Abstract:
The Jacobian J0(N) of the modular curve X0(N) has received much attention within arithmetic geometry for its relation with cusp forms and elliptic curves. In particular, the group of Q-rational points on X0(N) controls the cyclic N-isogenies of elliptic curves. A conjecture of Ogg predicted that, for N prime, the torsion of this group comes all from the cusps. The statement was proved by Mazur and later generalised to arbitrary level N into what we call generalised Ogg’s conjecture.
Consider now the generalised Jacobian J0(N)m with respect to a modulus m. This algebraic group also seems to be related to the arithmetic of X0(N) through the theory of modular forms. In the talk we will present new results that compute the Q-rational torsion of J0(N) for N an odd integer with respect to a cuspidal modulus m. These generalise previous results of Yamazaki, Yang and Wei. In doing so, we will also discuss how our results relate to generalised Ogg’s conjecture.