1.7. A THEOREM OF GROTHENDIECK AND A CONSTRUCTION OF SERRE 79
Example 1.7.1.2 shows that this guess is false, since A /G as built there admits
a CM structure since supersingular elliptic curves over K always admit a CM
structure (as they descend to elliptic curves over the algebraic closure of Fp inside
K). Allowing isogenies does not eliminate the need for a finite extension when the
ground field K is not algebraically closed:
1.7.2.4. Example. For a finite field κ, Example 1.7.2.2 adapts to work over K =
κ(t) by beginning with an elliptic curve over κ (for which complex multiplication
by an imaginary quadratic field exists in general; see Corollary 1.6.2.5). One can
do likewise over κ(t) with κ an algebraic closure of Fp.
Motivated by the above examples, Grothendieck proved a reasonable analogue
of Theorem 1.7.2.1 in positive characteristic:
1.7.2.5. Theorem (Grothendieck). Let A be an abelian variety over a field K with
char(K) = p 0, and assume A admits sufficiently many complex multiplications.
Then there exists a finite extension K of K, a finite subfield κ K , and an
abelian variety B over κ such that the scalar extensions A ×Spec(K) Spec(K ) and
B ×Spec(κ) Spec(K ) over K are isogenous.
For an exposition of Grothendieck’s proof, see [89]. The essential difficulty in
the proof in contrast with characteristic 0 is that the isogeny cannot be avoided,
even when K = K (due to Example 1.7.1.2). The proof of Theorem 1.7.2.5 is
immediately reduced to the case when K is finitely generated over Fp. Grothendieck
used the theory of potentially good reduction to find the required K /K and made
a descent from K to a finite subfield via a Chow trace (in the sense of [23, §6]).
1.7.3. There is a refinement of Grothendieck’s theorem, due to C-F. Yu, that clar-
ifies the role of the isogeny and proceeds in a simpler way by using moduli spaces
of abelian varieties. This refinement is given in 1.7.5. We will not need that result,
but the main ingredient in its proof is a technique to modify the endomorphism
ring that will be very useful later, so we now explain that technique.
As motivation, consider an abelian variety A of dimension g 0 over a field K
such that A admits sufficiently many complex multiplications, and let P
End0(A)
be a commutative semisimple Q-subalgebra with [P : Q] = 2g. The intersection
O := P End(A) is an order in P that may not be maximal (i.e., it may not equal
OP := OLi , where Li is the decomposition of P into a finite product of number
fields). It is natural to ask if there is an isogenous abelian variety for which the
non-maximality problem goes away. The following discussion addresses this issue.
1.7.3.1. Example. Consider the preceding setup with K = C. In this case we
have an analytic uniformization
Aan
= V/Λ in which V is a C-vector space equipped
with a C-linear action by P and Λ is a lattice in V stable under the order O. Then
Λ := OP · Λ is an OP -stable lattice in V and V/Λ is an isogenous quotient of
Aan
on which OP naturally acts. This algebraizes to an isogenous quotient A of A such
that under the identification
End0(A
) =
End0(A)
we have P End(A ) = OP .
We need an algebraic variant of the analytic construction in Example 1.7.3.1.
Observe that OP · Λ is the image of the natural map OP ⊗O Λ V . Inspired
by this, we are led to ask if these is a way to enlarge an endomorphism ring via
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