[Paper Review] From potential modularity to modularity for integral Galois representations and rigid Calabi-Yau threefolds

Luís Dieulefait|ArXiv.org|Sep 7, 2004

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper establishes modularity for irreducible, odd, two-dimensional $σ_{\ell}$-adic Galois representations with rational coefficients, crystalline at $\ell$, and satisfying a trace condition at 3, by combining compatible family existence (Dieulefait) with potential modularity (Taylor) and ordinarity control (Wiles). The key result proves modularity of rigid Calabi-Yau threefolds over $\mathbb{Q}$ with good reduction at 3 and $3 \nmid a_3$, extending modularity criteria beyond previous requirements at 5 or 7.

ABSTRACT

We prove modularity for any irreducible crystalline $\ell$-adic odd 2-dimensional Galois representation (with finite ramification set) unramified at 3 verifying an "ordinarity at 3" easy to check condition, with Hodge-Tate weights $\{0, w \}$ such that $2 w < \ell$ (and $\ell > 3$) and such that the traces $a_p$ of the images of Frobenii verify $\Q(\{a_p \}) = \Q $. This result applies in particular to any motivic compatible family of odd two-dimensional Galois representations of $\Gal(\bar{\Q}/\Q)$ if the motive has rational coefficients, good reduction at 3, and the "ordinarity at 3" condition is satisfied. As a corollary, this proves that all rigid Calabi-Yau threefolds defined over $\Q$ having good reduction at 3 and satisfying $ 3 mid a_3$ are modular.

Motivation & Objective

• To establish modularity for odd, irreducible, two-dimensional $\ell$-adic Galois representations with rational coefficients and finite ramification.
• To prove modularity of rigid Calabi-Yau threefolds defined over $\mathbb{Q}$ under a trace condition at 3.
• To extend modularity criteria beyond previous requirements at primes like 5 or 7, focusing on the condition $3 \nmid a_3$.
• To bridge potential modularity with full modularity via control of ordinarity at 3 using Wiles' results.
• To show that compatible families of motives with rational coefficients and good reduction at 3 are modular if $3 \nmid a_3$.

Proposed method

• Use the existence of a compatible family of Galois representations containing $\sigma_\ell$, as established in Dieulefait (2004).
• Apply Taylor’s potential modularity theorems to show that the family becomes modular over a totally real extension $F$ of $\mathbb{Q}$ where 3 splits completely.
• Translate the condition $3 \nmid a_3$ into ordinarity of the $3$-adic representation in the family using Wiles’ criterion for ordinarity in Hilbert modular forms.
• Use solvable base change to ensure that ordinarity at all primes above 3 in $F$ implies ordinarity of the full $3$-adic representation.
• Apply Skinner-Wiles modularity lifting theorems to deduce modularity of the $3$-adic representation, hence of $\sigma_\ell$ via compatibility.
• Leverage the fact that the residual mod 3 representation is either modular or reducible, and use ordinarity to rule out reducibility and ensure modularity.

Experimental results

Research questions

• RQ1 Under what conditions is a two-dimensional, odd, irreducible $\ell$-adic Galois representation with rational coefficients and finite ramification modular?
• RQ2 Can the modularity of rigid Calabi-Yau threefolds over $\mathbb{Q}$ be established using only good reduction at 3 and a trace condition at 3?
• RQ3 How does the condition $3 \nmid a_3$ relate to the ordinarity of the $3$-adic representation in a compatible family?
• RQ4 Can potential modularity results be combined with coefficient field control and trace conditions to deduce full modularity?
• RQ5 What is the role of Wiles’ theorem on ordinarity in lifting modularity from residual representations?

Key findings

• The paper proves that any irreducible, odd, two-dimensional $\ell$-adic Galois representation with rational coefficients, crystalline at $\ell$, and satisfying $\ell > 2w$ and $3 \nmid a_3$, is modular.
• The condition $3 \nmid a_3$ implies that the $3$-adic representation in the compatible family is ordinary, enabling application of modularity lifting theorems.
• The $3$-adic representation is shown to be modular via Skinner-Wiles results, given its ordinarity and residual mod 3 modularity.
• The modularity of the $\ell$-adic representation follows from compatibility with the modular $3$-adic representation.
• The result applies to all rigid Calabi-Yau threefolds defined over $\mathbb{Q}$ with good reduction at 3 and $3 \nmid a_3$, proving their modularity.
• The paper provides a new modularity criterion for rigid Calabi-Yau threefolds that does not require good reduction at 5 or 7, unlike prior results.