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Let $f$ be a newform of weight $k \geq 2$ and level $N \geq 1$ without complex multiplication. A prime $p$ is said to be ordinary for $f$ if the $p$-th Fourier coefficient $a_p(f)$ is a $p$-adic unit (to make sense of this in general, one needs to choose a prime ideal above $p$ in the field $K_f$ of Fourier coefficients of $f$, equivalently an embedding of $K_f$ into $\overline{\mathbf{Q}_p}$).

Non-ordinary primes seem to be rare. For example, if $f$ has coefficients in $\mathbf{Z}$, then a very rough guess is that $a_p(f)$ modulo $p$ is randomly distributed, so we may expect that

\begin{equation*} \# \{p \leq x : a_p \equiv 0 \textrm{ mod } p \} \stackrel{?}{\approx} \sum_{p \leq x} \frac{1}{p} \sim \log \log x. \end{equation*} On the other hand, for the modular form $\Delta = \sum_{n \geq 1} \tau(n) q^n$, it seems not to be known that there are infinitely many primes $p$ such that $\tau(p) \not\equiv 0$ mod $p$, so estimating the number of ordinary or non-ordinary primes is difficult in general.

If we consider elliptic curves, we may ask whether every prime $p$ is non-ordinary for some elliptic curve $E$ over $ \mathbf{ Q } $, which means that the reduction of $E$ mod $p$ is supersingular. In fact, Deuring has shown that given any integer $a$ such that $|a|<2\sqrt{p}$, there exists an elliptic curve $E_p$ over the finite field $\mathbf{F}_p$ with exactly $p+1-a$ points over $\mathbf{F}_p$. Now take any elliptic curve $E$ over $\mathbf{Q}$ whose reduction mod $p$ is $E_p$ and use the modularity theorem. We get a modular form $f$ of weight $2$ with integral coefficients satisfying $a_p(f)=a$, and we may choose it to be non-CM (in fact, we get infinitely many such modular forms).

My question is whether this kind of result is known or even expected in higher weight. Certainly, if you have a finite set of newforms without CM, the above heuristics suggest that there exist infinitely many primes which are ordinary for all these newforms. But I don't know what happens for an infinite set of newforms, even if this set is "thin" in some sense.

Here is another example of question which arises.

Fix a weight $k \geq 3$. Is every prime $p$ non-ordinary for some non-CM newform of weight $k$ and level not dividing $p$? At least, are there infinitely many such primes?

Hida's theory implies that if $p \geq 5$ and $k$ is equal to $\{3,4,5,\ldots,10,14\}$ modulo $p-1$, then all newforms of weight $k$ and level $1$ are non-ordinary at $p$. This gives a positive answer to the question for some primes $p$ which are small with respect to $k$ (in particular, these primes satisfy $p \leq k-2$).

One may try to use the theory of congruences between modular forms, like the theory of Hida families, but usually one starts with a modular form which is ordinary at $p$, and I don't know to which extent the theory has been generalized to the non-ordinary case.

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Given p and k, it's clear we can find a CM-type newform of weight k and some level which is supersingular at p (just choose an imaginary quadratic field in which p is inert, and some sufficiently large conductor away from p).

So it suffices to find a second newform that is congruent to the first one mod p and is not of CM type. This can always be done (after adding some auxiliary primes to the level) by the answer to this old question of mine: Congruences between CM and non-CM modular forms.

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  • $\begingroup$ Very nice and helpful answer, thank you! $\endgroup$ – François Brunault Sep 10 '18 at 20:58
  • $\begingroup$ Of course, this leads immediately to a much harder question: can one always find an $f$ satisfying the conditions of your question such that the mod p Galois representation of f has large image? $\endgroup$ – David Loeffler Sep 11 '18 at 13:17

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