BUNTES The Honda Tate Theorem (Angus)

Section 1.14 The Honda Tate Theorem (Angus)

$q = p^n\text{,}$ $A$ a simple abelian variety over $\FF_q\text{,}$ $\pi_A$ the frobenius on $A\text{,}$ $\End^0(A) = \QQ\otimes \End(A)\text{,}$ $f_A$ is the charpoly of $A$ (i.e. of $\pi_A$ ).

Fact 1.14.1.

$\End^0(A)$ is a division ring.
$\QQ\lb \pi\rb$ is a field.
$\displaystyle Z(\End^0(A)) = \QQ\lb \pi_A\rb$

Lemma 1.14.2. The Weil Conjectures.

The roots of $f_A$ all have absolute value $\sqrt q\text{.}$ Alternatively, under all embeddings

$\begin{equation*} \iota \colon \QQ\lb \pi_A\rb \hookrightarrow \CC,\,|\iota(\pi_A)| = \sqrt q\text{.} \end{equation*}$

Definition 1.14.3. $q$ -Weil numbers.

A $q$ -Weil number is an algebraic integer $\pi$ s.t.

$\begin{equation*} \forall \iota \colon \QQ\lb \pi\rb \hookrightarrow \CC,\,|\iota(\pi)| = \sqrt q \end{equation*}$

we say that two $q$ -Weil numbers are conjugate if they have the same minimal polynomial over $\QQ\text{,}$ and write $\pi \sim \pi'\text{.}$

From the facts so far we have a map

$\begin{equation*} \{\text{simple AVs}/\FF_q\}\to \{q\text{-Weil numbers}\} \end{equation*}$

$\begin{equation*} A \mapsto \pi_A \end{equation*}$

Theorem 1.14.4.

We have a bijection

$\begin{equation*} \{\text{isogeny classes of simple AVs}/\FF_q\}\xrightarrow{\sim} \{\text{conjugacy classes of }q\text{-Weil numbers}\} \end{equation*}$

$\begin{equation*} A \mapsto \pi_A\text{.} \end{equation*}$

We need to show this is well-defined, injectivity and surjectivity.

Subsection 1.14.1 Honda-Tate map

Recall:

Corollary 1.14.5.

Let $A,B$ be abelian varieties over $\FF_q$ with rational Tate modules $V_l A, V_lB$ then

$\begin{equation*} A\sim_{\text{isog}} B \iff V_l A \simeq V_l B \forall l \ne p\text{.} \end{equation*}$

Corollary 1.14.6.

$\begin{equation*} A\sim_{\text{isog}} B \iff f_A = f_B \end{equation*}$

Proof.

By above $V_l A \simeq V_lB$ for all $l \ne p$ but $f_A$ (resp. $f_B$) is the charpoly of $\pi_a$ ($\pi_B$) on $V_l A$ ($V_l(B)$).

The Galois modules $V_lA$ and $V_l B$ are semisimple. The Brauer-Nesbitt theorem says $f_A = f_B \implies V_lA \simeq V_lB $ for $l\ne p\text{.}$

Recalling that $f_A$ is a power of the minimal polynomial of $\pi_A\text{,}$

$\begin{equation*} A \sim_{\text{isog}} B\implies f_A =f_B \implies \pi_A \sim \pi_B\text{.} \end{equation*}$

So the Honda-Tate map is well defined.

This doesn't quite give injectivity because a priori $f_A$ and $f_B$ could be powers of the minpolys of $\pi_A, \pi_B\text{.}$

Subsection 1.14.2 Injectivity and Brauer groups

From last time:

Proposition 1.14.7.

There exists a certain quantity $r(f_A, f_B)$ such that

$\begin{equation*} r(f_A, f_B) = \rank \Hom(A,B)\text{.} \end{equation*}$

Corollary 1.14.8.

Let $d = \lb \End^0(A) : \QQ(\pi_A) \rb^{1/2}\text{,}$ let $h_A = \operatorname{minpoly}_\QQ(\pi_A)$ then $f_A = h_A^d\text{.}$

Proof.

Study the formula for $r(f_A, f_A)$ Edixhoven-van der Geer-Moonen 16.22.

So the next step is to try and recover $\End^0(A)$ from $\pi\text{.}$

Definition 1.14.9. Central simple algebras.

