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 $$\mathbb Q$$ 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.