A homomorphism f : A
is symmetric when the map
ιA/S : A

is equal to f. Writing f

:= f
the equality ιA/S
= ιAt/S
and the functoriality
of ιA/S in A (applied with respect to f) implies f †† = f, so if we abuse notation by
writing f t rather than f then (f t)t = f. We say f is symmetric when f t = f (or
more accurately, f = f). This property holds if it does so on fibers over S, because
homomorphisms f, f : A B between abelian schemes coincide if fs = fs for all
s S. Indeed, for noetherian S such rigidity is [83, Cor. 6.2], and the general case
reduces to this because equality on all fibers descends through direct limits (since
it says that the finitely presented ideal of (f, f )−1(ΔA/S) in OA is nilpotent).
A polarization of A is a homomorphism f : A
that is a polarization on
geometric fibers. Any such f is necessarily symmetric. The properties of polar-
izations are developed in [83, §6.2] for projective abelian schemes, but the only
purpose of imposing projectivity at the outset (even though it is a consequence of
the definition, due to [34, IV3, 9.6.4]) is to ensure the existence of the dual abelian
scheme, so such an assumption may be eliminated. Definition. A homomorphism ϕ : A B between abelian schemes over
a scheme S is an isogeny when it is surjective with finite fibers. (Equivalently, the
homomorphims ϕs are isogenies in the sense of abelian varieties for each s S.)
Since quasi-finite proper morphisms are finite by [34, IV4, 18.12.4] (or by [34,
IV3, 8.11.1] with finite presentation hypothesis, which suffices for us), any isogeny
between abelian schemes is a finite morphism. Moreover, by the fibral flatness
criterion [34, IV3, 11.3.11], such maps are flat. Hence, if ϕ as above is an isogeny
then it is finite locally free (and surjective), so the closed subgroup ker(ϕ) is a finite
locally free commutative S-group scheme. Thus, B represents the fppf quotient
sheaf A/ker(ϕ). For example, setting ϕ = [n]A for n 1 gives A/A[n] A.
Turning this around, suppose we are given the abelian scheme A and a closed
S-subgroup N A that is finite locally free over S. Consider the fppf quotient
sheaf A/N. We claim that this quotient is (represented by) an abelian scheme, so
the map A A/N with kernel N is an isogeny. It suffices to work Zariski-locally
on S, so we may assume that N S has all fibers with the same order n 1. We
then have N A[n], due to the following result (proved in [123, §1]): Theorem (Deligne). Let S be a scheme and let H be a commutative S-
group scheme for which the structural morphism H S is finite and locally free.
If the fibers Hs
have rank n for all s S then H is killed by n.
The quotient sheaf A/N is an fppf torsor over A/A[n] A with fppf covering group
A[n]/N that is finite (and hence affine) over S. It then follows from effective fppf
descent for affine morphisms that the quotient A/N is represented by a scheme
finite over A/A[n] = A, and the map A A/N is an fppf A[n]/N-torsor, so the
S-proper S-smooth A is finite locally free over A/N (as A[n]/N is finite locally free
over S). Hence, A/N is proper and smooth since A is, and likewise its fibers over
S are geometrically connected. Thus, A/N is an abelian scheme as desired.
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