The Smallest Invariant Factor of Elliptic Curves, and Coincidences

TL;DR

This paper investigates the positivity of the density constant C_E,j for primes with a specific invariant factor in elliptic curves over Q, using group theory and experimental evidence of division field coincidences.

Contribution

It extends prior results by characterizing when C_E,j is positive for non-CM elliptic curves, linking it to division field coincidences through adelic Galois representations.

Findings

C_E,j appears to vanish only with division field coincidences

Experimental data shows families of division field coincidences from abelian division fields

Strengthens the understanding of prime distributions related to elliptic curve invariants

Abstract

For an elliptic curve E over Q and a natural number j, Cojocaru has shown that there is an explicit constant C_E,j giving (under GRH) the density of primes p of good reduction such that the smallest invariant factor of E(F_p) is j. For E without complex multiplication, we study the question of when C_E,j is positive (a necessary and, on GRH, sufficient condition for there to be infinitely many such p), strengthening a result by Kim. Our arguments are group-theoretic using the image of the adelic Galois representation of E. Experimentally, C_E,j appears to vanish only when there is a coincidence of division fields; we document a number of families of such coincidences arising from abelian division fields.