Fix an odd prime $p$.

For concreteness, let $N$ be coprime to $p$, and let $2 \leq k \leq p$. Let $S^+(N,k)$ be the newforms in $S_k(\Gamma_1(N))$.

Let $f = \sum a_n q^n \in S^+(N,k)$. We say that $f$ is $p$-ordinary if $v_p(a_p)=0$. (If $p$ splits in the field of coefficients of $f$, we require that $a_p$ not be divisible by any prime $\mathfrak{p}$ above $p$.)

For a "random" $N$ and $k$ satisfying the conditions I set out at the beginning, what can be said about the proportion of $p$-ordinary forms in $S^+(N,k)$?

If we let $N$ and $k$ vary, can we say anything about the density of $p$-ordinary forms in the union $\bigcup_{N,k} S_k^+(\Gamma_1(N))$?


An ordinary modular form needs to be $p$-stable (by definition). If you have a modular form of level $N$ co-prime to $p$, you need to associate a $p$-stabilization to this latter to get an ordinary modular form with level $Np$.

If you take $T$ the Hecke algebra acting on Katz p-adic modular forms, it is known that the classical points are dense in $\operatorname{Spec}T$, and $T$ has krull dimension at least $4$, but the ordinary Hecke algebra has a dimension equal to $2$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.