On Congruence classes of orders of reductions of elliptic curves

“Doc in Progress” is pleased to introduce you to:

• Antigona Pajaziti - PhD in Mathematics - University of Luxembourg

Let E be an elliptic curve defined over Q and Ẽ_p(F_p) denote the reduction of E modulo a prime p of good reduction for E. Given an integer m ≥ 2 and any a modulo m, we consider how often the congruence |Ẽ_p(F_p)| ≡ a (mod m) holds. We show that the greatest common divisor of the integers |Ẽ_p(F_p)| over all rational primes p cannot exceed 4. We then exhibit elliptic curves over Q(t) with trivial torsion for which the orders of reductions of every smooth fiber modulo primes of positive density at least 1/2 are divisible by a fixed small integer. We also show that if the torsion of E grows over a quadratic field K, then one may explicitly compute |Ẽ_p(F_p)| modulo |E(K)_{tors}|. More precisely, we show that there exists an integer N ≥ 2 such that |Ẽ_p(F_p)| is determined modulo |E(K)_{tors}| according to the arithmetic progression modulo N in which p lies. It follows that given any a modulo |E(K)_{tors}|, we can estimate the density of primes p such that the congruence |Ẽ_p(F_p)| ≡ a (mod |E(K)_{tors}|) occurs.