[Paper Review] Serre's conjecture over F_9

Jordan S. Ellenberg|ArXiv.org|Jul 20, 2001

Algebraic Geometry and Number Theory8 references4 citations

TL;DR

This paper proves Serre's conjecture for mod 9 Galois representations under specific local conditions at primes 3 and 5, using a geometric method based on solvable points on Hilbert modular surfaces. The key result establishes modularity of odd Galois representations over 𝔽₉ and, as a corollary, modularity of Hilbert-Blumenthal abelian surfaces with good ordinary reduction at 3 and 5.

ABSTRACT

In this paper, we show that an odd Galois representation rhobar: Gal(Qbar/Q) --> GL_2(F_9) satisfying certain local conditions at 3 and 5 is modular. Our main tool is an idea of Taylor, which reduces the problem to that of exhibiting points on a Hilbert modular surface which are defined over a solvable extension of Q, and which satisfy certain reduction properties. As a corollary, we show that Hilbert-Blumenthal abelian surfaces over Q with good ordinary reduction at 3 and 5 are modular.

Motivation & Objective

• To prove Serre’s conjecture for mod 9 Galois representations satisfying local conditions at 3 and 5.
• To extend the modularity lifting techniques of Taylor and Skinner-Wiles to the case of 𝔽₉.
• To establish the modularity of Hilbert-Blumenthal abelian surfaces with good ordinary reduction at 3 and 5.
• To demonstrate that certain Hilbert modular surfaces have Zariski-dense sets of points over solvable extensions, introducing the notion of 'property S'.

Proposed method

• Reduces the modularity of a Galois representation to the existence of rational points on a twisted Hilbert modular surface over a solvable extension of ℚ.
• Applies Taylor’s idea of cyclic descent to descend automorphic forms from solvable extensions back to ℚ.
• Uses the theorem of Skinner and Wiles to lift modular representations and establish modularity of 3-adic Galois representations.
• Employs the Langlands–Tunnell theorem and Diamond’s refinement of the modularity lifting theorem to handle the 3-adic and 5-adic components.
• Constructs a Hilbert-Blumenthal abelian surface over a solvable extension F/ℚ such that its 3-torsion realizes the given mod 9 Galois representation.
• Imposes conditions on inertia groups at 3 and 5 to ensure the representation is Dₚ-distinguished and ordinary, enabling descent.

Experimental results

Research questions

• RQ1 Can Serre’s conjecture be extended to the case of 𝔽₉, where GL₂(𝔽₉) is non-solvable?
• RQ2 Do Hilbert modular surfaces associated to small discriminant fields have Zariski-dense sets of points over solvable extensions (property S)?
• RQ3 Under what local conditions on Galois representations at 3 and 5 is modularity over 𝔽₉ achievable?
• RQ4 Can the modularity of Hilbert-Blumenthal abelian surfaces with good ordinary reduction at 3 and 5 be established via Galois representation methods?
• RQ5 To what extent do the techniques relying on solvability of GL₂(𝔽ₚ) and property S generalize beyond small finite fields?

Key findings

• The paper proves that an odd Galois representation ρ̄: Gal(ℚ̄/ℚ) → GL₂(𝔽₉) is modular if its restriction to D₃ is upper-triangular with characters satisfying a specific inertia condition and if the image of I₅ lies in SL₂(𝔽₉) with odd order in PSL₂(𝔽₉).
• The modularity of the representation is established by realizing it as the 3-torsion of a Hilbert-Blumenthal abelian surface over a solvable extension F/ℚ, which is itself modular.
• The key geometric insight is that certain twisted Hilbert modular varieties have Zariski-dense sets of points over solvable extensions, a property the paper calls 'property S'.
• As a corollary, every Hilbert-Blumenthal abelian surface A/ℚ with good ordinary reduction at 3 and 5 is a quotient of J₀(N) for some N, hence modular.
• The proof relies on a cyclic descent argument applied to 3-adic Galois representations, using the fact that the 3-adic representation lifts to a modular one via Theorem 3.1 and the modularity lifting theorems.
• The result provides evidence for Serre’s conjecture in the non-solvable case, though the authors caution that such techniques may not generalize to larger finite fields due to the rarity of property S in higher-genus varieties.