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 \(\mathbb Q\) with a rational point P of infinite order, we may count the primes p of good reduction for which \((P \bmod p)\) generates \(E(\mathbb F_p)\) . In this work, we formulate and investigate two natural variants of the Lang-Trotter conjecture. For one of them, we require that the group \(E(\mathbb F_p)\) and its subgroup \(< (P \bmod p)>\) 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 \(\ell \) of the natural densities (which we prove to exist and be rational) of those p such that the exponents of \(E(\mathbb F_p)\) and \(< (P \bmod p)>\) have the same \(\ell \) -adic valuation. Numerical examples support the validity of our conjectures.
