[Paper Review] Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms

A. O. L. Atkin, Wen-Ching Winnie Li+2 more|arXiv (Cornell University)|May 22, 2010

Advanced Algebra and Geometry30 references4 citations

TL;DR

This paper establishes the automorphy of 4-dimensional $ρ_\ell$ Galois representations associated with noncongruence cusp forms of weight $\kappa > 2$ that admit quaternion multiplication over a biquadratic extension of $\mathbb{Q}$. Using modern modularity techniques, including Serre's conjecture and Tate's vanishing theorem, it proves that these representations arise from automorphic forms on $\mathrm{GL}_4(\mathbb{A}_\mathbb{Q})$, and verifies the Atkin-Swinnerton-Dyer congruences linking coefficients of noncongruence and congruence modular forms.

ABSTRACT

In this paper we study the compatible family of degree-4 Scholl representations $ρ_{\ell}$ associated with a space $S$ of weight $κ> 2$ noncongruence cusp forms satisfying Quaternion Multiplications over a biquadratic field $K$. It is shown that when either $K$ is totally real or $κ$ is odd, $ρ_\ell$ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $GL_4(\mathbb Q)$. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in $S$ and those of certain automorphic forms via the three-term Atkin and Swinnerton-Dyer congruences.

Motivation & Objective

• To establish the automorphy of degree-4 Scholl representations associated with noncongruence cusp forms that possess quaternion multiplication (QM) over a biquadratic extension of $\mathbb{Q}$.
• To demonstrate that these representations arise from automorphic forms on $\mathrm{GL}_4(\mathbb{A}_\mathbb{Q})$ via tensor product constructions and modularity lifting techniques.
• To verify the three-term Atkin-Swinnerton-Dyer congruences between Fourier coefficients of noncongruence cusp forms and those of associated congruence modular forms.
• To construct explicit examples of such automorphic lifts, including a new case with QM over $\mathbb{Q}(\sqrt{2},\sqrt{3})$.
• To provide a concrete Hecke eigenform $f$ in $S_3(\Gamma_0(576), (-6/\cdot))$ whose coefficients reflect the structure of the Galois representation via $\rho_\ell^\text{new} \cong \eta \otimes \mathrm{Ind}_{G_{\mathbb{Q}(\sqrt{-2})}}^G \chi_{-2}^{-1}$.

Proposed method

• Leverages the recently proven Serre's conjecture over $\mathbb{Q}$ to establish modularity of 2-dimensional $\ell$-adic representations arising from geometry.
• Uses Tate's vanishing theorem $H^2(G, \mathbb{C}^\times) = 0$ to lift a projective representation $\tilde{\gamma}$ with finite image to an ordinary representation $\gamma$ of $G$.
• Constructs the full 4-dimensional representation $\rho_\ell$ as a tensor product $\eta \otimes \gamma$, where $\eta$ is modular and $\gamma$ is induced from a finite character on a quadratic subfield of the biquadratic field $K$.
• Applies the result of Ramakrishnan to show that the tensor product corresponds to an automorphic representation of $\mathrm{GL}_4(\mathbb{A}_\mathbb{Q})$.
• Employs the Faltings-Serre modularity technique in a generalized setting, adapted to representations with quaternion multiplication, to bypass ineffective Hecke operators on noncongruence forms.
• Uses explicit Hecke operators and normalizers of $\Gamma^1(6)$ to define the quaternion multiplication operators $J_{-2}, J_{-3}$, which act over $\mathbb{Q}(\sqrt{-2}, \sqrt{-3})$ and generate the quaternion algebra.

Experimental results

Research questions

• RQ1 Can 4-dimensional Scholl representations with quaternion multiplication over a biquadratic field be shown to be automorphic?
• RQ2 Do the Atkin-Swinnerton-Dyer congruences hold for noncongruence cusp forms in spaces with quaternion multiplication?
• RQ3 Can the Galois representation associated with such noncongruence forms be realized as a tensor product of two 2-dimensional modular representations?
• RQ4 Is there a concrete Hecke eigenform in a congruence space that realizes the automorphic lift of the Scholl representation?
• RQ5 How does the structure of the quaternion multiplication operators $J_{-2}, J_{-3}$ relate to the Galois representation and its decomposition?

Key findings

• The 4-dimensional Scholl representation $\rho_\ell^\text{new}$ associated with the space $\langle F_1, F_5 \rangle$ of noncongruence cusp forms of weight 3 is automorphic, corresponding to an automorphic representation of $\mathrm{GL}_4(\mathbb{A}_\mathbb{Q})$.
• The representation $\rho_\ell^\text{new}$ is isomorphic to $\eta \otimes \mathrm{Ind}_{G_{\mathbb{Q}(\sqrt{-2})}}^G \chi_{-2}^{-1}$, where $\eta$ arises from a weight 3 newform on a congruence subgroup.
• The Atkin-Swinnerton-Dyer congruences are verified numerically for the space $\langle F_1, F_5 \rangle$, with characteristic polynomials of Frobenius factoring into quadratic terms over $\mathbb{Q}(\sqrt{-6})$, $\mathbb{Q}(\sqrt{-3})$, etc.
• A Hecke eigenform $f(z)$ in $S_3(\Gamma_0(576), (-6/\cdot))$ is explicitly constructed as a linear combination of eta-products with coefficients involving $i = \sqrt{-1}, j = \sqrt{2}, k = \sqrt{3}$, and its Fourier coefficients match those of the Galois representation.
• The representation space over $\mathbb{Q}_\ell \otimes \mathbb{Q}(\sqrt{2}, \sqrt{3})$ carries quaternion multiplication over $\mathbb{Q}(\sqrt{-2}, \sqrt{-3})$, confirming the algebraic structure of the endomorphism algebra.
• For primes $p = 5, 7, 11, 13, 17, 19, 23, 29$, the characteristic polynomials of $\mathrm{Frob}_p$ factor into quadratics with coefficients in $\mathbb{Q}(\sqrt{-6}), \mathbb{Q}(\sqrt{-3})$, etc., supporting the tensor product structure and the ASD congruences.