Journal of the Ramanujan Mathematical Society

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Average Frobenius Distribution for Inerts in β„š(i)


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

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Given an integer r, we consider the problem of enumerating the inert prime ideals 𝔭 of β„š(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 cr log log x where cr 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 β„š.
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  • Average Frobenius Distribution for Inerts in β„š(i)

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Authors

Chantal David
Department of Mathematics and Statistics, Concordia University, 1455 De Maisonneuve West Montreal, Quebec H3G 1M8, Canada
Francesco Pappalardi
Dipartimento Di Matematica, Universita Roma Tre, Largo S. L. Murialdo, 1, I–00191 Rome, Italy

Abstract


Given an integer r, we consider the problem of enumerating the inert prime ideals 𝔭 of β„š(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 cr log log x where cr 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 β„š.