I've been carrying out some experiments on the computer and I noticed the following congruence phenomenon: fixing a prime $p$, it seems that any modular form over $SL_2(\mathbb{Z})$ and of weight $k \geq p+1$ is congruent $\pmod{p}$ to some modular form over $\Gamma_0(p)$ and of weight $k-(p-1)$.

(Edit: by congruent I mean the formal q-expansions at infinity are congruent).

Of course the case where $k = p+1$ is a result of Serre in his "p-adic zeta functions" paper.

I can prove the statement above is true when $p \equiv 11 \pmod{12}$ and $k \geq p+5$. I'm interested to know if someone knows if this is true in general and possibily a proof or a reference. Thanks!

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    $\begingroup$ Did you look at Jochnowitz's two papers in Trans AMS 270 (1982)? There might be something relevant there. $\endgroup$ Sep 18, 2013 at 12:16
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    $\begingroup$ Scott Ahlgren and I considered some extensions of Serre's results in higher weights in our paper "Higher Weierstrass points on $X_0(p)$" in Transactions of the AMS 355 (2003), 1521-1535. See section 4 there. We work out things more in the opposite direction though from your question, relating forms on $\Gamma_0(p)$ to ones on $SL_2(\mathbb{Z})$ rather than the other way around, but maybe it will be helpful. $\endgroup$ Sep 20, 2013 at 16:33
  • $\begingroup$ Isn't this Hida's "control theorem"? $\endgroup$
    – Will Sawin
    Mar 4, 2016 at 15:41


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