An Elliptic Curve Analogue of Pillai's Lower Bound on Primitive Roots

Steven Jin; Lawrence C. Washington

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arXiv · 2104.13256

An Elliptic Curve Analogue of Pillai's Lower Bound on Primitive Roots

Steven Jin·Lawrence C. Washington

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Abstract

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. This can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime.

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Steven Jin, Lawrence C. Washington. 2021-04-27. An Elliptic Curve Analogue of Pillai's Lower Bound on Primitive Roots. https://arxiv.org/abs/2104.13256

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