OFFSET
1,1
COMMENTS
a(1) = 7 is due to Mazur; a(2) = 13 to Kamienny, Kenku, and Momose; a(3) = 13 to Parent; and a(4) = 17 to Kamienny, Stein, and Stoll. See Derickx 2011.
For each n = 1..32, an explicit elliptic curve with a point of order p(n) has been found over a number field of degree n where p(n) = 7, 13, 13, 17, 19, 37, 23, 23, 31, 37, 31, 43, 37, 43, 43, 37, 43, 43, 43, 61, 47, 67, 47, 73, 53, 79, 61, 53, 53, 73, 61, 97. So p(n) is a lower bound for a(n). I suspect most of them are sharp but that would be difficult to prove. - Mark van Hoeij, May 21 2014
LINKS
Maarten Derickx, Torsion points on elliptic curves over number fields of small degree, UW Number Theory Seminar, 2011
Mark van Hoeij, Low Degree Places on the Modular Curve X1(N)
Wikipedia, Mazur's torsion theorem
EXAMPLE
Mazur proved that elliptic curves over the rationals can have p-torsion only for p = 2, 3, 5, 7, so a(1) = 7.
CROSSREFS
KEYWORD
nonn,more,hard,bref
AUTHOR
Jonathan Sondow, Oct 12 2013
STATUS
approved