Potentially 𝐺𝐿₂-type Galois representations associated to noncongruence modular forms

Li, Wen Ching Winnie; Liu, Tong; Лонг, Линг

doi:10.1090/tran/7364

Cited by 4 publications

(4 citation statements)

References 27 publications

(12 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These properties include transformation laws, explicit evaluations, and contiguous relations. These functions have played central roles in the study of combinatorial supercongruences [1,3,36,43,46,47,51,54,55,56,57,58], Dwork hypersurfaces [9,45], Galois representations [40,41], L-functions of elliptic curves [6,10,11,25,39,44,52,60,63], hyperelliptic curves [7,8], K3 surfaces [4,19,52], Calabi-Yau threefolds [2,3,64], the Eichler-Selberg trace formula [24,25,26,27,38,48,58,59], among other topics.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Distribution of values of Gaussian hypergeometric functions

Ono¹,

Saad²,

Saikia³

2021

Preprint

View full text Add to dashboard Cite

In the 1980's, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Apéry-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the 2 F 1 functions, the limiting distribution is semicircular, whereas the distribution for the 3 F 2 functions is the more exotic Batman distribution.

show abstract

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Distribution of values of Gaussian hypergeometric functions

Ono¹,

Saad²,

Saikia³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…When λ = 1, A 1 has Picard number 20 and ρ H D, is 2dimensional. As shown in [59,Sect. 5] by the first two authors and Liu, the action of the Galois group G Q on the 2-dimensional subspace of the cohomological space It is explained in [54] by Stienstra and Beukers (see also [4,Sect.…”

Section: An Examplementioning

confidence: 93%

A Whipple $$_7F_6$$ Formula Revisited

Лонг

2022

La Matematica

Self Cite

View full text Add to dashboard Cite

A well-known formula of Whipple relates certain hypergeometric values 7 F 6 (1) and 4 F 3 (1). In this paper, we revisit this relation from the viewpoint of the underlying hypergeometric data H D, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple's formula when the hypergeometric data H D are primitive and self-dual. If the data are also defined over Q, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of -adic representations of the absolute Galois group of Q attached to H D. For specialized choices of H D, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values 7 F 6 (1) in Whipple's formula to the periods of these modular forms. KeywordsHypergeometric functions • Whipple's 7 F 6 formula • Hypergeometric character sums • Galois representations and modular forms Mathematics Subject Classification 11F11 • 33C20 • 11F80 • 11F67 • 11T24 B Ling Long

show abstract

“…Since HD 1 is defined over Q and HD 2 can be extended to Q, the representations ρ HD1, | G(N ) and ( ⊗ ρ HD2, )| G(N ) can be extended to G Q as ρ BCM HD1, and ⊗ ρ BCM HD2, respectively. Hence by (36) where σ HD1,sym, and σ HD1,alt, are both 2-dimensional. Here we use sym and alt in the subscripts in order to keep track of their origin.…”

Section: 3mentioning

confidence: 99%

A Whipple $_7F_6$ formula revisited

Li¹,

Лонг²,

Tu³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

A well-known formula of Whipple relates certain hypergeometric values 7F6(1) and 4F3(1). In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data HD, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple's formula when the hypergeometric data HD are primitive and defined over Q. In this case, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of -adic representations of the absolute Galois group of Q attached to HD. For specialized choices of HD, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values 7F6(1) in Whipple's formula to the periods of modular forms occurred.

show abstract

Potentially 𝐺𝐿₂-type Galois representations associated to noncongruence modular forms

Cited by 4 publications

References 27 publications

Distribution of values of Gaussian hypergeometric functions

Distribution of values of Gaussian hypergeometric functions

A Whipple $$_7F_6$$ Formula Revisited

A Whipple $_7F_6$ formula revisited

Contact Info

Product

Resources

About