CHAPTER 1

Variants

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 ([25]).

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

function is

(1.1) G(z, ζ; Ev) = inf

Hv⊂Ev

Hv compact

G(z, ζ; Hv) .

If ζ is not in the closure of Ev, the upper Robin constant V ζ(Ev) is finite and is

defined by

(1.2) V ζ(Ev) = lim

z→ζ

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

([51], Theorem 4.4.4) G(z, ζ; Ev) = G(z, ζ; Ev) and V

ζ

(Ev) = Vζ(Ev). For nonar-

chimedean v, if Ev is assumed to be algebraically capacitable in the sense of ([51]),

then G(z, ζ; Ev) = G(z, ζ; Ev) and V

ζ

(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 =

v

Ev

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

Cw

∼

=

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) .

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http://dx.doi.org/10.1090/surv/193/01