Short average distribution of a prime counting function over families of\n elliptic curves

2016/09/27 by Sumit Giri, Giri, Sumit
Mathematics · #11G05 #11G20 #11N05 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1609.08549

openalex publication_date 2016/09/27 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Let E be an elliptic curve defined over \ℚ and let N be a\npositive integer. Now, M_E(N) counts the number of primes p such that the\ngroup E_p( mathbbF_p) is of order N. In an earlier joint work with\nBalasubramanian, we showed that M_E(N) follows Poisson distribution when an\naverage is taken over a family of elliptic curve with parameters A and B\nwhere A, , B\≥ N^\(\ℓ)/(2)(\log N)^1+\γ and\nAB>N^\(3\ℓ)/(2)(\log N)^2+\γ for a fixed integer \ℓ and any\n\γ>0. In this paper, we show that for sufficiently large N, the same\nresult holds even if we take A and B in the range\n\exp(N^{\(\ε2)/(20\ℓ)})\≥ A, B>N^\ε and\nAB>N^\(3\ℓ)/(2)(\log N)^6+\γ for any \ε>0.\n

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