Computing the Mazur and Swinnerton-Dyer critical subgroup of elliptic curves

Source abstract

Let E E be an optimal elliptic curve defined over Q \mathbb {Q} . The critical subgroup of E E is defined by Mazur and Swinnerton-Dyer as the subgroup of E ( Q ) E(\mathbb {Q}) generated by traces of branch points under a modular parametrization of E E . We prove that for all rank two elliptic curves with conductor smaller than 1000, the critical subgroup is torsion. First, we define a family of critical polynomials attached to E E and develop two algorithms to compute such polynomials. We then give a sufficient condition for the critical subgroup to be torsion in terms of the factorization of critical polynomials. Finally, a table of critical polynomials is obtained for all elliptic curves of rank two and conductor smaller than 1000, from which we deduce our result.