A229831 Largest prime p such that some elliptic curve over an extension of the rationals of degree n has a point of finite order p.

7, 13, 13, 17

Offset: 1

Author

Jonathan Sondow, Oct 12 2013

Keywords

• nonn
• more
• hard
• bref

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

Examples

			Mazur proved that elliptic curves over the rationals can have p-torsion only for p = 2, 3, 5, 7, so a(1) = 7.
		

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

Crossrefs

Cf. A221362.