# Taniyama-Shimura 1

It’s time to return to plan A. I started this year by saying I’d post on some fundamental ideas in arithmetic geometry. The local system thing is hard to get motivated about, since the way I was going to use it in my research seems irrelevant at the moment. My other option was to blog some stuff about class field theory, since there is a reading group on the topic that I belong to this quarter.

The first goal of this new series is to understand the statement of the famous Taniyama-Shimura conjecture that led to the proof of Fermat’s Last Theorem. A lot of people can probably mumble something about the conjecture if they have any experience in algebraic/arithmetic geoemtry or any of the number theory type fields, but most people probably can’t say anything precise about what the conjecture says (I’ll continue to call it a “conjecture” even though it has been proved).

The statement of the conjecture is that every elliptic curve over ${\mathbb{Q}}$ is modular. Simple enough, but to unravel what it means to be modular we are going to have to take many posts just for the definition. If you’ve seen this explained before, it might still be interesting to read this series because I’m going to set up the machinery in a slightly different (but equivalent) way so that it will generalize to varieties other than elliptic curves in the future.

We’ll first define modular forms. A modular form of weight ${k}$ and level ${N}$ is an element of the vector space ${M_k(\Gamma_0(N))}$ which consists of holomorphic functions on the upper half plane ${f:\mathcal{H}\rightarrow \mathbf{C}}$ satisfying the additional transformation property that

$\displaystyle \displaystyle f \left(\frac{az+b}{cz+d}\right)=(cz+d)^kf(z)$

for all matrices ${\left(\begin{matrix} a & b \\ c & d \end{matrix}\right)\in SL_2(\mathbf{Z})}$ such that ${c\equiv 0 \mod N}$ (plus something else that we’ll get to shortly).

This is an analytic object if there ever was one. If this is the first time you’ve seen this, then the thing to pay attention to is that these depend on a choice of weight, ${k}$, and level, ${N}$. To get a feel for the level, note that it becomes “easier” to satisfy this transformation law as the level increases, because the amount of matrices we have to check is less. For example, when ${N=1}$ this says our ${f}$ has to behave nicely under every single linear fractional transformation that sends the upper half plane to the upper half plane. One might reasonably guess that ${0}$ is the only holomorphic function with this property. More on this later. The weight is a little harder to get a feel for.

The map ${z\mapsto e^{2\pi i z}}$ is a holomorphic map from the upper half plane onto the punctured unit disk. Note that ${e^{2\pi i z}\rightarrow 0}$ as ${z}$ tends to infinity along the imaginary axis. We can compose with this map and consider our modular form to be a holomorphic function on the punctured disk. This is well-defined because if ${e^{2\pi i z}=e^{2\pi i w}}$, then ${z}$ and ${w}$ differ by an integer and ${f(z+n)=f\left(\left(\begin{matrix}1 & n \\ 0 & 1 \end{matrix}\right)\cdot z\right)=(1)^kf(z)=f(z)}$.

We say ${f}$ extends to be holomorphic at infinity if there is a holomorphic extension to the whole disk. We require modular forms to have this property. Thus a modular form has a Fourier expansion called a ${q}$-expansion denoted

$\displaystyle \displaystyle f=\sum_{n=0}^\infty a_nq^n \ \text{where} \ q=e^{2\pi i z}$

(note that a Fourier series in general involves negative powers, but these would give a pole at infinity). The cusp forms are the subspace denoted $S_k(\Gamma_0(N))$ of the modular forms that vanish at all cusps. To define cusp, just think of the extended upper half plane as ${\mathbf{H}\cup \mathbb{P}^1_{\mathbb{Q}}}$. We stick all the rational numbers along the real line in and also throw in a point at infinity. In practice, we only have to check holomorphic extension across finitely many of these cusps because due to the transformation law we only need to pick on cusp in each equivalence class under the action of the matrix group. When for instance $N=1$ again, all we have to check is that $f$ vanishes at infinity, or upon composing to the disk we get that $a_0=0$.

Do any of these things exist? Well, as we’ve already noted, for small N it seems very hard to satisfy these properties. In fact, our guess was right, $\dim_\mathbb{C} S_k(\Gamma_0(1))=0$ for $1. So until we bump the weight up to 12, we actually only have the 0 function satisfying our properties. For weight 12, there is only one up to scalar multiple. This doesn't look good, but actually when we allow the level to grow we get a lot (even of low weight). But before next time, just ponder how severe the symmetry condition we are imposing is. Somehow every elliptic curve is closely related to one of these which is why the result is so surprising.

Now we have our basic analytic object of the conjecture. The next several posts will go back to the algebraic side of things. Depending on how much detail I decide to give to define the terms in the Taniyama-Shimura conjecture could take anywhere from 4 to 8 or so posts, just to give you an idea of how long you have to hold out for the statement.