BUNTES Tate's Isogeny Theorem (Sachi)

Section 1.13 Tate's Isogeny Theorem (Sachi)

Subsection 1.13.1 The Theorem

Theorem 1.13.1. Tate.

Let \(A,B/\FF_q = k\text{,}\) \(q = p^n\text{,}\) \(l\ne p\) be abelian varieties and \(G = \Gal{k^s}{k}\text{,}\) then

\begin{equation*} \Hom_{k}(A,B)\otimes \ZZ_l \to \Hom_G(T_l A, T_lB) = \Hom_{\ZZ_l}(T_lA,T_lB)^G \end{equation*}

(where the \(G\) action on \(\Hom_{\ZZ_l}(T_l A, T_lB)\) is \((gf)(x) = gf(g^{-1}x)\)) is an isomorphism.

Remark 1.13.2.

Tate's theorem is also true for function fields over finite fields (Zarhin) and fields that are finitely generated over their prime field (Faltings), e.g. number fields. Not true over algebraically closed fields though.

Subsection 1.13.2 Motivation

Let \(\pi_A\) and \(\pi_B\) be the (relative) Frobenii on \(V_l(A), V_l(B)\)

\begin{equation*} \Hom_{k}(A,B)\otimes \QQ_l \to \Hom_G(V_l A, V_lB) \end{equation*}

\(P_A,P_B\) characteristic polynomials of \(\pi_A,\pi_B\text{.}\)

Toy Weil conjectures: \(P_A, P_B\) have \(\ZZ\)-coefficients, don't depend on the choice of \(l\text{.}\) Provided that induced action of Frobenii are semisimple, we can find a number \(r(P_A,P_B)\) then Tate implies

\begin{equation*} r(P_A,P_B) = \dim_{\QQ_l} \Hom_G (V_l(A), V_l(B)) = \rank \Hom_k(A,B) \end{equation*}

Corollary 1.13.3.

Let \(A, B\) be abelian varieties over \(\FF_q\) and \(P_A, P_B\) as above

\begin{equation*} \rank \Hom_k(A,B) = r(P_A,P_B) \end{equation*}
TFAE
1. \(B\) is \(k\)-isogenous to an abelian subvariety of \(A\)
2. \(V_lB\) is \(G\)-isomorphic to a \(G\)-subrepresentation of \(V_lA\) for \(l \ne \characteristic k\)
3. \begin{equation*} P_B | P_A \end{equation*}

we also have similar statements for equivalence, but get a nice statement about counting points over all extensions determining an abelian variety.

Proof.

\begin{equation*} \alpha \colon V_l(B)\hookrightarrow V_l(A) \end{equation*}

the surjectivity in Tate's theorem means we can choose \(u \in \Hom_k(B,A) \otimes \QQ_l\text{.}\) \(V_l(u) = \alpha\text{.}\) Choose \(u \in \Hom_k(B,A) \otimes \QQ\) arbitrarily close to \(\alpha\text{.}\) Lower semicontinuity implies if \(V_l(u)\) is close enough to \(\alpha\text{,}\) can ensure \(V_l(u)\) is injective (\(\ker (V_l (u)) = 0\)) take multiple to get \(u \in \Hom_k(B,A)\text{.}\) Since \(T_l (u)\) is injective \(u \) is an isogeny to an abelian subvariety.

Subsection 1.13.3 Isogeny category

Recall: The isogeny category, Theorem 1.8.1, Corollary 1.8.3. So we have a category \(\cat{Isog}\) of abelian varieties with

\begin{equation*} \Hom_{\cat{Isog}}(A,B) = \Hom_\cat{AV}(A,B)\otimes \QQ\text{.} \end{equation*}

Now if \(f \colon A \to B\) there exists \(g\colon B \to A\) an isogeny and \(n\in \ZZ_{\ge 1}\) s.t. \(gf = \lb n \rb\text{.}\) So \(\frac 1n g\) is an inverse for \(f \in \cat{Isog}\) so isogenies are isomorphisms in \(\cat{Isog}\text{.}\)

\(\cat{Isog}\) is a semisimple abelian category. The simples are simple abelian varieties.

Decomposition up to isogeny into a product of simple abelian varieties is unique.
If \(A\) is simple \(\End A \otimes \QQ\) is a division algebra over \(\QQ\text{.}\) Reason: If \(A\) is simple in an abelian category, if \(\End A \supseteq k\) a field implies it's a division algebra.

