1.7. A THEOREM OF GROTHENDIECK AND A CONSTRUCTION OF SERRE 85
We conclude that if ker(h) is not ´ etale then h factors through FrE0/Fq , so there
is a commutative diagram
E0
FrE0/Fq
h
OL ⊗O E0
1⊗FrE0/Fq
E0p) (
h(p)
OL ⊗O
E0p)(
of O-linear isogenies. Since
h(p)
is the initial O-linear map from
E0p) (
to an OL-
linear module scheme over Fq, the right vertical map must be an isomorphism. This
latter map is the relative Frobenius isogeny for the elliptic curve OL ⊗O E0, as that
also makes the outside square commute (and commutativity uniquely determines
the right vertical map in terms of the other maps on the outside of the diagram).
The Frobenius isogeny for an elliptic curve over Fq is not an isomorphism, so we
have a contradiction. Thus, ker(h) is ´ etale and hence has order not divisible by p.
We conclude that h induces an isomorphism on p-divisible groups, so by the
Serre–Tate deformation theorem it follows that the formal deformation theory of
E0 is the same as that of E0. Any formal deformation of E0 over a complete
local noetherian W (Fq)-algebra is a scheme, since the inverse ideal sheaf of the
identity section provides a canonical algebraization, so by using formal GAGA for
morphisms [34, III1, 5.4.1] to keep track of the CM structure we may replace E0
with E0 to arrange that OL End(E0).
The OL-action on Lie(E0) selects a prime over p in OL and embeds its residue
field into Fq. Pre-composing the OL-action on E0 with the involution of L if
necessary, we can arrange that OL acts on Lie(E0) through an embedding OL/p
Fq, so the formal group corresponding to E0[p∞] is a formal OL,p-module over Fq of
dimension 1. Both Lubin–Tate theory and the deformation theory of 1-dimensional
formal modules [49, 22.4.4] ensure that this lifts to a formal OL,p-module over R,
so we obtain the desired OL-linear formal lift of E0 over R.
1.7.5. Variant on Grothendieck’s theorem. C-F. Yu’s variant on Theorem
1.7.2.5 asserts that we can first apply an isogeny and then pass to a finite extension
on K (with no further isogeny involved) to get to a situation that descends to a finite
field. This goes as follows. Consider the setup in Theorem 1.7.2.5. By Proposition
1.3.2.1, the simple factors have sufficiently many complex multiplications, so we
may focus on the case of simple abelian varieties A. Choose a polarization, so the
division algebra D =
End0(A)
is endowed with a positive involution. By [133, 2.2],
there is a maximal commutative subfield L D that is stable under the involution,
so L is either totally real or CM. We claim that L is a CM field, or in other words
L is not totally real.
To prove this property of L, first note that by Proposition 1.3.6.4 (in positive
characteristic) the division algebra D is either of Type III or Type IV (in the sense
of Theorem 1.3.6.2). Since L contains the center Z of D, for Type IV we get the
CM property for L from the fact that Z is CM in such cases. For Type III, the key
point is that Z is totally real and D is non-split at all real places of Z. We know
that DL is split over L since L is a maximal commutative subfield of D, so L is not
totally real. Hence, once again L is a CM field.
Applying Proposition 1.7.4.5, we can pass to an isogenous abelian variety so
that OL End(A). In this special case, we may conclude via the following theorem.
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