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 \(\pi^N_{A_{\FF_{q'}}} = \pi^{N'}\text{.}\)