Densities of bounded primes for hypergeometric series with rational parameters

Franc, Cameron; Gill, Brandon; Goertzen, Jason; Pas, Jarrod; Tu, Frankie

doi:10.48550/arxiv.1803.10607

Cited by 1 publication

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“…Our analysis of the Fourier coefficients of a basis of solutions to the relevant modular linear differential equation boils down to understanding the number-theoretic properties of the coefficients of the hypergeometric series study the arithmetic of the coefficients of the hypergeometric series 2 F 1 at arguments in a quadratic field. The arithmetic of the coefficients of 2 F 1 with rational parameters are examined in the works of Dwork [8], [9], Christol [6], Franc, Gannon, and Mason [10], and Franc, Gill, Goertzen, Pas, and Tu [11].…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

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

Gottesman

2019

Int. J. Number Theory

View full text Add to dashboard Cite

show abstract

Densities of bounded primes for hypergeometric series with rational parameters

Cited by 1 publication

References 8 publications

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

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

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