Skip to main content

Section 1.11 The Rosati involution (Alex)

Let \(A/k\) be an abelian variety and \(f \in \End(A)\text{.}\) Via pullback we get \(\hat f \in \End(\hat A)\text{,}\) in the case where \(A\) is polarized i.e. we have an isogeny \(\phi \colon A\to \hat A\) we might wonder what the relation is between \(\hat f\) and \(f\text{.}\) E.g. \(\hat \id = \id\) but here we have \(\hat \phi \id \phi = \lb \deg \phi\rb\text{,}\) this is a little ugly, depends on the degree of our polarization. If we work with \(\Hom^0(A,B) = \Hom(A,B) \otimes \QQ\) rather than \(\Hom(A,B)\) we have a bona fide inverse \(\phi\inv\) of an isogeny \(\phi\text{.}\) So now we can ask precisely, what is the relationship of the endomorphism \(f^\dagger = \phi^{-1}\circ \hat f \circ \phi\in \End^0(A)\) with \(f\text{?}\)

What sort of properties does this map \(f \mapsto f^\dagger\) have?

Definition 1.11.1. The Rosati involution.

The map \(\phi^{-1} \hat{-} \phi = {-}^\dagger \colon \End^0(A) \to \End^0(A)\) is called the Rosati involution.

\begin{equation*} e_l^\phi(f a ,a') = e_l(fa, \phi a')= e_l(a, \hat f \phi a') = e_l(a, \phi\phi\inv \hat f \phi a') = e_l^\phi(a, f^\dagger a') \end{equation*}

We apply the previous proposition and skew-symmetry of a polarization (over some extension)

\begin{equation*} e_l^\lambda(\alpha a,a') = e_l^\lambda(a, \alpha^\dagger a') = e_l^\lambda({\alpha^\dagger}^{\dagger} a, a') \end{equation*}

for all \(a,a'\in V_l A\text{.}\)

So we have a weird algebra with a weird operation, what can we do? Perhaps inspired by the killing form of a lie algebra:

We can form a bilinear form using the trace

\begin{equation*} \End^0(A) \times \End^0(A) \to \QQ \end{equation*}
\begin{equation*} (f,g) \mapsto \tr(fg^\dagger)\text{.} \end{equation*}

So given a simple abelian variety we have a division algebra \(/ \QQ\) equipped with a positive definite involution.

Definition 1.11.7. Albert algebras?

A division algebra \(D\) finite over \(\QQ\) with an involution \('\) such that \(\tr_{D/\QQ}(xx') > 0\ \forall x\in D^\times\) is called an Albert algebra.

Such algebras were studied by Albert who proved an important classification theorem.

There is a fascinating table in Mumford, page 200 or something.

As one might hope, changing the polarization does not change the type of the algebra + involution pair.

One might wonder which endomorphisms are invariant under this process? I.e. what is

\begin{equation*} \{f \in \End^0(A) : f^\dagger = f\}\text{.} \end{equation*}

Equivalently, for which \(f\) is the dual given by conjugating by our polarization.

We can map

\begin{equation*} \QQ \otimes_\ZZ \NS(X) = \QQ \otimes_\ZZ \Pic X/\Pic^0 X \to \Hom(A, \hat A) \end{equation*}
\begin{equation*} \sheaf M \mapsto \phi_{\sheaf M}\text{,} \end{equation*}

however we also have an isomorphism

\begin{equation*} \Hom^0(A, \hat A) \xrightarrow{\sim} \End^0(A) \end{equation*}
\begin{equation*} \phi \mapsto \lambda\inv \phi \end{equation*}

for some fixed polarization \(\lambda\text{,}\) hence we can view \(\NS(A)\otimes \QQ\) inside \(\End^0(A)\text{.}\)

Fix \(\alpha \in \End^0(A)\) and \(l\ne \characteristic(k)\) odd. Applying Proposition 1.10.8 we see that \(\lambda \alpha = \phi_{\sheaf L}\) for some \(\sheaf L\) iff \(e^{\lambda\alpha}_l\) is skew-symmetric, but we also have

\begin{equation*} e_l^{\lambda\alpha}(a,a')= e_l^{\lambda}(a,\alpha a') = -e_l^\lambda(\alpha a',a) = -e_l(a', \hat\alpha \lambda a) \end{equation*}

for all \(a,a' \in V_lA\) this is the same as requiring \(\lambda\alpha = \hat \alpha \lambda\) i.e. \(\alpha = \alpha^\dagger\text{.}\) .

Another cool result we can now prove (in fact this was the reason Weil introduced the notion of a polarization).

Let \(\alpha\) be an automorphism of \((A, \lambda)\) i.e. \(\lambda =\hat \alpha \lambda \alpha\text{,}\) then \(\alpha^\dagger \alpha= 1\) and so

\begin{equation*} \alpha \in \End(A)\cap \{\beta \in \End(A) \otimes \RR: \trace(\alpha^\dagger \alpha) = 2g\} \end{equation*}

but \(\End(A)\) is discrete inside the compact RHS.