The Conjecture

Let $k=2m$ be an even integer and $p$ a prime. Let $\mathbb{T}$ be the Hecke algebra associated to $S_k(\Gamma_0(p))$ and let $\tilde{\mathbb{T}}$ be the normalization of $\tilde{\mathbb{T}}$ in $\mathbb{T}\otimes \mathbb{Q}$.

Conjecture 5.1  

$$\ord_p([\tilde{\mathbb{T}}: \mathbb{T}]) = \left\lfloor\frac{p}{12}\right\rfloor\cdot \binom{m}{2} + a(p,m),$$

where

$$a(p,m) = \begin{cases} 0 & \text{if $p\equiv 1\pmod{12}$,}... ... a(5,m)+a(7,m) & \text{if $p\equiv 11\pmod{12}$.} \end{cases}$$

In particular, when $k=2$ we conjecture that $[\tilde{\mathbb{T}}:\mathbb{T}]$ is not divisible by $p$.

Here $\binom{x}{y}$ is the binomial coefficient ``$x$ choose $y$'', and floor and ceiling are as usual. We have checked this conjecture against significant numerical data. (Will describe here.)