Let $K$ be an imaginary quadratic field. Let $f \in S_2(\mathfrak{n})$ be a weight $2$ cuspidal cof level $\Gamma_0(\mathfrak{n})$ over $K$ (for definitions one can see http://www.lmfdb.org/knowledge/show/mf.bianchi.bianchimodularforms ), consider its $L$-function

$L(f,s)=\sum_{\mathcal{P} \not =0 \in Spec(O_K)} c(\mathcal{P}) (N\mathcal{P})^{-s}$

here $c(\mathcal{P})$ are the Fourier coefficients at the cusp $\infty$ and we assume they are all rational numbers.

The question is, do we know that $c(\mathcal{P})=0$ for infinitely many prime ideals $\mathcal{P}$? How about $c(\mathcal{P})=0 \mod N(\mathcal{P})$?

Motivation: if we are considering similar problems over $\mathbb Q$, then we know $a_p=0$ for infinitely many $p$ because the corresponding elliptic curve over $\mathbb Q$ must have supersingular reduction at infinitely many primes.

  • $\begingroup$ Do these (conjecturally) correspond to elliptic curves over imaginary quadratic field? I don't think there is a single non-CM elliptic curve over a field with no real places where we know there are infinitely many supersingular primes. If you pick your favorite no-CM elliptic curve over an imaginary quadratic field, and check it is modular, then I think we have no clue how to check it has infinitely many vanishing Fourier coefficients. $\endgroup$ – Will Sawin Sep 11 at 19:30
  • $\begingroup$ @Will Sawin I hope there is an automorphic approach to this problem, because modularity is so difficult.. $\endgroup$ – sawdada Sep 11 at 23:32
  • $\begingroup$ I'm trying to argue that there is no automorphic approach. Checking modularity for a given elliptic curve is easy nowadays (I think). But there is no progress on this problem in either the geometric or automorphic settings. $\endgroup$ – Will Sawin Sep 12 at 1:40
  • $\begingroup$ There is indeed no automorphic approach, AFIAK, which is why the corresponding statement is completely unknown for elliptic modular forms of weight $> 2$. $\endgroup$ – David Loeffler Sep 12 at 9:43

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