[Paper Review] Uniform behavior of families of Galois representations on Siegel modular forms and the Endoscopy Conjecture

Luís Dieulefait|ArXiv.org|Apr 27, 2003

Advanced Algebra and Geometry8 references3 citations

TL;DR

This paper establishes a uniformity principle for Galois representations attached to genus two Siegel cusp forms of weight $k > 3$ with multiplicity one and semistable reduction: if one $\ell$-adic representation in the compatible family is reducible for an odd prime $\ell_0$, then all are reducible. Combined with Serre’s modularity conjecture (now a theorem), this proves the Endoscopy Conjecture, showing that irreducible components arise from classical modular forms via twist.

ABSTRACT

We prove the following uniformity principle: if one of the Galois representations in the family attached to a genus two Siegel cusp form of weight $k>3$, "semistable" and with multiplicity one, is reducible (for an odd prime $p$),then all the representations in the family are reducible. This, combined with Serre's conjecture (which is now a theorem) gives a proof of the Endoscopy Conjecture.

Motivation & Objective

• To establish a uniformity principle for reducibility of Galois representations in compatible families attached to genus two Siegel cusp forms.
• To extend previous results on irreducibility and large image for level 1 forms to the general semistable case.
• To prove that reducibility at one prime implies reducibility for all primes in the compatible family.
• To show that the Endoscopy Conjecture holds under semistability, by combining uniformity with Serre’s conjecture (now a theorem).

Proposed method

• Use Taylor’s results on Fontaine-Mazur conjecture and meromorphic $L$-function continuation for odd two-dimensional Galois representations.
• Apply $p$-adic Hodge theory to analyze inertia action and extend Ribet’s results on semistable two-dimensional representations to higher weights.
• Employ Chebotarev density theorem and Galois group theory to deduce global uniformity from local reducibility.
• Work with the field of coefficients of characteristic polynomials (rather than field of definition) to handle non-absolute irreducibility.
• Construct compatible families of two-dimensional representations from the components of the 4-dimensional Galois representation.
• Apply Serre’s conjecture (now a theorem) to deduce modularity of the 2-dimensional components, proving they arise from classical modular forms.

Experimental results

Research questions

• RQ1 Under what conditions does reducibility of a single $\ell$-adic Galois representation in a compatible family imply reducibility for all others?
• RQ2 Can the uniformity of reducibility be established for compatible families of four-dimensional symplectic Galois representations attached to Siegel modular forms?
• RQ3 How does the combination of uniform reducibility and Serre’s modularity conjecture lead to the Endoscopy Conjecture?
• RQ4 To what extent can the assumption $\ell_0 > 4k - 5$ be removed in the uniformity result?
• RQ5 What is the role of semistability in ensuring that reducibility is a uniform property across the compatible family?

Key findings

• If the $\ell_0$-adic Galois representation attached to a genus two Siegel cusp form of weight $k > 3$ is reducible for some odd prime $\ell_0 \nmid N$, then all representations in the compatible family are reducible.
• The uniformity result holds for all primes $\ell_0 > 2$, after removing the earlier restriction $\ell_0 > 4k - 5$.
• When the form is not of Saito-Kurokawa type, the reducible components must be two irreducible 2-dimensional representations with the same determinant.
• The 2-dimensional components of the Galois representation are shown to be modular via Serre’s conjecture, arising from classical modular forms of weight 2 and $2k-2$.
• The Endoscopy Conjecture is proven in the semistable case: the irreducible components of the 4-dimensional Galois representation are modular up to twist.
• The result extends to the non-semistable case via recent work of Skinner and Urban, confirming the Endoscopy Conjecture in full generality for these families.