On the Lang-Trotter conjecture

par Antonella Perucca (Université du Luxembourg)

jeudi 30 janv. 2025, 16:00 → 17:00 Europe/Paris

Salle Pellos (1R2)

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 $(Pmodp)$ generates $E({\mathbb{F}}_p)$. We formulate and investigate some natural variants of the Lang-Trotter Conjecture.
For example, we require that the order of the point $(Pmodp)$ equals the exponent of the group $E({\mathbb{F}}_p)$: this means that the subgroup generated by the point is as large as possible, and the condition is meaningful also for non-cyclic groups. This is joint work with Alexandre Benoist.