A central simple algebra $B/k$ is a $k$ -algebra $B$ with no two-sided ideals and $Z(B) = k\text{.}$

Theorem 1.14.10. Artin-Wedderburn.

Any such algebra is isomorphic to $M_n(D)$ for $D$ a division ring over $k\text{.}$

Definition 1.14.11. Brauer groups.

The Brauer group of $k$ $\operatorname{Br}(k)$ is the set of central simple algebras under $\otimes$ modulo the algebras $M_n(k)\text{.}$

Fact 1.14.12.

If $k = \overline k\text{,}$ $\operatorname{Br}(k) = 0\text{.}$
$k$ complete nonarchimidean $\operatorname{Br}(k) = \QQ/\ZZ$
$\displaystyle \operatorname{Br}(\RR) = \ZZ/2\ZZ$

Given a place $v$ of $k$ we get a map

$\begin{equation*} \operatorname{Br}(k) \to \operatorname{Br}(k_v) \end{equation*}$

$\begin{equation*} D \mapsto D\otimes k_v \end{equation*}$

in fact we get an injection

$\begin{equation*} \operatorname{Br}(k) \hookrightarrow \prod_v \operatorname{Br}(k_v) \simeq \prod_{v\text{ nonarch}} \QQ/\ZZ \times \prod_{v\text{ real}} \ZZ/2\ZZ \end{equation*}$

$\begin{equation*} D\mapsto (\operatorname{inv}_v(D))_v \end{equation*}$

these $\operatorname{inv}_v(D)$ are called the local invariants.

Proposition 1.14.13.

Let $A/\FF_q$ be an elementary abelian variety. Let $K = \QQ(\pi_A)$ then

$\begin{equation*} \operatorname{inv}_v(\End^0(A)) = \begin{cases} \frac{v(\pi_A)}{v(q)} [k_v: \QQ_p], \amp v|p\\ \frac 12,\amp v\text{ real}\\ 0, \amp \text{else}\end{cases} \end{equation*}$

Proof.

Edixhoven-van der Geer-Moonen 16.30.

Proposition 1.14.14.

Let $d= \lb \End^0(A) : \QQ(\pi_A) \rb^{1/2}$ then $d$ is the least common denominator of all the $\operatorname{inv}_v(\End^0(A))\text{.}$

Corollary 1.14.15.

$\begin{equation*} \pi_A\sim \pi_B \iff f_A = f_B\text{.} \end{equation*}$

Proof.

$\Leftarrow$ done.

$\Rightarrow$ Let $D_{\pi_A} , D_{\pi_B}$ be the division rings with invariants specified as in Proposition 1.14.13. $\pi_A\sim \pi_B \implies D_{\pi_A} \simeq D_{\pi_B} \implies f_A = \operatorname{minpoly}(\pi_A) ^d = f_B\text{.}$

Subsection 1.14.3 Surjectivity and CM theory

We need to show that for $\pi$ a $q$ -Weil number there exists an abelian variety $A/\FF_q$ such that $\pi_A \sim \pi\text{.}$

Definition 1.14.16.

Such a $q$ -Weil number $\pi$ is called effective.

Proposition 1.14.17.

A $q$ -Weil number $\pi$ is effective if and only if $\pi^N$ is effective for some $N\in \ZZ_{\ge 1}\text{.}$

Proof.

$\Rightarrow$ clear.

$\Leftarrow$ By assumption we have $A'/k$ a simple abelian variety s.t. $\pi_{A'} \sim \pi^N$ for $k$ a degree $N$ extension of $\FF_q\text{.}$ Let

\begin{equation*} A = \Res_{k/\FF_q}(A') \end{equation*}

on the rational Tate modules we have

\begin{equation*} V_l A = \Ind_{G_k}^{G_{\FF_q}} (V_lA') \end{equation*}

where

\begin{equation*} G_k = \Gal{\overline{\FF_q}}{k} \end{equation*}

\begin{equation*} G_{\FF_q} = \Gal{\overline{\FF_q}}{\FF_q} \end{equation*}

since $G_k\text{,}$ $G_{\FF_q}$ are abelian, by studying the induced action, one can see

\begin{equation*} \Ind_{G_k}^{G_{\FF_q}} (\pi_{A'}) = \pi_A^N \end{equation*}

in particular $f_A(T) = f_{A'}(T^N)\text{.}$ Choosing a simple factor $A_i$ one gets $\pi_{A_i} \sim \pi\text{.}$

So it is sufficient to show $\pi^N$ is effective.

