Questions About the Reductions Modulo Primes of an Elliptic Curve

This is largely a survey paper in which we discuss new and old problems about the reductions Ep modulo primes p of a fixed elliptic curve E defined over the field of rational numbers. We investigate, in particular, how the “noncyclic” part of the group of points of Ep is distributed, thus making progress toward a conjecture of R. Takeuchi. The new result is Theorem 2 of Section 3. Many interesting questions that resemble classical problems in number theory, such as Artin’s primitive root conjecture, the twin prime conjecture, the Buniakowski–Schinzel hypothesis, can be formulated using the group of points of the reduction modulo a prime of a fixed elliptic curve defined over a global field, say over Q. Before discussing these questions in detail, let us recall the basic definitions and properties of elliptic curves. For precise references or more detailed facts, the reader is referred to [33,34]. An elliptic curve E defined over Q is the locus of an equation of the form