Modular Forms on Noncongruence Subgroups and Atkin–Swinnerton-Dyer Relations

Fang, Liqun; Hoffman, J. William; Linowitz, Benjamin; Rupinski, Andrew; Verrill, Helena

doi:10.1080/10586458.2010.10129064

Cited by 5 publications

(4 citation statements)

References 15 publications

(10 reference statements)

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“…This observation inspired the folklore "Unbounded denominators conjecture" saying that having Fourier coefficients with unbounded denominators precisely characterizes modular functions and forms for noncongruence subgroups. This conjecture has been proven in special cases, see, for example [15,34,35,37].…”

Section: Modular Forms On Noncongruence Subgroupsmentioning

confidence: 84%

Noncongruence subgroups and Maass waveforms

Strömberg

2019

Journal of Number Theory

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The main topic of the paper is spectral theory for noncongruence subgroups of the modular group. We have studied a selection of the main conjectures in the area: the Roelcke-Selberg and Phillips-Sarnak conjectures and Selberg's conjecture on exceptional eigenvalues. The first two concern the existence and nonexistence of an infinite discrete spectrum for certain types of Fuchsian groups and last states that there are no exceptional eigenvalues for congruence subgroups, or in other words, there is a specific spectral gap in the cuspidal spectrum.Our main theoretical result states that if the corresponding surface has a reflectional symmetry which preserves the cusps then the Laplacian on this surface has an infinite number of "new" discrete eigenvalues. We define old and new spaces of Maass cusp forms for noncongruence subgroups in a way that provides a natural generalization of the usual definition from congruence subgroups and give a method for determining the structure of the old space algorithmically.In addition to the theoretical result we also present computational data, including a table of subgroups of the modular group and eigenvalues of Maass forms for noncongruence subgroups. We also present, for the first time, numerical examples of both exceptional and (non-trivial) residual eigenvalues. To be able to (even heuristically) cerify lists of computed eigenvalues we also proved an explicit average Weyl's law in this setting.

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Section: Modular Forms On Noncongruence Subgroupsmentioning

confidence: 84%

Noncongruence subgroups and Maass waveforms

Strömberg

2019

Journal of Number Theory

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“…GO 4 -type) of d = 2 and odd κ ≥ 3 obtained by Liu and Yu in [LY]. Other known automorphy results are also for low degree Scholl representations including [LLY,ALL,Lon,FHLRV,HLV,AL 3 ]. Applying the results in this paper, we prove the potential automorphy for an infinite family of explicitly constructed Scholl representations with unbounded degrees, extending the automorphy results shown in [LLY, ALL, Lon] alluded above.…”

Section: Introductionmentioning

confidence: 78%

Potentially 𝐺𝐿₂-type Galois representations associated to noncongruence modular forms

Liu

Лонг

2018

Trans. Amer. Math. Soc.

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In this paper, we consider representations of the absolute Galois group Gal(Q/Q) attached to modular forms for noncongruence subgroups of SL2(Z). When the underlying modular curves have a model over Q, these representations are constructed by Scholl in [Sch1] and are referred to as Scholl representations, which form a large class of motivic Galois representations. In particular, by a result of Belyi, Scholl representations include the Galois actions on the Jacobian varieties of algebraic curves defined over Q. As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. Using recent developments on automorphy lifting theorem, we obtain various automorphy and potential automorphy results for potentially GL2-type Galois representations associated to noncongruence modular forms. Our results are applied to various kinds of examples. Especially, we obtain potential automorphy results for Galois representations attached to an infinite family of spaces of weight 3 noncongruence cusp forms of arbitrarily large dimensions.2000 Mathematics Subject Classification. 11F11,11F80.

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“…As observed above, for almost all primes p, V p,±,u contain a nonzero f ±,u = n≥1 a ±,u (n)q n/µ ∈ S with p-adically integral Fourier coefficients. They form a basis of S. It follows from [Sch85,FHL+] that the ASD congruences hold, namely,…”

Section: Degree 4 Scholl Representations With Qm and Atkin-swinnertonmentioning

confidence: 99%

Galois representations with quaternion multiplication associated to noncongruence modular forms

Atkin¹,

Liu

et al. 2013

Trans. Amer. Math. Soc.

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In this paper we study the compatible family of degree-4 Scholl representations ρ ℓ associated with a space S of weight κ > 2 noncongruence cusp forms satisfying Quaternion Multiplication over a biquadratic extension of Q. It is shown that ρ ℓ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for GL4 over 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.

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Modular Forms on Noncongruence Subgroups and Atkin–Swinnerton-Dyer Relations

Abstract: Abstract. We give new examples of noncongruence subgroups Γ ⊂ SL2(Z) whose space of weight 3 cusp forms S3(Γ) admits a basis satisfying the Atkin-Swinnerton-Dyer congruence relations with respect to a weight 3 newform for a certain congruence subgroup.

Cited by 5 publications

References 15 publications

Noncongruence subgroups and Maass waveforms

Noncongruence subgroups and Maass waveforms

Potentially 𝐺𝐿₂-type Galois representations associated to noncongruence modular forms

Galois representations with quaternion multiplication associated to noncongruence modular forms

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