Special Hypergeometric Motives and Their <i>L</i>-Functions: Asai Recognition

Dembélé, Lassina; Panchishkin, Alexei A.; Voight, John; Zudilin, Wadim

doi:10.1080/10586458.2020.1737990

Cited by 10 publications

(8 citation statements)

References 33 publications

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“…which is (c). Similar recent discussions relating L-values of hypergeometric motives to L-functions of modular forms or Hilbert modular forms include Osbrun and Straub [46], and Dembélé et al [19].…”

Section: An Examplementioning

confidence: 54%

“…Based on this work, McCarthy and Papanikolas relate hypergeometric sums with Siegel eigenforms in [43]. Dembélé, Panchishkin, Voight, and Zudilin in [19] provided numerical evidence relating special values of certain hypergeometric L-functions and Asai L-functions of Hilbert modular forms. In [27], the authors interpreted hypergeometric character sums in a manner parallel to the development of classical hypergeometric functions.…”

Section: Hypergeometric Datamentioning

confidence: 94%

“…In recent years, rapid progress has been made in hypergeometric character sums under the framework of hypergeometric motives. It gives a unifying perspective to many hypergeometric structures and identities, see [8,12,18,19,[27][28][29]32,38,42,49]. Classical hypergeometric formulas such as Clausen's formula have a finite field analogue by Evans and Greene [22].…”

Section: Hypergeometric Datamentioning

confidence: 99%

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A Whipple $$_7F_6$$ Formula Revisited

Лонг

2022

La Matematica

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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

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Section: An Examplementioning

confidence: 54%

Section: Hypergeometric Datamentioning

confidence: 94%

Section: Hypergeometric Datamentioning

confidence: 99%

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A Whipple $$_7F_6$$ Formula Revisited

Лонг

2022

La Matematica

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“…Formulas #1 and #2 of Table 4 were discovered by B. Gourevitch and the author respectively, and the formula #3 of Table 4 was discovered by Y. Zhao [14]. More precisely he gives a reverse convergent version of it for π 4 . Formula #1 of Table 5 was discovered by J. Cullen, and the formulas #2 and #3 by Y. Zhao [14].…”

Section: Introductionmentioning

confidence: 99%

Bilateral Ramanujan-like series for $1/π^k$ and their congruences

Guillera¹

2019

Preprint

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“…For brevity, we call these hypergeometric sheaves. Their Frobenius trace functions, known as finite hypergeometric functions, were discovered independently by J. Greene [Gre87], and have found applications in describing motives of elliptic curves and other varieties [Ono98,BCM15,DPVZ19]. 1.2.3.…”

mentioning

confidence: 99%

Geometric Langlands for hypergeometric sheaves

Kamgarpour

2020

Preprint

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Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler-Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. In this paper, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems, thus confirming a central conjecture of the geometric Langlands program for hypergeometrics. The key new concept is the notion of hypergeometric automorphic data. We prove that this automorphic data is generically rigid (in the sense of Zhiwei Yun) and identify the resulting Hecke eigenvalue with hypergeometric sheaves. The definition of hypergeometric automorphic data in the tame case involves the mirabolic subgroup, while in the wild case, semistable (but not necessarily stable) vectors coming from principal gradings intervene.

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Special Hypergeometric Motives and Their L-Functions: Asai Recognition

Abstract: We recognize certain special hypergeometric motives, related to and inspired by the discoveries of Ramanujan more than a century ago, as arising from Asai L-functions of Hilbert modular forms.

Cited by 10 publications

References 33 publications

A Whipple $$_7F_6$$ Formula Revisited

A Whipple $$_7F_6$$ Formula Revisited

Bilateral Ramanujan-like series for $1/π^k$ and their congruences

Geometric Langlands for hypergeometric sheaves

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