An elliptic curve analogue of Pillai’s lower bound on primitive roots

Let $E/\mathbb {Q}$ be an elliptic curve. For a prime p of good reduction, let $r(E,p)$ be the smallest non-negative integer that gives the x-coordinate of a point of maximal order in the group $E(\mathbb {F}_p)$ . We prove unconditionally that $r(E,p)> 0.72\log \log p$ for infinitely many p, and...

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Bibliographic Details
Published inCanadian mathematical bulletin Vol. 65; no. 2; pp. 496 - 505
Main Authors Jin, Steven, Washington, Lawrence C.
Format Journal Article
LanguageEnglish
Published Canada Canadian Mathematical Society 01.06.2022
Cambridge University Press
Subjects
Curves
Hypotheses
Lower bounds
Numbers
finite fields
14H52
Elliptic curves
11G20
primitive root
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ISSN0008-4395
1496-4287
DOI10.4153/S0008439521000448

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Summary:Let $E/\mathbb {Q}$ be an elliptic curve. For a prime p of good reduction, let $r(E,p)$ be the smallest non-negative integer that gives the x-coordinate of a point of maximal order in the group $E(\mathbb {F}_p)$ . We prove unconditionally that $r(E,p)> 0.72\log \log p$ for infinitely many p, and $r(E,p)> 0.36 \log p$ under the assumption of the Generalized Riemann Hypothesis. These can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime.
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ISSN:0008-4395
1496-4287
DOI:10.4153/S0008439521000448