Strategy for proving surjectivity

Construct a division algebra $D_\pi\text{.}$
Choose a CM field $L$ splitting $D_\pi\text{.}$
Find an abelian variety $A/\CC$ of type $(L, \Phi)\text{.}$
In fact $A$ is defined over a number field $K$ and has good reduction at $v|p\text{.}$
Apply the Shimura-Taniyama formula to relate $\pi_A$ to $\Phi\text{.}$
Choose $\Phi$ wisely (in retrospect in 3) to relate $\pi$ to $\pi_A\text{.}$
Show $\pi_A^N = \pi^{N'}\text{.}$

$D_\pi$ is given by the invariants described by $\pi$ (and $K = \QQ(\pi)$ ).

Proposition 1.14.18.

There exists a CM field $L/\QQ(\pi)$ such that $L$ splits $D_\pi$ and further

$\begin{equation*} [L:\QQ(\pi) ] = [ D_\pi: \QQ(\pi)]^{1/2} \end{equation*}$

Proof.

Two cases:

$\QQ(\pi)$ is totally real, in which case $\QQ(\pi) = \QQ$ or $\QQ(\sqrt{p})\text{.}$
$\QQ(\pi)$ is a CM field with totally real subfield $\QQ(\pi + q/\pi)\text{.}$

In the case

Choose $L = \QQ(\pi)(\sqrt{-p})\text{.}$
Let $d= \lb D_\pi: \QQ(\pi)\rb^{1/2}\text{.}$ This $L$ splits $D_\pi\text{.}$

Definition 1.14.19. CM types.

For a CM field $L$ all the embeddings

$\begin{equation*} \iota\colon L \hookrightarrow \CC \end{equation*}$

come in complex conjugate pairs, choosing an embedding for each pair defines a subset $\Phi\subseteq \Hom(L, \CC)$ such that

$\begin{equation*} \Phi \cup \overline \Phi = \Hom(L,\CC) \end{equation*}$

$\begin{equation*} \Phi \cap \overline \Phi = \emptyset \end{equation*}$

such a choice of $\Phi$ is called a CM type.

Let $A/\CC$ be an abelian variety with CM by $L$ i.e.

$\begin{equation*} L\hookrightarrow \End^0(A) \end{equation*}$

then

$\begin{equation*} \CC\otimes L = \prod_\iota \CC \end{equation*}$

acts on the tangent space at the origin $\Lie(A)\text{.}$

Proposition 1.14.20.

The action of $\CC\otimes L$ factors through the quotient $\prod_{\iota\in \Phi} \CC$ for some CM type $\Phi\text{.}$ We then say $A/\CC$ is of type $(L,\Phi)\text{.}$

Theorem 1.14.21.

For any CM type $(L,\Phi)$ there exists an abelian variety $A/\CC$ of type $(L, \Phi)\text{.}$

Proof.

Found in Shimura-Taniyama.

The fact that $A$ is in fact defined over a number field $K$ is also in Shimura-Taniyama.

Theorem 1.14.22.

Let $A/K$ be an abelian variety which admits CM. Then $A/K$ admits potentially good reduction at all places $v$ of $K\text{.}$

Proof.

Highly nontrivial, Neron models, Chevalley decomposition, Neron-Ogg-Shafarevich criterion, result of Grothendieck on potentially stable reduction.

After passing to a finite extension we will assume $A/K$ has good reduction at places $v|p\text{.}$ So we have a reduction $A_{\FF_{q'}}/\FF_{q'}\text{.}$ For a place $w|p$ of $L$ let

$\begin{equation*} \Sigma_w = \Hom(L_w, \CC_p) \end{equation*}$

$\begin{equation*} \Phi_w = \Phi \cap \Sigma_w\text{.} \end{equation*}$

Theorem 1.14.23. Shimura-Taniyama formula.

For all places $w|p$ of $L\text{,}$

$\begin{equation*} \frac{w(\pi_{A_{\FF_{q'}}})}{w(q')} = \frac{\#\Phi_w}{\#\Sigma_w} \end{equation*}$

Proof.

