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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| Published in | Canadian mathematical bulletin Vol. 65; no. 2; pp. 496 - 505 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Canada
Canadian Mathematical Society
01.06.2022
Cambridge University Press |
| Subjects | |
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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. These can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime. |
|---|---|
| AbstractList | 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. 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. 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. |
| Author | Jin, Steven Washington, Lawrence C. |
| Author_xml | – sequence: 1 givenname: Steven surname: Jin fullname: Jin, Steven email: sjin6816@umd.edu organization: Department of Mathematics, University of Maryland, College Park, MD 20742, USA e-mail: lcw@umd.edu – sequence: 2 givenname: Lawrence C. surname: Washington fullname: Washington, Lawrence C. email: lcw@umd.edu organization: Department of Mathematics, University of Maryland, College Park, MD 20742, USA e-mail: lcw@umd.edu |
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| Cites_doi | 10.1112/plms/s3-12.1.179 10.2996/kmj/1138843605 10.1007/978-1-4684-0296-4 10.1016/j.ipl.2005.09.014 10.1007/BFb0060851 10.1002/mana.19490030104 10.1007/978-1-4612-3464-7_18 10.1090/S0002-9904-1942-07767-6 10.1090/S0002-9904-1945-08291-3 10.5802/aif.3274 10.1016/j.jnt.2022.03.012 10.1090/S0025-5718-1992-1106981-9 10.2140/pjm.1957.7.861 |
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| Keywords | finite fields 14H52 Elliptic curves 11G20 primitive root |
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| References | 1945; 55 1944; 8 1957; 7 1955; 7 1962; 12 2006; 97 1949; 66 1992; 58 1942; 48 1949; 3 1930; 1 2019; 69 S0008439521000448_r1 Kohel (S0008439521000448_r10) 2000 S0008439521000448_r3 S0008439521000448_r2 S0008439521000448_r5 S0008439521000448_r4 S0008439521000448_r7 S0008439521000448_r9 S0008439521000448_r8 Vinogradov (S0008439521000448_r19) 1930; 1 Pillai (S0008439521000448_r14) 1944; 8 Lagarias (S0008439521000448_r11) 1977 S0008439521000448_r13 S0008439521000448_r12 S0008439521000448_r18 Fridlender (S0008439521000448_r6) 1949; 66 S0008439521000448_r17 S0008439521000448_r16 S0008439521000448_r15 |
| References_xml | – volume: 1 start-page: 7 year: 1930 end-page: 11 article-title: On the least primitive root of a prime publication-title: Dokl. Akad. Nauk, S.S.S.R – volume: 55 start-page: 131 year: 1945 end-page: 132 article-title: Least primitive root of a prime publication-title: Bull. Amer. Math. Soc. – volume: 7 start-page: 861 issue: 1 year: 1957 end-page: 865 article-title: On the least primitive root of a prime publication-title: Pacific J. Math. – volume: 69 start-page: 1411 issue: 3 year: 2019 end-page: 1458 article-title: An explicit upper bound for the least prime ideal in the Chebotarev density theorem publication-title: Ann. l’Inst. Fourier – volume: 58 start-page: 369 year: 1992 end-page: 380 article-title: Searching for primitive roots in prime fields publication-title: Math. Comp. – volume: 48 start-page: 726 year: 1942 end-page: 730 article-title: On the least primitive root of a prime publication-title: Bull. Amer. Math. Soc. – volume: 8 start-page: 14 year: 1944 end-page: 17 article-title: On the smallest primitive root of a prime publication-title: J. Indian Math. Soc. – volume: 3 start-page: 7 year: 1949 end-page: 8 article-title: Über den kleinsten positiven quadratischen Nichtrest nach einer Primzahl publication-title: Math. Nachr. – volume: 12 start-page: 179 year: 1962 end-page: 192 article-title: On character sums and primitive roots publication-title: Proc. Lond. Math. Soc. (3) – volume: 66 start-page: 351 year: 1949 end-page: 352 article-title: On the least n-th power non-residue publication-title: Proc. USSR Acad. Sci. – volume: 7 start-page: 43 issue: 2 year: 1955 end-page: 44 article-title: A note on the different of the composed field publication-title: Kodai Math. Sem. Rep. – volume: 97 start-page: 41 issue: 2 year: 2006 end-page: 45 article-title: Efficient polynomial time algorithms computing industrial-strength primitive roots publication-title: Inform. Process. Lett. – volume-title: Algorithmic number theory year: 2000 ident: S0008439521000448_r10 – ident: S0008439521000448_r2 doi: 10.1112/plms/s3-12.1.179 – ident: S0008439521000448_r18 doi: 10.2996/kmj/1138843605 – volume: 1 start-page: 7 year: 1930 ident: S0008439521000448_r19 article-title: On the least primitive root of a prime publication-title: Dokl. Akad. Nauk, S.S.S.R – ident: S0008439521000448_r12 doi: 10.1007/978-1-4684-0296-4 – ident: S0008439521000448_r17 – ident: S0008439521000448_r3 doi: 10.1016/j.ipl.2005.09.014 – ident: S0008439521000448_r13 doi: 10.1007/BFb0060851 – ident: S0008439521000448_r15 doi: 10.1002/mana.19490030104 – volume: 8 start-page: 14 year: 1944 ident: S0008439521000448_r14 article-title: On the smallest primitive root of a prime publication-title: J. Indian Math. Soc. – ident: S0008439521000448_r7 doi: 10.1007/978-1-4612-3464-7_18 – ident: S0008439521000448_r8 doi: 10.1090/S0002-9904-1942-07767-6 – ident: S0008439521000448_r4 doi: 10.1090/S0002-9904-1945-08291-3 – start-page: 409 volume-title: Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) year: 1977 ident: S0008439521000448_r11 – ident: S0008439521000448_r1 doi: 10.5802/aif.3274 – ident: S0008439521000448_r9 doi: 10.1016/j.jnt.2022.03.012 – ident: S0008439521000448_r16 doi: 10.1090/S0025-5718-1992-1106981-9 – ident: S0008439521000448_r5 doi: 10.2140/pjm.1957.7.861 – volume: 66 start-page: 351 year: 1949 ident: S0008439521000448_r6 article-title: On the least n-th power non-residue publication-title: Proc. USSR Acad. Sci. |
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| Snippet | 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... 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... 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... |
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| SubjectTerms | Curves Hypotheses Lower bounds Numbers |
| Title | An elliptic curve analogue of Pillai’s lower bound on primitive roots |
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