Abstract
Let E be an elliptic curve defined over \({\mathbb{Q}}\). Let Γ be a subgroup of rank r of the group of rational points \({E(\mathbb{Q})}\) of E. For any prime p of good reduction, let \({\bar{\Gamma}}\) be the reduction of Γ modulo p. Under certain standard assumptions, we prove that for almost all primes p (i.e. for a set of primes of density one), we have
where f (x) is any function such that f (x) → ∞, at an arbitrary slow speed, as x → ∞. This provides additional evidence in support of a conjecture of Lang and Trotter from 1977.
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Akbary, A., Ghioca, D. & Murty, V.K. Reductions of points on elliptic curves. Math. Ann. 347, 365–394 (2010). https://doi.org/10.1007/s00208-009-0433-6
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DOI: https://doi.org/10.1007/s00208-009-0433-6