1,11

a(n) is the least d>0 for which there exists a plane curve f(x,y)=0 of degree d in x or y which is birationally equivalent to the modular curve X_1(n). There exists infinitely many non-isomorphic elliptic curves defined over number fields of degree a(n) which contain a point of order n. a(n)=1 if and only if X_1(n) has genus 0 and these values of n represent the possible finite orders of a point on an elliptic curve over Q.

By Mazur's theorem, these are 1,2,3,4,5,6,7,8,9,10 and 12. a(n)=2 if and only if X_1(n) is elliptic or hyperelliptic, which occurs only for n=11,13,14,15,16 and 18 [Mestre 1981]. The lower bound a(17)>3 follows from [Parent 1999] and the upper bound a(17)<=4 appears (for example) in [Reichert 1986]. a(20)=3 since it cannot be 1 or 2 and an explicit example of degree 3 is known (see below).

From [Jeon-Kim-Schweizer 2006] it follows that this is the only case when a(n)=3. The results a(21)=4 and a(22)=4 then follow from explicit examples [Sutherland 2008]. a(24) is either 4 or 5 and a(n) is not 4 for any n other than 17, 21, 22, or 24 by the results of [Jeon-Kim-Park 2006]. a(23) must be 5, 6, or 7. See [Sutherland 2008] for these and other upper bounds for n <= 50.

J.-F. Mestre, "Corps euclidiens, unites exceptionnelles et courbes elliptiques," J. Number Theory, vol. 13, 1981, pp. 123-137

Markus Reichert, "Explicit Determination of Nontrivial Torsion Structures of Elliptic Curves Over Quadratic Number Fields," Math. Comp. 46 (1986), pp. 637-658.

Daeyeol Jeon, Chang Heon Kim and Andreas Schweizer, "On the torsion of elliptic curves over cubic number fields," Acta Arithmetica 113 (2004), pp. 291-301.

Andrew V. Sutherland, Constructing elliptic curves with prescribed torsion over finite fields, preprint, 2008.

a(20)<=3 because y^3+(x^2+3)y^2+(x^3+4)y+2=0 is an explicit plane model for X_1(20) and a(20)=3 because it is not 1 or 2 (these are all known).

A029937

Adjacent sequences: A146876 A146877 A146878 this_sequence A146880 A146881 A146882

Sequence in context: A025801 A140426 A060548 this_sequence A058762 A029252 A094876

hard,more,nonn

Andrew V. Sutherland (drew(AT)math.mit.edu), Nov 03 2008