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For the space of modular forms and the space of cusp forms (here I only care about the level $1$ case), we have the action by Hecke algebras. Therefore, we can calculate the discriminant of this action. Let $D_{\mathrm{modular}}(k)$ and $D_{\mathrm{cusp}}(k)$ denote the discriminant of the hecke algebra over the space of modular/cusp forms of weight $k$.

It seems to me (by some calculation) that $$\frac{D_{\mathrm{modular}}(k)}{D_{\mathrm{cusp}}(k)}=B_{k}^2\cdot(\textrm{product of small primes})^2,$$ where $B_k$ denotes the $k$-th Bernoulli number.

What I want to ask is that is there any reference for this identity, and is it known that which small primes will show up on the right hand side of the above identity?

Thanks a lot.

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  • $\begingroup$ The primes showing up here should be the ones for which the cuspidal Hecke algebra does not split off from the full Hecke algebra as a direct summand, which happens exactly when there's a congruence between a cusp form and an Eisenstein series in weight $k$. Thus the appearance of the Bernoulli number is rather natural, isn't it? $\endgroup$
    – David Loeffler
    Commented May 23, 2015 at 10:57

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