$, they constructed an elliptic curve over $\mathbf Q$ such that $L(E,s) = L(f, s)$ (they found this curve sitting inside the Jacobian variety of $X_0(N)$).

Since the Hasse-Weil $L$-function of an elliptic curve over $\mathbf Q$ only depends on its isogeny class, there is a correspondence

$$\{E/\mathbf Q \text{ an elliptic curve of conductor $N$}\}/{\text{isogeny}} \cong \{\text{normalized newforms in $S_2(\Gamma_0(N))$}\}.$$

Since $S_2(\Gamma_0(N)) \cong \Omega^1(X_0(N))$ by the map $f \mapsto f \: d\tau$, the genus of $X_0(N)$ is at most equal to the number of isogeny classes of elliptic curves over $\mathbf Q$ of conductor $N$. For example, when $N=1$, $X_0(1) \cong \mathbf P^1$, so there are no elliptic curves of conductor $1$ (already a not-so-trivial fact).

So even if we restrict ourselves to forms of weight $2$ on $\Gamma_0(N)$, there are a couple of hundreds of years of mathematics to be learned. Many, many things to say.

In fact, one can prove that the modular curve $X_0(N)$ admits a smooth model over $\mathbf Z[1/n]$ (and in particular over $\mathbf Q$). This should be a very surprising fact - indeed, the curve $X_0(N)$ is initially defined over $\mathbf C$, i.e. it's a Riemann surface, and there is no reason a priori why it should admit a model over $\mathbf Q$ (in other words, the functor from smooth curves over $\mathbf Q$ to Riemann surfaces is neither full nor essentially surjective). This extra God-given arithmetic data is what makes these objects so rich and fascinating (peace be upon his Noodly Appendage).

Anyways - that's just a little bit of what we can say. If you'd like a good read, I recommend Rational Points on Modular Elliptic Curves by Henri Darmon. I'm reading through it myself and I share your fascination!

Answer 2

Some brief comments on your three questions.

1. Why care about all modular forms and not just, say, cusp forms? Well, why in real analysis do we use all real numbers if, in practice, so few of them are really of direct interest (not so many classical transcendental constants come up, just $\pi$, $e$, Euler's constant,$\Gamma(1/4)$,...)? The answer is that we need the whole real line to get a lot of the machinery of real analysis to work. Likewise, even if ultimately you may care more about cusp forms than other modular forms, the cusp forms are pretty subtle objects and it is usually easier to write down examples of modular forms first that may not necessarily be cusp forms. Differences of such modular forms may then turns out to be cusp forms, so the extent that it's subtle to write down cusp forms directly, you can still get at some examples of them using other modular forms.

2. See Serre's article "Modular forms of weight one and Galois representations" in the 1977 Durham conference proceedings Algebraic Number Theory (edited by Froehlich, but note this is not Cassels & Froehlich).

3. One of the great theorems about elliptic curves over $\mathbf Q$ is Mazur's theorem classifying all possible torsion subgroups in $E({\mathbf Q})$. The theorem amounts to a description of the rational points on certain modular curves, which is obtained using the Jacobians of these curves. An understanding of the geometry of these objects is a prerequisite for understanding the rational points on them. (The later work of Kamienny and Merel on uniform boundedness of torsion on elliptic curves over general number fields was an extension of Mazur's approach.)