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
b a
) 0 , (0.1)
b a
) · log(
b a
) log(√

b +


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

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)


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 =
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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