Journal of the Ramanujan Mathematical Society

Amir Akbary ¹

Affiliations
1 Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive West, Lethbridge, Alberta, T1K 3M4, CA

Abstract

Let E be an elliptic curve defined over Q. For any prime p of good reduction, let E_p be the reduction of E mod p. Denote by N_p the cardinality of E_p(F_p), where F_p is the finite field of p elements. Let P(N_p) be the greatest prime divisor of N_p. We prove that if E has CM then for all but o(x/ log x) of primes

                                      p ≤ x,P(Np) > p^ϑ(p),

where ϑ is any number less than ϑ₀ = 1 − 1/2 e^−1/4 = 0.6105 · · ·. As an application of this result we prove the following. Let Γ be a free subgroup of rank r ≥ 2 of the group of rational points E(Q), and Γp be the reduction of Γ mod p, then for a positive proportion of primes p ≤ x, we have

                                            |Γ_p| > p^ϑ₀−ϵ,

where ϵ > 0.

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Amir Akbary ¹

Affiliations
1 Department of Mathematics and Statistics, Concordia University 1455 de Maisonneuve Blvd. West Montreal, Quebec, H3G 1M8, CA

Abstract

We will establish lower bounds in terms of the level for the number of holomorphic cusp forms of weight k > 2 whose various L-functions do not vanish at the central critical point. This work generalizes the work of W. Duke [l] which was for the case of weight 2.

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