Alladi Ramakrishnan Hall

Order of cuspidal subgroup and index of Eisenstein ideal

Dipramit Majumdar

IIT Madras

Let $p$ be a prime, $J_0(p)$ denotes the Jacobian variety of the modular curve $X_0(p)$ and $C_0(p)$ is the cuspidal subgroup of $J_0(p)$ generated by $(0)-(\infty)$. Ogg's conjecture states that

$$J_0(p)(\mathbb{Q})_{tor} = C_0(p).$$

This conjecture of Ogg was proved by Mazur in his celebrated Modular Curve and Eisenstein Ideal paper. Key idea of his proof was to give $C_0(p)$ a Hecke algebra structure and prove that index of the Eisenstein ideal in Hecke algebra is same as the order of the cuspidal subgroup. In this talk, I'll talk about recent development of generalised Ogg's conjecture in the non-prime level case.