Subsection 1.13.4 Reductions

Lemma 1.13.4.

\begin{equation*} \ZZ_l \otimes \Hom_\cat{AV} (A,B) \to \Hom_H(T_l, T_lB) \end{equation*}
is an isomorphism if and only if
\begin{equation*} \QQ_l \otimes \Hom_\cat{AV} (A,B) \to \Hom_G(V_l A, V_l B) \end{equation*}
is an iso
If for every \(C\text{,}\)
\begin{equation*} \QQ_l \otimes \End_\cat{AV} (C) \to \End_G(V_l C) \end{equation*}
is an isomorphism then the above is an isomorphism for every pair \(A,B\text{.}\)

Proof.

The first map is always injective, the cokernel is torsion free, hence free. It's an isomorphism if and only if \(\QQ_l \otimes \coker = 0\) As \(\QQ_l\) is flat over \(\ZZ_l\) the second map injective and its cokernel is \(\QQ_l \otimes\) the cokernel of the first map.
\begin{equation*} C = A\times B \end{equation*}
then
\begin{equation*} \End^0(C) = \End^0(A)\oplus \Hom^0(A,B) \oplus \Hom^0(B,A) \oplus \End^0(B) \end{equation*}
and
\begin{equation*} \End_G(V_lC) = \End_G(V_lA)\oplus \Hom_G(V_lA,V_lB) \oplus \Hom_G(V_lB,V_lA) \oplus \End_G(V_lB) \end{equation*}
which the injection above preserves, in particular if the last map is an isomorphism, so are the rest.

One more reduction!

\begin{equation*} E_l = \End_k(A) \otimes \QQ_l\subseteq \End_{\QQ_l} (V_lA) \end{equation*}

\begin{equation*} F_l = \QQ_l[G] \subseteq \End_{\QQ_l} (V_lA) \end{equation*}

automorphisms of \(V_l(A) \) coming from \(G\text{.}\)

Note 1.13.5.

\(E_l\) coming from \(k\)-rational endomorphisms commute with the Galois action

\begin{equation*} F_l\subseteq C_{\End_{\QQ_l}(V_l(A))}(E_l) \end{equation*}

want equality.

Lemma 1.13.6.

The last map of the reduction lemma is an isomorphism if and only if
\begin{equation*} C(C(E_l)) = \End_G(V_l(A)) \end{equation*}
If \(F_l\) is semisimple the map is an isomorphism if and only if
\begin{equation*} C(E_l) = F_l \end{equation*}

Proof.

Double centralizer theorem, if \(E_l\) is semisimple then \(C(C(E_l))= E_l\text{.}\) Poincaré reducibility implies
\begin{equation*} A\sim \prod A_i^{m_i} \end{equation*}

\begin{equation*} \End^0(A) = \End^0(\prod A_i^{m_i}) = \prod \Mat_{m_i}(\End^0(A_i)) \end{equation*}
a finite dimensional division algebra \(/\QQ\text{.}\) A matrix algebra over a finite dimensional division algebra is semisimple.
If \(F_l\) is semisimple
\begin{equation*} C(E_l) = F_l \iff E_l = C(C(E_l)) \end{equation*}
so
\begin{equation*} E_l = C(F_l) = \End_G(V_l(A))\text{.} \end{equation*}

Subsection 1.13.5 Proof of Tate using finiteness

We introduce a hypothesis: \(\operatorname{Hyp}(k,A,l)\) there exist only finitely many (up to \(k\)-isomorphism) abelian varieties \(B\) s.t. there is a \(k\)-isogeny of \(l\)-power degree from \(B\to A\text{.}\)

\(D =C(E_l)\) want that \(C(D) = \End_G(V_l(A))\) know \(C(D) \subseteq E_l \subseteq \End_G(V_l(A))\) want \(C(D) \supseteq \End_G(V_l(A))\text{.}\) Let \(\alpha \in \End_G(V_l(A))\) show that it commutes with everything in \(D\text{.}\) Equivalently let \(W\) be the graph of \(\alpha\)

\begin{equation*} W = \{(x,\alpha x) \in V_l(A)\times V_l(A)\} \subseteq V_l(A)\times V_l(A) \end{equation*}

note \(g\in G\) then \(g\acts (x,\alpha x) = (gx,g\alpha x ) = (gx, \alpha(gx))\text{.}\)

\begin{equation*} \alpha\in C(D) \iff \forall x\in V_l(A), d\in D \end{equation*}

\begin{equation*} \alpha d x = d\alpha x \iff (d\oplus d)W \subseteq W \forall d\in D \end{equation*}

\begin{equation*} W\ni (dx, d\alpha x) = (dx, \alpha d x) \end{equation*}

Lemma 1.13.7. Technical lemma.

