Introduction

The prototype for the Fekete-Szeg¨ o theorem with local rationality is Raphael

Robinson’s theorem on totally real algebraic integers in an interval:

Theorem (Robinson [48], 1964). Let [a, b] ⊂ R. If b − a 4, then there are

infinitely many totally real algebraic integers whose conjugates all belong to [a, b].

If b − a 4, there are only finitely many.

Robinson also gave a criterion for the existence of totally real units in [a, b]:

Theorem (Robinson [49], 1968). Suppose 0 a b ∈ R satisfy the conditions

log(

b − a

4

) 0 , (0.1)

log(

b − a

4

) · log(

b − a

4ab

) − log(√

√

b +

√

a

b −

√

a

)

2

0 . (0.2)

Then there are infinitely many totally real units α whose conjugates all belong to

[a, b]. If either inequality is reversed, there are only finitely many.

David Cantor’s “Fekete-Szeg¨ o theorem with splitting conditions” on P1 ([14],

Theorem 5.1.1, 1980) generalized Robinson’s theorems, reformulated them adeli-

cally, and set them in a potential-theoretic framework.

In this work we prove a strong form of Cantor’s result, valid for algebraic curves

of arbitrary genus over global fields of any characteristic.

Let K be a global field, a number field or a finite extension of Fp(T ) for some

prime p. Let K be a fixed algebraic closure of K, and let Ksep ⊆ K be the sep-

arable closure of K. We will write Aut(K/K) for the group of automorphisms

Aut(K/K)

∼

=

Gal(Ksep/K). Let MK be the set of all places of K. For each v ∈

MK , let Kv be the completion of K at v, let Kv be an algebraic closure of Kv, and

let Cv be the completion of Kv. We will write Autc(Cv/Kv) for the group of continu-

ous automorphisms of Cv/Kv; thus Autc(Cv/Kv)

∼

=

Aut(Kv/Kv)

∼

=

Gal(Kv sep/Kv).

Let C/K be a smooth, geometrically integral, projective curve. If F is a field

containing K, put CF = C ×K Spec(F ) and let C(F ) = HomF (Spec(F ), CF ) be the

set of F -rational points; let F (C) be the function field of CF . When F = Kv, we

write Cv for CKv . Let X = {x1,...,xm} be a finite, Galois-stable set points of C(K),

and let E = EK =

v∈MK

Ev be a K-rational adelic set for C, that is, a product of

sets Ev ⊂ Cv(Cv) such that each Ev is stable under Autc(Cv/Kv). For each v, fix

an embedding K → Cv over K, inducing an embedding C(K) → Cv(Cv). In this

way X can be regarded as a subset of Cv(Cv): since X is Galois-stable, its image is

independent of the choice of embedding. The same is true for any Galois-stable set

of points in C(K), such as the set of Aut(K/K)-conjugates of a point α ∈ C(K).

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