A cubic counterpart of Jacobi’s identity and the AGM

Borwein, Jonathan M.; Borwein, Peter

doi:10.1090/s0002-9947-1991-1010408-0

Cited by 95 publications

(91 citation statements)

References 3 publications

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“…Another aspect of our approach is that we find a deep connection between the allorders P/NP relations (1.5) and (1.6), and Ramanujan's theory of elliptic functions with respect to alternative bases [93][94][95][96][97][98][99][100][101][102], and extensions to modular functions [103]. These number theoretic functions are also associated with topological c = 3 Landau-Ginzburg models [104,105] and certain superconformal quantum field theories [106].…”

Section: Jhep05(2017)087mentioning

confidence: 99%

“…This modular structure can be formulated within Ramanujan's theory of elliptic functions in alternative bases [93][94][95][96][97][98][99][100][101][102][103] and it plays an important role in the corresponding quantum theories, as we discuss below in section 4. But here we first review some number theoretic results concerning the classical theories.…”

Section: Ramanujan's Theory Of Elliptic Functions In Alternative Basementioning

confidence: 99%

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Quantum geometry of resurgent perturbative/nonperturbative relations

Başar

Dunne

Ünsal

2017

J. High Energ. Phys.

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For a wide variety of quantum potentials, including the textbook 'instanton' examples of the periodic cosine and symmetric double-well potentials, the perturbative data coming from fluctuations about the vacuum saddle encodes all non-perturbative data in all higher non-perturbative sectors. Here we unify these examples in geometric terms, arguing that the all-orders quantum action determines the all-orders quantum dual action for quantum spectral problems associated with a classical genus one elliptic curve. Furthermore, for a special class of genus one potentials this relation is particularly simple: this class includes the cubic oscillator, symmetric double-well, symmetric degenerate triple-well, and periodic cosine potential. These are related to the Chebyshev potentials, which are in turn related to certain N = 2 supersymmetric quantum field theories, to mirror maps for hypersurfaces in projective spaces, and also to topological c = 3 Landau-Ginzburg models and 'special geometry'. These systems inherit a natural modular structure corresponding to Ramanujan's theory of elliptic functions in alternative bases, which is especially important for the quantization. Insights from supersymmetric quantum field theory suggest similar structures for more complicated potentials, corresponding to higher genus. Our approach is very elementary, using basic classical geometry combined with all-orders WKB.

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Section: Jhep05(2017)087mentioning

confidence: 99%

Section: Ramanujan's Theory Of Elliptic Functions In Alternative Basementioning

confidence: 99%

Quantum geometry of resurgent perturbative/nonperturbative relations

Başar

Dunne

Ünsal

2017

J. High Energ. Phys.

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“…The functions defined in (2.8)-(2.10) are the "cubic" theta-functions, first introduced by the Borweins [22], who proved that…”

Section: Ramanujan's Alternative Theories Of Elliptic Functionsmentioning

confidence: 99%

“…is the key to proving the inversion formula (2.5), while in the cubic theory, Ramanujan's cubic transformation [22], [11], [9, p. 97, Cor. 2.4]…”

Section: Ramanujan's Alternative Theories Of Elliptic Functionsmentioning

confidence: 99%

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Flowers Which We Cannot Yet See Growing in Ramanujan’s Garden of Hypergeometric Series, Elliptic Functions, and q’S

Berndt

2001

Special Functions 2000: Current Perspective and Future Directions

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Some results about zero‐cycles on abelian and semi‐abelian varieties

Gazaki

2019

Mathematische Nachrichten

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In this short note we extend some results obtained in [7]. First, we prove that for an abelian variety A with good ordinary reduction over a finite extension of Qp with p an odd prime, the Albanese kernel of A is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi‐abelian variety G over a perfect field k, we construct a decreasing integral filtration false{Frfalse}r≥0 of Suslin's singular homology group, H0singfalse(Gfalse), such that the successive quotients are isomorphic to a certain Somekawa K‐group.

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A cubic counterpart of Jacobi’s identity and the AGM

Abstract: Abstract. We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is al + anh + bl\

Cited by 95 publications

References 3 publications

Quantum geometry of resurgent perturbative/nonperturbative relations

Quantum geometry of resurgent perturbative/nonperturbative relations

Flowers Which We Cannot Yet See Growing in Ramanujan’s Garden of Hypergeometric Series, Elliptic Functions, and q’S

Some results about zero‐cycles on abelian and semi‐abelian varieties

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