Tate has a proof using CM theory of $p$-divisible groups.

Recall we fixed $\pi$ and from this we deterministically formed $\QQ(\pi), D_{\pi}, L$ however we have no restriction on our choice of $\Phi\text{.}$

Lemma 1.14.24.

We can choose $\Phi$ such that for all places $w|p$ of $L\text{,}$

$\begin{equation*} \frac{w(\pi)}{w(q)} = \frac{\#\Phi_w}{\#\Sigma_w} \end{equation*}$

Proof.

Let $v = w|_{\QQ(\pi)}$ be the place of $\QQ(\pi)$ below $w\text{.}$ Let

\begin{equation*} n_w = \frac{w(\pi)}{w(q)}\#\Sigma_w = \frac{w(\pi)}{w(q)}[L_w :\QQ_p] \end{equation*}

\begin{equation*} = \frac{w(\pi)}{w(q)}[L_w :\QQ(\pi)_v ][ \QQ(\pi)_v:\QQ_p] \end{equation*}

by recalling the formula for the local invariants of $D_\pi$ we get

\begin{equation*} n_w = \operatorname{inv}_w(D_\pi\otimes_{\QQ(\pi)} L)\text{.} \end{equation*}

But $L$ splits $D_\pi$ so $n_w \in \ZZ\text{,}$ further

\begin{equation*} n_w + n_{\overline w} = \left( \frac{w(\pi)}{w(q)} + \frac{\overline w(\pi)}{\overline w(q)} \right) \#\Sigma_w \end{equation*}

\begin{equation*} = \left( \frac{w(\pi\overline \pi)}{w(q)} \right) \#\Sigma_w = \#\Sigma_w \end{equation*}

check the CM type $\Phi = \bigcup_w \Phi_w$ where for each $w$ $\# \Phi_w = n_w\text{.}$ Then the formula follows.

Combining the previous result with the Shimura-Taniyama formula we get that for all places $w|p$

$\begin{equation*} \frac{w(\pi_{A_{\FF_{q'}}})}{w(q')} = \frac{w(\pi)}{w(q)}\text{.} \end{equation*}$

Taking the correct power,

$\begin{equation*} w\left( \frac {\pi^m_{A_{\FF_{q'}}}}{\pi^{m'}}\right) = 0 \forall w|p \end{equation*}$

$\begin{equation*} \pi,\pi_{A_{\FF_{q'}}}|q^{m'} \end{equation*}$

$\begin{equation*} \implies w(\cdots) = 0 \forall w \nmid p \end{equation*}$

since $|\pi^{m'}|_w = |\pi_{A_{\FF_{q'}}}^m|_w = (q^{m'})^{1/2} \forall \text{ infinite places}$

$\begin{equation*} \pi_{A_{\FF_{q'}}}/ \pi_A^{m'} \end{equation*}$

is a root of unity $\pi^N_{A_{\FF_{q'}}} = \pi^{N'}\text{.}$

Section 1.14 The Honda Tate Theorem (Angus)

Fact 1.14.1.

Lemma 1.14.2. The Weil Conjectures.

Definition 1.14.3. qq-Weil numbers.

Theorem 1.14.4.

Subsection 1.14.1 Honda-Tate map

Corollary 1.14.5.

Corollary 1.14.6.

Proof.

Subsection 1.14.2 Injectivity and Brauer groups

Proposition 1.14.7.

Corollary 1.14.8.

Proof.

Definition 1.14.9. Central simple algebras.

Theorem 1.14.10. Artin-Wedderburn.

Definition 1.14.11. Brauer groups.

Fact 1.14.12.

Proposition 1.14.13.

Proof.

Proposition 1.14.14.

Corollary 1.14.15.

Proof.

Subsection 1.14.3 Surjectivity and CM theory

Definition 1.14.16.

Proposition 1.14.17.

Proof.

Proposition 1.14.18.

Proof.

Definition 1.14.19. CM types.

Proposition 1.14.20.

Theorem 1.14.21.

Proof.

Theorem 1.14.22.

Proof.

Theorem 1.14.23. Shimura-Taniyama formula.

Proof.

Lemma 1.14.24.

Proof.

Definition 1.14.3. $q$ -Weil numbers.