The title refers to the paper of Faltings:

Hodge-Tate structures and modular forms.

Math. Ann. 278 (1987), no. 1-4, 133–149.

The main theorem in the paper says that the associated Galois rep to a modular form (of weight $k+2$), when restricted to $G_{Qp}$, has Hodge-Tate weights $\{0, k+1\}$.

My question is, does there exist any more easier-to-understand expositions of this result? In particular, since $p$-adic Hodge theory has so far developed so much, maybe a modern exposition could have better notations and more insights, etc??

  • 2
    $\begingroup$ In the ordinary case, page 164 of Mazur's "Infinite fern" paper sketches a way to see this, provided you already know that the determinant of the representation is $\chi \epsilon_{p}^{k+1}$. $\endgroup$ – Jeff H Jun 23 '15 at 16:46

In the ordinary case, one can see that $\rho_{f}$ is an extension of an unramified character $\chi'$ by $\chi\epsilon_p^{k-1}$ (where $\epsilon_p$ is cyclotomic character). And it is easy to see that $\rho_f$ is potentially semi-stable at $p$ of hodge tate weight (k-1,0), and even crystalline when $k\geq 3$ and $\chi=1$ or when this extension is flat. But when we don't assume the ordinariness the proof become more hard and it is a result of Saito that any modular form of weight $k$ is potentially semi-stable of weight (k-1,0). Moreover, when $p$ doesn't divide the level, $\rho_f$ will be crystalline (result of scholl).


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