2
$\begingroup$

Is [Scholl, Motives for modular forms, Theorem 1.2.4 (ii)] proven for any $p$ independent of the weight?

Concretely, let $f$ be a normalized eigenform of weight $w$. Let $p$ be a prime not dividing the level of $f$. Then the $\pi$-adic ($\pi \mid p$) realization is crystalline at $p$ and the characteristic polynomial equal to the Hecke polynomial of $f$ for $p$.

[Scholl] says that forthcoming work of Faltings should remove his condition $p \geq w$, but I read that there might be a mistake in Faltings' article. Is this true?

If it is true and known (which is what several posts like Galois representations attached to newforms here suggest), what is a citable reference (article and theorem therein)?

$\endgroup$

1 Answer 1

3
$\begingroup$

Blasius and Rogawski's paper "Motives for Hilbert modular forms" (1993) proves a more general result for Hilbert modular forms over any totally-real field, which includes this as a special case.

$\endgroup$
6
  • $\begingroup$ Thanks. But Corollary 2.5.3 is only for large $p$. $\endgroup$
    – user471019
    Jun 2, 2023 at 17:29
  • 1
    $\begingroup$ Good point. The more general statement is certainly true and well-known nowadays, it's just a question of pinpointing the earliest appearance of it in print. Saito's "Modular forms and p-adic Hodge theory" (Inventiones 129, 1997) proves something considerably more general (while ascribing the $p \nmid N$ case to Scholl, without any $p \ge w$ assumption). $\endgroup$ Jun 5, 2023 at 15:38
  • $\begingroup$ Thanks! What do I need to know about $p$-adic Hodge theory/what should I read to understand why Saito's result implies $p \nmid N \implies$ crystalline? $\endgroup$
    – user471019
    Jun 5, 2023 at 16:22
  • 3
    $\begingroup$ @user471019 I think the first complete published proof removing the condition for large $p$ and independent of Faltings's work (which does contain a gap), is in Tsuji, Inventiones 1999 (not that this paper was available already to Saito in 1997 and that Saito's paper crucially relies on it, even to state its main theorem). Falting's proof was completed in 2002 (Almost étale extensions) but most human beings find it hard to digest nevertheless. $\endgroup$
    – Olivier
    Jun 5, 2023 at 20:05
  • 1
    $\begingroup$ @user471019 Somewhat unfortunately, I believe the answer to the question "what do I need to know about $p$-adic Hodge theory to understand Saito's result ?" is basically "all of it": when the weight of the modular form is large compared to $p$, I don't think there is any significant short cut to obtain Ccrys (the statement that good reduction implies crystalline) just for the Kuga-Sato variety (rather than for a general propre smooth scheme with good reduction). On the other hand, you can certainly understand what Saito does and what it means while taking the proof of Ccrys as a black box. $\endgroup$
    – Olivier
    Jun 6, 2023 at 7:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.