Fourier coefficients of three-dimensional vector-valued modular forms

Marks, Christopher

doi:10.4310/cntp.2015.v9.n2.a5

Cited by 11 publications

(18 citation statements)

References 16 publications

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“…Chris Marks verified this result earlier in [14] using a different but related method, in all but finitely many cases. The exceptional cases (there are probably many) were not made explicit in [14].…”

Section: Vector-valued Modular Forms Of Minimal Weightmentioning

confidence: 54%

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Three-dimensional imprimitive representations of the modular group and their associated modular forms

Franc¹,

Mason²

2016

Journal of Number Theory

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ABSTRACT. This paper uses previous results of the authors [6] to study certain noncongruence modular forms. We prove that these forms have unbounded denominators, and in certain cases we verify congruences of Atkin-Swinnerton-Dyer type [2] satisfied by the Fourier coefficients of these forms. Our results rest on group-theoretic facts about the modular group Γ, a detailed study of imprimitive three-dimensional representations of Γ, and the theory of their associated vector-valued modular forms. For the proof of the congruences we also make essential use of a result of Katz [7].

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Section: Vector-valued Modular Forms Of Minimal Weightmentioning

confidence: 54%

“…However, in that setting one does not encounter noncongruence modular forms. Chris Marks first studied the 3-dimensional case in [14], and obtained results about unbounded denominators for vector-valued modular forms of large enough weight.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional imprimitive representations of the modular group and their associated modular forms

Franc¹,

Mason²

2016

Journal of Number Theory

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“…In this section, we will prove the analogous result for vector-valued modular forms of any weight in Theorem 5.35. Our method of proof follows very closely Chris Marks' paper [28]. In [28], Marks proved a similar result for three-dimensional vector-valued modular forms with respect to certain three-dimensional representations of Γ.…”

Section: 3mentioning

confidence: 63%

“…Mason [30] proved that for all but a finite number of two-dimensional irreducible representations ρ of Γ, every vector-valued modular form for ρ whose Fourier coefficients are algebraic numbers has the property that the denominators of the Fourier coefficients of each of its component functions is unbounded. Marks [28] has proven the analogous result for all but a finite number of three-dimensional representations ρ of Γ. Franc and Mason [12] solved a modular linear differential equation to prove the same result for all two-dimensional representations ρ such that ker ρ is noncongruence.…”

Section: Introductionmentioning

confidence: 75%

“…Our method of proof follows very closely Chris Marks' paper [28]. In [28], Marks proved a similar result for three-dimensional vector-valued modular forms with respect to certain three-dimensional representations of Γ. Our proof that the denominators of the Fourier coefficients of each of the component functions of F ′ are unbounded uses entirely different ideas than those in [28].…”

Section: 3mentioning

confidence: 81%

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The arithmetic of vector-valued modular forms on Γ0(2)

Gottesman

2019

Int. J. Number Theory

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Let ρ denote an irreducible two-dimensional representation of Γ 0 (2). The collection of vector-valued modular forms for ρ, which we denote by M (ρ), form a graded and free module of rank two over the ring of modular forms on Γ 0 (2), which we denote by M (Γ 0 (2)). For a certain class of ρ, we prove that if Z is any vector-valued modular form for ρ whose component functions have algebraic Fourier coefficients then the sequence of the denominators of the Fourier coefficients of both component functions of Z is unbounded. Our methods involve computing an explicit basis for M (ρ) as a M (Γ 0 (2))module. We give formulas for the component functions of a minimal weight vector-valued form for ρ in terms of the Gaussian hypergeometric series 2 F 1 , a Hauptmodul of Γ 0 (2), and the Dedekind η-function.

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Discreteness and integrality in Conformal Field Theory

Kaidi

Perlmutter

2021

J. High Energ. Phys.

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Various observables in compact CFTs are required to obey positivity, discreteness, and integrality. Positivity forms the crux of the conformal bootstrap, but understanding of the abstract implications of discreteness and integrality for the space of CFTs is lacking. We systematically study these constraints in two-dimensional, non-holomorphic CFTs, making use of two main mathematical results. First, we prove a theorem constraining the behavior near the cusp of integral, vector-valued modular functions. Second, we explicitly construct non-factorizable, non-holomorphic cuspidal functions satisfying discreteness and integrality, and prove the non-existence of such functions once positivity is added. Application of these results yields several bootstrap-type bounds on OPE data of both rational and irrational CFTs, including some powerful bounds for theories with conformal manifolds, as well as insights into questions of spectral determinacy. We prove that in rational CFT, the spectrum of operator twists $$ t\ge \frac{c}{12} $$ t ≥ c 12 is uniquely determined by its complement. Likewise, we argue that in generic CFTs, the spectrum of operator dimensions $$ \Delta >\frac{c-1}{12} $$ Δ > c − 1 12 is uniquely determined by its complement, absent fine-tuning in a sense we articulate. Finally, we discuss implications for black hole physics and the (non-)uniqueness of a possible ensemble interpretation of AdS3 gravity.

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Fourier coefficients of three-dimensional vector-valued modular forms

Cited by 11 publications

References 16 publications

Three-dimensional imprimitive representations of the modular group and their associated modular forms

Three-dimensional imprimitive representations of the modular group and their associated modular forms

The arithmetic of vector-valued modular forms on Γ0(2)

Discreteness and integrality in Conformal Field Theory

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