TY - JOUR

T1 - Short average distribution of a prime counting function over families of elliptic curves

AU - Giri, Sumit

N1 - Publisher Copyright:
© 2019

PY - 2020/7

Y1 - 2020/7

N2 - Let E be an elliptic curve defined over Q and let N be a positive integer. Then, ME(N) counts the number of primes p such that the group Ep(Fp) is of order N. For a given positive integer ℓ, we study the probability of the event {ME(N)=ℓ}. In an earlier joint work with Balasubramanian, we showed that ME(N) follows the Poisson distribution when an average is taken over a family of elliptic curves with parameters A and B where [Formula presented] and [Formula presented] for a positive integer ℓ and any γ>0. In this paper, we show that for sufficiently large N, the same result holds even if we take A and B in the range [Formula presented] and [Formula presented] for any ϵ>0.

AB - Let E be an elliptic curve defined over Q and let N be a positive integer. Then, ME(N) counts the number of primes p such that the group Ep(Fp) is of order N. For a given positive integer ℓ, we study the probability of the event {ME(N)=ℓ}. In an earlier joint work with Balasubramanian, we showed that ME(N) follows the Poisson distribution when an average is taken over a family of elliptic curves with parameters A and B where [Formula presented] and [Formula presented] for a positive integer ℓ and any γ>0. In this paper, we show that for sufficiently large N, the same result holds even if we take A and B in the range [Formula presented] and [Formula presented] for any ϵ>0.

KW - Barban-Davenport-Halberstam conjecture

KW - Elliptic curve

KW - Finite field

KW - Group order

KW - Prime k-tuple

UR - http://www.scopus.com/inward/record.url?scp=85077564576&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2019.11.011

DO - 10.1016/j.jnt.2019.11.011

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AN - SCOPUS:85077564576

SN - 0022-314X

VL - 212

SP - 376

EP - 408

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -