A function related to the Mordell–Weil rank of elliptic curves

Abstract

Let [Formula: see text] be a prime number. For each natural number [Formula: see text], we study the behavior of the function [Formula: see text] which enumerates the number of factorizations [Formula: see text] with [Formula: see text] a perfect square (mod [Formula: see text]). The study of this function is inspired by the cognate function [Formula: see text] which enumerates the number of factorizations [Formula: see text] with [Formula: see text] a perfect square. The descent theory of elliptic curves would show that if [Formula: see text] is unbounded for squarefree values of [Formula: see text], then there are elliptic curves over the rational number field with arbitrarily large rank. In this note, we show for every prime [Formula: see text], [Formula: see text] is unbounded as [Formula: see text] ranges over squarefree values, thus providing some evidence for the conjecture that [Formula: see text] is unbounded for squarefree [Formula: see text].