Two natural variants of the Lang-Trotter conjecture on primitive points for elliptic curves

The Lang-Trotter conjecture on primitive points is the analogue for elliptic curves of Artin’s conjecture on primitive roots. Indeed, if we have an elliptic curve E over Q with a rational point P of infinite order, we may count the primes p of good reduction for which (P p) generates E (Fₚ). In this work, we formulate and investigate two natural variants of the Lang-Trotter conjecture. For one of them, we require that the group E (Fₚ) and its subgroup have the same exponent, namely the cyclic subgroup is as large as possible. We conjecture that the set of primes p such that this condition holds admits a natural density, whose value is a rational multiple of the product over all primes of the natural densities (which we prove to exist and be rational) of those p such that the exponents of E (Fₚ) and have the same -adic valuation. Numerical examples support the validity of our conjectures.