In this chapter we give six variants of Theorem 0.4, strengthening it in different
ways. In Chapter 4, Theorem 0.4, Corollary 0.5 and the variants stated here will
be reduced to yet another variant (Theorem 4.2) and we will spend most of the
book proving the theorem in that form.
Our first variant is similar to the original theorem of Fekete and Szeg¨ o ().
In that theorem the sets Ev ⊂ C were compact, and the conjugates of the al-
gebraic integers produced were required to lie in arbitrarily small open neighbor-
hoods Uv of the Ev. In Theorem 1.2 below, we lift the assumption of compactness
and replace the Cantor capacity with inner Cantor Capacity γ(E, X), which is de-
fined for arbitrary adelic sets. We also replace the neighborhoods Uv with “quasi-
neighborhoods”, which are finite unions of open sets in Cv(Cv) and open sets in
Cv(Fw), for algebraic extensions Fw/Kv in Cv.
The inner Cantor capacity γ(E, X) is similar to Cantor capacity except that it
is defined in terms of upper Green’s functions G(z, xi; Ev). Here, we briefly recall
the definitions of G(z, xi; Ev) and γ(E, X) and some of their properties; they are
studied in detail in §3.9 and §3.10 below.
Upper Green’s functions are gotten by taking decreasing limits of Green’s func-
tions of compact sets. For an arbitrary Ev ⊂ Cv(Cv), if ζ / ∈ Ev the upper Green’s
(1.1) G(z, ζ; Ev) = inf
G(z, ζ; Hv) .
If ζ is not in the closure of Ev, the upper Robin constant V ζ(Ev) is finite and is
(1.2) V ζ(Ev) = lim
G(z, ζ; Ev) + logv(|gζ(z)|v) ,
where gζ(z) is the uniformizer from (0.3). By (0.6), if Ev is compact then by
(, Theorem 4.4.4) G(z, ζ; Ev) = G(z, ζ; Ev) and V
(Ev) = Vζ(Ev). For nonar-
chimedean v, if Ev is assumed to be algebraically capacitable in the sense of (),
then G(z, ζ; Ev) = G(z, ζ; Ev) and V
(Ev) = Vζ(Ev). The upper Green’s function is
symmetric and nonnegative: for all z, ζ / ∈ Ev, G(z, ζ; Ev) = G(ζ, z; Ev) ≥ 0. It has
functoriality properties under pullbacks and base change like those of G(z, ζ; Ev).
Now assume that each Ev is stable under Autc(Cv/Kv), and that E =
is compatible with X. Let L/K be a finite normal extension containing K(X). For
each place v of K and each place w of L with w|v, after fixing an isomorphism
Cv, we can pull back Ev to a set Ew ⊂ Cw(Cw), which is independent of the
isomorphism chosen. If we identify Cv(Cv) with Cw(Cw), then for z, ζ / ∈ Ev
(1.3) G(z, ζ; Ew) log(qw) = [Lw : Kv] · G(z, ζ; Ev) log(qv) .