An $abcd$ theorem over function fields and applications

Abstract

Abstract. -We provide a lower bound for the number of distinct zeros of a sum 1 + u + v for two rational functions u, v, in term of the degree of u, v, which is sharp whenever u, v have few distinct zeros and poles compared to their degree. This sharpens the "abcd-theorem" of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface x a + y a + z c = 1 contains only finitely many rational or elliptic curves, provided a ≥ 10 4 and c ≥ 2; this provides special cases of a known conjecture of Bogomolov. Résumé (Un théorème abcd sur les corps de fonctions et applications)Nous démontrons une minoration pour le nombre de zéros distincts d'une somme 1 + u + v, u, v étant deux fonctions rationnelles, en fonction du degré de u et v; cette minoration est forte si le nombre de zéros et poles de u, v est suffisament petit par rapport à leur degré. Dans certains cas, on obtient une amélioration de l'inégalité de Voloch et Brownawell-Masser, qui entraîne des nouveaux résultats de finitude sur les équations diophantiennes sur les corps de fonctions.Par exemple, nous démontrons que la surface de type Fermat définie par l'équation x a +y a +z c = 1 ne contient qu'un nombre fini de courbes rationnelles ou elliptiques, dès que a ≥ 10 4 et c ≥ 2. Ce résultat constitue un cas particulier d'une célèbre conjecture de Bogomolov.

Greatest common divisors of $u-1, v-1$ in positive characteristic and rational points on curves over finite fields

“…Still another point of view is to interpret the left-hand side as a gcd of u − 1, v − 1, viewed as functions on X \ S; in turn, Silverman [14] interpreted it as a height, with respect to the exceptional divisor in a blow-up of G 2 m . Theorem 1 admits various applications, for instance to a special case of a conjecture of Vojta concerning integral points for the complement in P 2 of certain curves (see [4], [7]) and to rational curves on projective surfaces [6].…”

Unit equations and Fermat surfaces in positive characteristic

“…If v 2 ∈ K p , then also v 3 ∈ K p and we conclude that 3 . This is again a contradiction, so suppose that v 2 , v 3 ∈ K p .…”

“…In fact, this is a rather easy consequence from his abc Theorem. However, it is a more difficult task to find the smallest N using abc Theorems, see for example [3]. Our Theorem 4 is also based on abc type arguments and for this reason it should not be surprising that we can not distinguish between the case N = p r + 1, giving unirational surfaces [13], and N = ap r + b with 0 < a, b small.…”

“…In this case it is not a priori clear what the structure of the solution set should be, but Derksen and Masser give a completely explicit description that we repeat here in the special case that n = 3. They define so-called one dimensional Frobenius families to be F(u) := {(u 1 , u 2 , u 3 ) p e : e ≥ 0} for u = (u 1 , u 2 , u 3 ) ∈ (K * ) 3 and two dimensional Frobenius families 3 , where all multiplications of tuples are taken coordinate-wise. Then Derksen and Masser prove that the solution set of…”

On the Győry-Sárközy-Stewart conjecture in function fields

“…We first recall the following result of Corvaja and Zannier in [8], Corollary 2.3, see also [9], Theorem CZ, or [21], Theorem 2.2. Furthermore, Zannier in [21], Section 2.4, also gives several applications of this result.…”