If \(W\subseteq V_l(A)\) is \(G\)-stable subspace then there exists \(u\in E_l\) s.t. \(u V_l(A) = W\text{.}\)

Proof.

For \(n \in \ZZ_{\ge 0}\) let \(U_n = (W\cap T_l(A)) + l^n T_A\) which is a \(G\)-stable lattice in \(V_l A\text{,}\)

\begin{equation*} l^nT_lA \subseteq U_n \subseteq T_l A \end{equation*}

let \(\mathcal K_n \subseteq A\lb l^n \rb (k^s) = T_l A/ l^n T_l A\) be the image of \(U_n\text{.}\) \(\mathcal K_n\) is stable under \(G\)-action on \(A\lb l^n \rb (k^s)\) which implies \(\mathcal K_n = K_n (k^s)\text{.}\) Let \(\pi_n \colon A \to B_n = A/ K_n\text{,}\) \(\iota_n \colon B_n \to A\) unique isogeny s.t.

\begin{equation*} \iota_n \circ \pi_n = \lb [l^n]_A \end{equation*}

then \(T_lB \cong U_n\) as \(\ZZ_l\)-modules with \(G\)-action. As \(T_l(\iota_n) \colon U_n =T_l B \to T_l A\) is the inclusion map. Assuming \(\operatorname{Hyp}(k,A,l)\) we can find \(n = n_1 \lt n_2 \lt \cdots\) s.t. we have

\begin{equation*} \alpha_i \colon B_n \xrightarrow{\sim} B_{n_i} \end{equation*}

\begin{equation*} \xymatrix{ B_n \ar[r]^{\alpha_i} & B_{n_i} \ar[d]_{\iota_{n_i}}\\ A\ar[u]_{\pi_n} \ar@{-->}_{u_i}[r] & A } \end{equation*}

\(u_i = \iota_{n_i} \circ \alpha_i \circ \pi_n\) is an endomorphism of \(A\) on Tate modules \(T_l(u_i)\) is induced map

\begin{equation*} T_l A \xrightarrow{[l^n]} U_n \xrightarrow{T_l\alpha_i} U_{n_i} \hookrightarrow T_l A \end{equation*}

because \(\ZZ_l \otimes \End A\) is a free \(\ZZ_l\)-module of finite rank compact in \(l\)-adic topology subsequence of \(u_i \to u\) in \(\ZZ_l \otimes \End A\)

\begin{equation*} U_{n_1} \supseteq U_{n_2} \supseteq \cdots \end{equation*}

the endomorphism of \(T_l u\) maps \(T_l A\) to \(\bigcap_{i=1}^\infty U_{n_i} = W\cap T_l A\) passing to \(\QQ_l\)-coefficients, note \(\QQ_l(W\cap T_l A) = \QQ_l(l^n(W\cap T_l A)) = W\) so \(\im(V_l(u)) =W\text{.}\)

Why does the hypothesis hold.

Fact 1.13.8.

There exists a moduli space of \(d\)-polarised abelian varieties of \(\dim = g\) \(A_{g,d}\) which is a stack of finite type \(/k\text{.}\)

\begin{equation*} A_{g,d}( k) = \{(A,\lambda) : A , \lambda \colon A \to A^\vee,\ \deg d \} \end{equation*}

Zahrin's trick: \(A\) abelian variety \((A\times A^\vee)^4\) is principally polarized. Finiteness of direct factors \(B\subseteq A\) \(A\simeq B\times C\text{.}\)

Corollary 1.13.9.

If \(k = \FF_q\) exists only finitely many isogeny classes of abelian varieties of \(\dim g\text{.}\)

Proof.

\(A\) is a direct factor \((A\times A^\vee)^4 \in A_{8g,1}\text{.}\)

Proof.

of Tate.

Apply technical lemma to \(V_l(A\times A)\) and \(W\) so

\begin{equation*} (d\oplus d) W = (d \oplus d) uV_l(A\times A) = u(d\oplus d) V_l(A\times A) \subseteq uV_l(A\times A) = W \end{equation*}

\begin{equation*} \implies C(D) \supseteq \End_G(V_l(A))\text{.} \end{equation*}