Year: 2014
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Cornelissen, Gunther, Reynolds, Jonathan
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1521634330