Bounds for the integral points on elliptic curves over function fields

2017/10/01 by Alisa Sedunova, Sedunova, Alisa
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1710.00430

Abstract

In this paper we give an upper bound for the number of integral points on an elliptic curve E over F_q[T] in terms of its conductor N and q. We proceed by applying the lower bounds for the canonical height that are analogous to those given by Silverman and extend the technique developed by Helfgott-Venkatesh to express the number of integral points on E in terms of its algebraic rank. We also use the sphere packing results to optimize the size of an implied constant. In the end we use partial Birch Swinnerton-Dyer conjecture that is known to be true over function fields to bound the algebraic rank by the analytic one and apply the explicit formula for the analytic rank of E.

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