## 64.33 Precise form of Chebotarev

As a first application let us prove a precise form of Chebotarev for a finite étale Galois covering of curves. Let $\varphi : Y \to X$ be a finite étale Galois covering with group $G$. This corresponds to a homomorphism

$\pi _1(X) \longrightarrow G = \text{Aut}(Y/X)$

Assume $Y_{\overline{k}} =$ irreducible. If $C\subset G$ is a conjugacy class then for all $n > 0$, we have

$| \# \{ x \in X(k_ n) \mid F_ x \in C\} - \frac{\# C}{\# G} \cdot \# X(k_ n) | \leq (\# C)(2g - 2) \sqrt{q^ n}$

(Warning: Please check the coefficient $\# C$ on the right hand side carefully before using.)

Sketch. Write

$\varphi _*(\overline{\mathbf{Q}_ l}) = \oplus _{\pi \in \widehat{G}} \mathcal{F}_{\pi }$

where $\widehat{G}$ is the set of isomorphism classes of irred representations of $G$ over $\overline{\mathbf{Q}}_ l$. For $\pi \in \widehat{G}$ let $\chi _{\pi }: G \to \overline{\mathbf{Q}}_ l$ be the character of $\pi$. Then

$H^*(Y_{\overline{k}}, \overline{\mathbf{Q}}_ l) = \oplus _{\pi \in \widehat{G}} H^*(Y_{\overline{k}}, \overline{\mathbf{Q}}_ l)_\pi =_{(\varphi \text{ finite })} \oplus _{\pi \in \widehat{G}} H^*(X_{\overline{k}}, \mathcal{F}_\pi )$

If $\pi \neq 1$ then we have

$H^0(X_{\overline{k}}, \mathcal{F}_\pi ) = H^2(X_{\overline{k}}, \mathcal{F}_\pi ) = 0,\quad \dim H^1(X_{\overline{k}}, \mathcal{F}_\pi ) = (2g_ X - 2)d_\pi ^2$

(can get this from trace formula for acting on ...) and we see that

$|\sum _{x \in X(k_ n)} \chi _\pi (\mathcal{F}_ x)| \leq (2g_ X - 2) d_\pi ^2\sqrt{q^ n}$

Write $1_ C = \sum _\pi a_\pi \chi _\pi$, then $a_\pi = \langle 1_ C, \chi _\pi \rangle$, and $a_1 = \langle 1_ C, \chi _1\rangle = \frac{\# C}{\# G}$ where

$\langle f, h\rangle = \frac{1}{\# G}\sum _{g \in G} f(g)\overline{h(g)}$

Thus we have the relation

$\frac{\# C}{\# G} = ||1_ C||^2 = \sum |a_\pi |^2$

Final step:

\begin{align*} \# \left\{ x \in X(k_ n) \mid F_ x \in C\right\} & = \sum _{x \in X(k_ n)} 1_ C(x) \\ & = \sum _{x \in X(k_ n)} \sum _\pi a_\pi \chi _\pi (F_ x) \\ & = \underbrace{\frac{\# C}{\# G} \# X(k_ n)}_{ \text{term for }\pi = 1} + \underbrace{\sum _{\pi \neq 1}a_\pi \sum _{x\in X(k_ n)}\chi _\pi (F_ x)}_{ \text{ error term (to be bounded by }E)} \end{align*}

We can bound the error term by

\begin{align*} |E| & \leq \sum _{\pi \in \widehat{G}, \atop \pi \neq 1} |a_\pi | (2g - 2) d_\pi ^2 \sqrt{q^ n} \\ & \leq \sum _{\pi \neq 1} \frac{\# C}{\# G} (2g_ X - 2) d_\pi ^3 \sqrt{q^ n} \end{align*}

By Weil's conjecture, $\# X(k_ n)\sim q^ n$. $\square$

Comment #867 by Emmanuel Kowalski on

There is a TeXing issue here, at east in the online version.

Comment #868 by on

Indeed, thanks!

It's probably due to the presence of greater-than and lesser-than symbols. On the other hand, the correct LaTeX in this case would be \langle and \rangle, so I hope making the LaTeX more correct also fixes this render bug. If not I'll look into the parsing code.

Comment #870 by on

OK, that seems to have fixed it (refresh the page if it doesn't for you). The change is here. Thanks!

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).