The perfect power problem for elliptic curves over function fields

TL;DR

This paper extends the Siegel-Voloch theorem to elliptic curves over function fields, establishing finiteness results for certain rational points based on valuation divisibility conditions.

Contribution

It generalizes the theorem to non-isotrivial elliptic curves with specific j-invariant properties over function fields, providing new finiteness results and effective bounds.

Findings

Finiteness of rational points with valuation divisibility conditions

Effective bounds for elliptic curves over rational function fields

Potential extension of results to number fields under abc-hypothesis

Abstract

We generalise the Siegel-Voloch theorem about S-integral points on elliptic curves as follows: let K/F denote a global function field over a finite field F of characteristic p>3, let S denote a finite set of places of K and let E/K denote a non-isotrivial elliptic curve over K with j-invariant a p^s power but not p^(s+1) power in K. Fix a non-constant function f in K(E) with a pole of order N>0 at the zero element of E. We prove that there are only finitely many rational points P in E(K) such that for any valuation outside S for which f(P) is negative, that valuation of f(P) is divisible by some integer not dividing p^sN. We also present some effective bounds for certain elliptic curves over rational function fields, and indicate how a similar result can be proven over number fields, assuming the number field abc-hypothesis.