# Taniyama-Shimura 3: L-Series

For today, we assume our ${d}$-dimensional variety ${X/\mathbb{Q}}$ has the property that its middle etale cohomology is 2-dimensional. It won’t hurt if you want to just think that ${X}$ is an elliptic curve. We will first define the L-series via the Galois representation that we constructed last time. Fix ${p}$ a prime not equal to ${\ell}$ and of good reduction for ${X}$. Let ${M=\overline{\mathbb{Q}}^{\ker \rho_X}}$. By definition the representation factors through ${{Gal} (M/\mathbb{Q})}$. For ${\frak{p}}$ a prime lying over ${p}$ the decomposition group ${D_{\frak{p}}}$ surjects onto ${{Gal} (\overline{\mathbf{F}}_p/\mathbf{F}_p)}$ with kernel ${I_{\frak{p}}}$. One of the subtleties we’ll jump over to save time is that ${\rho_X}$ acts trivially on ${I_{\frak{p}}}$ (it follows from the good reduction assumption), so we can lift the generator of ${{Gal} (\overline{\mathbf{F}}_p/\mathbf{F}_p)}$ to get a conjugacy class ${{Frob}_p}$ whose image under ${\rho_X}$ has well-defined trace and determinant.

We define

$\displaystyle L(X,s) =(*)\prod_{p \ good} \frac{1}{1-{tr}(\rho_X({Frob}_p))p^{-s}+\det(\rho_X({Frob}_p))p^{-2s}}$

where ${(*)}$ is a product of terms at the bad primes. Note that since this is a two-dimensional representation basic linear algebra tells us that the product is over the simpler expression ${(\det(I-\rho_X({Frob}_p)p^{-s}))^{-1}}$.

If you don’t like all this Galois representation stuff, we can describe this L-series without reference to the Galois representation at all. In order to ease notation we will denote the reduction of ${X}$ at a fixed good prime ${p}$ by ${Y:=X_{\mathbf{F}_p}}$ and base changing to the algebraic closure ${\overline{Y}:=X_{\overline{\mathbf{F}_p}}}$. To simplify notation let ${k=\mathbf{F}_p}$.

We have several natural Frobenius actions on ${\overline{Y}}$. The first we will call the absolute Frobenius which we will denote ${F_{ab}:\overline{Y}\rightarrow \overline{Y}}$. This is the identity on the topological space and the ${p}$-th power map on the structure sheaf. On affine patches ${{Spec} A\rightarrow {Spec} \overline{k}}$ the map is the one induced by ${a\mapsto a^p}$ on ${A}$. We can check directly that the map on topological spaces is the identity. For any prime ideal ${\frak{q}\in{Spec} A}$ the contraction ${\frak{q}^c=\{a\in A: a^p\in \frak{q}\}=\{a\in A: a\in\frak{q}\}=\frak{q}}$ by the property of ${\frak{q}}$ being prime. This map translates in the language of schemes to ${(id, F): (\overline{Y}, \mathcal{O}_{\overline{Y}})\rightarrow (\overline{Y}, \mathcal{O}_{\overline{Y}})}$ where ${F}$ is raising sections of the sheaf to the ${p}$-th power.

Note that the absolute Frobenius is not a map of ${\overline{Y}}$ over ${\overline{k}}$. The map is also not the pullback despite making the same commutative diagram.

Let the standard structure map be ${\phi: \overline{Y}\rightarrow {Spec}\overline{k}}$ and define ${\overline{Y}^{(p)}=\overline{Y}\otimes\overline{k}}$ to be the pullback of Frobenius acting on the base field. Since we have a commutative diagram we get by the universal property of a pullback diagram some map ${F_r: \overline{Y}\rightarrow \overline{Y}^{(p)}}$ called the relative Frobenius. We define the arithmetic Frobenius to be the projection on the first factor ${F_{ar}:\overline{Y}^{(p)}\rightarrow \overline{Y}}$. A nice exercise to see if you understand these would be to write down a big commuting diagram that relates all these. Due to wordpress constraints, I won’t actually do this.

