Rational points on twists of X₀(63)
The rational points on in the case where is a composite number are considered. A computational study of some of the cases not covered by the results of Momose is given. Exceptional rational points are found in the cases and and the -invariants of the corresponding quadratic -curves are exhibited.
Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of , for and a prime number exceeding . This includes the case of the curves . We then prove, with the help of computer calculations, that the same holds true for in the range , . The combination of those results completes the qualitative study of rational points on undertook in our previous work, with the only exception of .
We derive a relation between induced representations on the group which implies a relation between the jacobians of certain modular curves of level . The motivation for the construction of this relation is the determination of the applicability of Mazur’s method to the modular curve associated to the normalizer of a non-split Cartan subgroup of .