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

Franc, Cameron; Mason, Geoffrey

doi:10.1016/j.jnt.2015.08.013

Cited by 16 publications

(11 citation statements)

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“…In [11] and [13], the authors showed explicitly how to describe vector-valued modular forms of rank two for SL 2 (Z) in terms of hypergeometric series 2 F 1 , and in [11] these series were then used to verify the extension of the UBD conjecture to vector-valued modular forms of rank two for SL 2 (Z). Shortly after, a similar result was proved [12] for some vector-valued modular forms of rank three, using generalized hypergeometric series 3 F 2 . In both of these papers, only a very conservative use of hypergeometric series was made.…”

Section: Introductionmentioning

confidence: 64%

On unbounded denominators and hypergeometric series

Franc¹,

Gannon²,

Mason³

2018

Journal of Number Theory

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We study the question of when the coefficients of a hypergeometric series are p-adically unbounded for a given rational prime p. Our first main result is a necessary and sufficient criterion (applicable to all but finitely many primes) for determining when the coefficients of a hypergeometric series with rational parameters are p-adically unbounded. This criterion is then used to show that the set of unbounded primes for a given series is, up to a finite discrepancy, a finite union of primes in arithmetic progressions. This set can be computed explicitly. We characterize when the density of the set of unbounded primes is 0, and when it is 1. Finally, we discuss the connection between this work and the unbounded denominators conjecture concerning Fourier coefficients of modular forms.

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Section: Introductionmentioning

confidence: 64%

On unbounded denominators and hypergeometric series

Franc¹,

Gannon²,

Mason³

2018

Journal of Number Theory

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“…Γ(N ) then all components are. Indeed, in [70] significant effort was expended in proving this statement for a narrow class of d = 3 representations. The integrality conjecture is usually stated in the math literature with the assumption that all components of v(q) have bounded denominator.…”

Section: Uniqueness Of T > Cmentioning

confidence: 99%

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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“…In weight zero, the case of interest to us thanks to Zhu's Theorem, such modular linear differential equations are equivalent to ordinary differential equations on the modular curve, which is the perspective of [7] and [40]. In low weights it is easy to use the known structure of M (ρ, L) to determine and solve these equations explicitly: see for example [28,29,30], and Sections 7, 8.1 and 8.2 below.…”

Section: Theorem 32 (Free Module Theorem) Thementioning

confidence: 99%