we noted in Example 1.6.4). In Proposition 3.1.10 we will prove that this property
in general descends from the perfect closure (set r = 1 there).
Fix a choice of K-subgroup inclusion αp E over K, so we get a canonical
copy of αp 2 in A = E2 (as the kernel of the Frobenius isogeny A A(p)). Over an
arbitrary field of characteristic p 0 the Frobenius and Verschiebung morphisms
of αp 2 vanish, so over any such field the non-trivial proper subgroups of αp 2 are
naturally parameterized by lines in the 2-dimensional tangent space Lie(αp); 2 this
parameterization is given by the tangent line of the subgroup (see [23, Thm. 3.18]
for more details). In particular, the non-trivial proper K-subgroups of αp 2 are
parameterized by P1(K), and if K /K is an extension field then the non-trivial
proper K -subgroups of (αp)K 2 are parameterized by P1(K ) (with the subset P1(K)
consisting of the tangent lines to the K -subgroups defined over K).
We conclude that if K /K is a non-trivial extension field then there are K -
subgroups G A := AK of order p that are contained in (αp)K 2 and do not arise
from a K-subgroup of A. In contrast with what we saw in Example for
isogeny classes over algebraically closed fields of characteristic 0, we claim that if
K is separably closed (or more generally if K is separably closed in K , with G not
defined over K inside A = AK ) then the isogenous quotient A /G of A = AK
cannot be defined over K as an abstract abelian variety!
Indeed, if there were an isomorphism A /G BK for an abelian variety B
over K then the resulting isogeny
AK = A A /G BK
descends to an isogeny A B over K by Lemma (since K /K is primary).
The kernel of this latter isogeny is a K-subgroup of A that descends G A ,
contrary to how G was chosen. Thus, no such B exists.
One lesson we learn from Example and Example is that fields of
definition for abelian varieties in positive characteristic are rather more subtle than
in characteristic 0, even when working over algebraically closed base fields.
1.7.2. Grothendieck’s theorem. To fully appreciate the significance of Exam-
ple, we turn our attention to a striking result of Grothendieck concerning
the field of definition of an abelian variety with sufficiently many complex multipli-
cations in positive characteristic. Before stating Grothendieck’s result, we record
the analogue in characteristic 0 that is a source of inspiration. Theorem (Shimura–Taniyama). Every non-zero abelian variety A with
sufficiently many complex multiplications over an algebraically closed field K of
characteristic 0 is defined (along with its entire endomorphism algebra) over a num-
ber field inside K.
Proof. By Example, without loss of generality we may replace A with an
isogenous abelian variety. Thus, by Proposition we can pass to the isotypic
(and even simple) case, and so by Theorem 1.3.4 the abelian variety A over K
admits complex multiplication by a CM field L. Let Φ be the resulting CM type on
L. Letting Q denote the algebraic closure of Q in K, we may view Φ as a Q-valued
CM type on L.
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