Instead, we’ll do an example. Let ${Y={Spec} k[t]}$ (recall that ${k=\mathbf{F}_p}$). This means that ${\overline{Y}={Spec} \overline{k} [t]={Spec} (k[t]\otimes_k \overline{k})}$. The descriptions in terms of the ring homomorphism that induces the map on the spectra are as follows. The absolute Frobenius is still just ${f\mapsto f^p}$. The relative Frobenius is ${F_{ab}\otimes id}$. Since the absolute raises elements of ${k[t]}$ to the ${p}$, everything in ${k}$ is fixed by this map, and on ${\overline{k}}$ it is defined to be fixed. This means that the relative Frobenius only alters the ${t}$ by ${t\mapsto t^p}$. This is sometimes referred to as “raising coordinates to the ${p}$-th power”. The arithemtic Frobenius does nothing to the ${k[t]}$ part, but raises the ${\overline{k}}$ coefficients to the ${p}$, so ${\sum a_nt^n\mapsto \sum a_n^p t^n}$. Likewise, the geometric Frobenius takes the ${p}$-th root of the coefficients.

Straightforward (but non-trivial) computations also give that the map that the absolute Frobenius induces on the \'{e}tale site is trivial. If we look at our diagram we see that ${F_{ab}=F_{ar}\circ F_r}$. Since the induced map on cohomology is contravariant this gives ${F^*_r\circ F^*_{ar}=id}$. This means that on cohomology ${F^*_r=(F^*_{ar})^{-1}=F^*_{ge}}$ by definition of the geometric Frobenius.

Now the smooth, proper base change theorem for \'{e}tale cohomology tells us that ${H^d_{\textit{\'{e}t}}(\overline{Y}, \mathbb{Q}_\ell)\simeq H^d_{\textit{\'{e}t}}(\overline{X}, \mathbb{Q}_\ell)}$ which is two-dimensional. Since the ${F_r}$ action here is a linear operator on a vector space it makes sense to take the trace and determinant. We can define the L-series without use of the Galois representation as:

$\displaystyle L(X,s)=(*)\prod_{p \ good} \frac{1}{1-{tr}(F_r^*)p^{-s}+\det(F_r^*)p^{-2s}}$

where again the ${(*)}$ is a product of terms involving primes of bad reduction. Since there are only finitely many this is irrelevant for the definition of modularity. Of course we could have defined this without all the different Frobenius actions (we only used the relative one), but now we can get to the punchline. These two L-series are actually the same.
We just sketched above that the action of ${F_r}$ and ${F_{ge}}$ were the same on the \'{e}tale site. But ${F_{ge}=1\times {Frob}_p^{-1}}$ where ${{Frob}_p}$ is the canonical generator of ${{Gal} (\overline{k}/k)}$. We have a surjection ${{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow {Gal} (\overline{k}/k)}$ and if we consider ${{Frob}_p}$ a lift of this element, then by the functoriality and equivariant isomorphisms above we get that ${{tr}(\rho_X({Frob}_p))={tr}(\rho_X'({Frob}_p^{-1}))={tr}(F_r^*)}$. The determinant term turns out to always be ${p^3}$ since it can be checked to be the third power of the ${\ell}$-adic cyclotomic character in both cases. Thus the two L-series are the same. This also tells us the representation is odd.

Note that they appear to be off by an inverse, but we actually took the contragredient representation of the one that acts on ${H^d_{\textit{\'{e}t}}(\overline{X}, \mathbb{Q}_\ell)}$, so the inverse corrects for this and they are actually the same.

This got a little technical at parts, so the one thing to take away from this post is that to any ${X/\mathbb{Q}}$ with two-dimensional middle cohomology we can produce some function which is just defined in terms of the trace and determinant of certain operators on the cohomology. This is called the L-series and will be crucial in the definition of modularity.