[Paper Review] Sch\\'emas en groupes et poids de Diamond-Serre

Xavier Caruso|arXiv (Cornell University)|May 9, 2007

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper corrects a flawed statement in a preprint by Gee (2007) concerning the weights of mod $p$ Hilbert modular forms via $p$-adic Hodge theory. It redefines the notion of class $J$-groups to fix an error in the original formulation, establishes a corrected version of Proposition 3.3.1, and proves it using crystalline and Breuil modules, thereby ensuring the validity of subsequent results in the modularity conjecture framework for totally real fields.

ABSTRACT

This note is a correction of (statement and proof of) proposition 3.3.1 of Toby Gee's preprint intitled *On the weights of mod p Hilbert modular forms*. The aim is to compare Galois representations arising from extensions of some group schemes (over the ring of integers of a p-adic field) endowed with a descent data, and extensions of some crystalline representations with given Hodge-Tate weights. The main tool of the proof is the theory of Breuil.

Motivation & Objective

• To correct a critical error in Proposition 3.3.1 of Gee's preprint on mod $p$ Hilbert modular forms.
• To redefine the notion of class $J$-groups by introducing an index shift to fix the flawed statement.
• To provide a complete and correct proof of the corrected version of Proposition 3.3.1 using $p$-adic Hodge theory and module-theoretic techniques.
• To ensure the validity of subsequent results in Gee's work that rely on this proposition.
• To clarify the interplay between crystalline representations, $E$-groups, and Breuil modules in the context of Diamond-Serre weights.

Proposed method

• Rederive the corrected statement of Proposition 3.3.1 by redefining the class $J$ condition with an index shift, based on feedback from Gee.
• Construct a module $\mathcal{C}$ in the category $\text{E-BrMod}_{\text{dd}}^{p-2}$ using $E[u]$-modules with filtration, Frobenius, and monodromy structures.
• Define morphisms $f_{\mathcal{M}}$ and $f_{\mathcal{N}}$ from source modules $\mathcal{M}$ and $\mathcal{N}$ to $\mathcal{C}$, using $u$-powers and weight shifts.
• Use congruences and inequalities from Lemma 5.8 and 5.9 to ensure well-definedness of the module structure and filtration.
• Verify that the morphisms commute with Frobenius, monodromy, and Galois action, confirming they are valid morphisms in the category.
• Leverage the theory of Breuil modules and crystalline representations to relate the corrected module structure to the original Galois representation.

Experimental results

Research questions

• RQ1 What is the correct formulation of Proposition 3.3.1 in Gee's preprint, given that the original version is flawed?
• RQ2 How does the introduction of an index shift in the definition of class $J$-groups resolve the error in the original statement?
• RQ3 Can the corrected version of the proposition be proven using the framework of $p$-adic Hodge theory and module categories?
• RQ4 What is the role of the module $\mathcal{C}$ in realizing the corrected correspondence between Galois representations and weights?
• RQ5 How do the morphisms $f_{\mathcal{M}}$ and $f_{\mathcal{N}}$ ensure compatibility with the structure of the category $\text{E-BrMod}_{\text{dd}}^{p-2}$?

Key findings

• The original statement of Proposition 3.3.1 in [6] is incorrect due to an error in the definition of class $J$-groups.
• The corrected version of the proposition is obtained by redefining class $J$-groups with an index shift, as suggested by Gee.
• The corrected statement is proven via the construction of a well-defined module $\mathcal{C}$ in the category $\text{E-BrMod}_{\text{dd}}^{p-2}$ with compatible Frobenius, filtration, and monodromy structures.
• The morphisms $f_{\mathcal{M}}$ and $f_{\mathcal{N}}$ are shown to be valid in the category, ensuring the commutativity of the key diagram (6).
• The proof relies on congruences and inequalities from Lemmas 5.8 and 5.9 to ensure that all terms, especially those involving $\lambda_i$, are well-defined and non-vanishing when needed.
• The corrected result is consistent with the use of the proposition in subsequent work by Gee, implying that the main theorems in [6] remain valid under the corrected formulation.