Elliptic curves over function fields with a large set of integral points

Ricardo P. Conceição ¹

¹ Oxford College of Emory University 100 Hamill Street Oxford, GA 30054, U.S.A.

Acta Arithmetica, Tome 161 (2013) no. 4, pp. 351-370

Résumé

We construct isotrivial and non-isotrivial elliptic curves over $\mathbb {F}_q(t)$ with an arbitrarily large set of separable integral points. As an application of this construction, we prove that there are isotrivial log-general type varieties over $\mathbb {F}_q(t)$ with a Zariski dense set of separable integral points. This provides a counterexample to a natural translation of the Lang–Vojta conjecture to the function field setting. We also show that our main result provides examples of elliptic curves with an explicit and arbitrarily large set of linearly independent points.

Détail

DOI : 10.4064/aa161-4-3

Keywords: construct isotrivial non isotrivial elliptic curves mathbb arbitrarily large set separable integral points application construction prove there isotrivial log general type varieties mathbb zariski dense set separable integral points provides counterexample natural translation lang vojta conjecture function field setting main result provides examples elliptic curves explicit arbitrarily large set linearly independent points