Two 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 mod 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 ⟨(P mod 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 ℓ of the natural densities (which we prove to exist and be rational) of those p such that the exponents of E(Fp) and ⟨(P mod p)⟩ have the same ℓ-adic valuation.Numerical examples support the validity of our conjectures.