Theorem 63.31.10. (See [Theorem 3.5, dJ-conjecture]) Suppose

$\rho _0: \pi _1(X)\to \text{GL}_ n(\mathbf{F}_ l)$

is a continuous, $l\neq p$. Assume

1. Conj. holds for $X$,

2. $\rho _0 |_{\pi _1(X_{\overline{k}})}$ abs. irred., and

3. $l$ does not divide $n$.

Then the universal deformation ring $R_{\text{univ}}$ of $\rho _0$ is finite flat over $\mathbf{Z}_ l$.

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, $\# X(k_ n)\sim q^ n$. $\square$

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