Journal of the Ramanujan Mathematical Society

Chantal David ¹, Francesco Pappalardi ²

Affiliations
1 Department of Mathematics and Statistics, Concordia University, 1455 De Maisonneuve West Montreal, Quebec H3G 1M8, CA
2 Dipartimento Di Matematica, Universita Roma Tre, Largo S. L. Murialdo, 1, I–00191 Rome, IT

Abstract

Given an integer r, we consider the problem of enumerating the inert prime ideals 𝔭 of ℚ(i) for which a given elliptic curve E has trace of Frobenius at 𝔭 equal to r. We prove that on average the number of such prime ideals up to x is asymptotic to c_r log log x where c_r is an explicit constant computed in terms of an Euler product. This result is in accordance with the standard heuristics. This problem generalises naturally the classical Lang-Trotter conjecture for elliptic curves over ℚ.

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