Explicit determination of non-trivial torsion structures of elliptic curves over quadratic number fields

Computing Galois representations and equations for modular curves $X_H(\ell)$

“…The modular curve Y 1 (N ) := X 1 (N )− {cusps} parameterizes isomorphism classes of pairs (E, P ) where E is an elliptic curve and P is an N -torsion point on E. Equations for X 1 (N ) have been extensively studied by many authors. The approach in Reichert [11], Baaziz [1] and Sutherland [14] not only gives an equation for X 1 (N ) but they also describe the moduli interpretation in terms of the equation, i.e. they describe how pairs (E, P ) correspond to solutions of their equations.…”

“…Here α is a root of x 2 + 2x − 1 and β is a root of x 2 + 5. The reductions of f 2 modulo 29 are f 2,1 = q + 16q 2 + 13q 3 + 26q 4 + 26q 5 + 5q 6 + 2q 7 + 13q 8 + 27q 9 + 10q 10 + 16q 11 + O(q 12 ), f 2,2 = q + 13q 2 + 16q 3 + 26q 4 + 26q 5 + 5q 6 + 2q 7 + 16q 8 + 27q 9 + 19q 10 + 13q 11 + O(q 12 ). We can check that ∆ 16 mod 29 = f 2,2 and m 16,29 = 29, T 2 − τ 16 (2) .…”

On the automorphism groups of some modular curves

“…By the remarks at the end of Section 3 we know that η is defined over Q(ζ 13 ) + , and hence η must be defined over Q(ζ 13 ) + Q(P 1 ). However, Billing and Mahler [2] (or, see [10]) give an explicit equation for X 1 (13), from which one sees, by explicit calculation, that Q(ζ 13 ) + Q(P 1 ) = Q. Thus, η is defined over Q.…”

Sequences of rational torsions on abelian varieties

“…This is slower than the quadratic sequences already found on Jacobians, but the method should be amenable to refinements by taking careful choices of E for each m (not necessarily with complex multiplication) so as to minimise [k : Q]. In particular, Richard Pinch has pointed out that, since X 1 (11), X 1 (13) have hyperelliptic models (given in [8]), this process will give abelian varieties over Q of dimension 2 with Q-rational 11-and 13-torsion points. It would be interesting to compare the resulting varieties with the Jacobians of the genus 2 curves obtained by specialising Results 2 and 3 of §2.…”