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