2014/11/28 by Cornelissen, Gunther, Reynolds, Jonathan
#11D41 #11G05 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1411.7787
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 ps 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 psN. 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.