Ramanujan’s Theories of Elliptic Functions to Alternative Bases

Berndt, Bruce C.

doi:10.1007/978-1-4612-1624-7_3

Cited by 66 publications

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“…23-39], Ramanujan records several elegant series for 1/π and asserts, "There are corresponding theories in which q is replaced by one or other of the functions" q r := q r (x) := exp −π csc(π/r) 2 [11], who gave these theories the appellation, the theories of signature r (r = 3, 4, 6). An account of this work may also be found in Berndt's book [9,Chap.…”

Section: Ramanujan's Alternative Theories Of Elliptic Functionsmentioning

confidence: 99%

“…This theorem was first proved in print by Berndt, Bhargava, and Garvan [11], [9, p. 99], with (2.11) being a necessary ingredient in their proof. The analogues of (2.7) in the theory of signature 3 are given by [9, p. 109, Lemma 5.1]…”

Section: Ramanujan's Alternative Theories Of Elliptic Functionsmentioning

confidence: 99%

“…In the theory of signature 4, or in the quartic theory, Berndt, Bhargava, and Garvan [11], [9, p. 146, eq. (9.7)] established a "transfer" principle of Ramanujan by which formulas in the theory of signature 4 can be derived from formulas in the classical theory.…”

Section: Ramanujan's Alternative Theories Of Elliptic Functionsmentioning

confidence: 99%

“…23-39] in 1914, but which only germinated in his notebooks [50]. These flowers were not seen by anyone but Ramanujan until Berndt, S. Bhargava, and F. G. Garvan [11] cultivated them in 1995. Since then, other gardeners have found further flowers, but Ramanujan has left a huge plot of land for still undiscovered related species to grow and be nurtured.…”

Section: Introductionmentioning

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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Section: Ramanujan's Alternative Theories Of Elliptic Functionsmentioning

confidence: 99%

Section: Ramanujan's Alternative Theories Of Elliptic Functionsmentioning

confidence: 99%

Section: Ramanujan's Alternative Theories Of Elliptic Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…Recall from [20] that for a, b ∈ N the series means that the summation does not include (m, n) = (0, 0).) Then Bloch introduced the regulator function R E (e 2πiu ) = Im(τ ) 2 π K 2,1 (τ ; u). It can be shown that Re(R E ) = D E , and that − Im(R E ) is the real-valued function given by…”

Section: Introductionmentioning

confidence: 99%

The elliptic trilogarithm and Mahler measures of K3 surfaces

Samart

2015

Forum Mathematicum

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Abstract. The aim of this paper is to derive explicitly a connection between the Zagier elliptic trilogarithm and Mahler measures of certain families of three-variable polynomials defining K3 surfaces. In addition, we prove some linear relations satisfied by the elliptic trilogarithm evaluated at torsion points on elliptic curves. The latter result can be viewed as a higher dimensional analogue of exotic relations of the elliptic dilogarithm. IntroductionFor m ∈ N, the classical m th polylogarithm function is defined byOne can obtain a multivalued function on C−{0, 1} from Li m (z) by extending it analytically. There are many versions of higher polylogarithms in the literature, including the followingwhereand B k is the k th Bernoulli number (B 0 = 1, B 1 = −1/2, B 2 = 1/6, B 3 = 0, . . .). It can be extended to a continuous function on P 1 (C) by the functional equationThe dilogarithm function and the function D(z) have been studied extensively and are known to satisfy several interesting properties. They are also found to have fruitful applications in algebraic K-theory and other related areas (see, for example, [23] where |z| ≤ 1. Now let us consider an "averaged" version of the function D(z), which will be described below. Let E be an elliptic curve defined over C. Then there exist τ ∈ H := {τ ∈ C | Im(τ ) > 0} and isomorphismswhere Λ = Z + Zτ , and ℘ Λ denotes the Weierstrass ℘-function. Using the transformations above, Bloch [4] defined the elliptic dilogarithmwhere q = e 2πiτ and x = e 2πiu is the image of P in C × /q Z . ( . It can be shown that Re(R E ) = D E , and that − Im(R E ) is the real-valued function given by3 log 2 |q|B 3 log |x| log |q| , where J(x) = log |x| log |1 − x|, and B n (X) denotes the n th Bernoulli polynomial. Similar to D E , the function J E is well-defined and invariant under x → qx. We can also extend R E , D E , and J E by linearity to the group of divisors on E(C).Recall that for any nonzero Laurent polynomial P ∈ C[X ±1 1 , . . . , X ±1 n ], the Mahler measure of P is defined byLet us denote m 2 (t) := 2m(x + x −1 + y + y −1 + t 1/2 ), m 3 (t) := 3m(x 3 + y 3 + 1 − t 1/3 xy). THE ELLIPTIC TRILOGARITHM AND MAHLER MEASURES OF K3 SURFACES 3Boyd [6] and Rodriguez Villegas [16] verified numerically that for many values of t ∈ Z, m 2 (t) and m 3 (t) appear to be of the formwhere E is the elliptic curve over Q defined by the zero locus of x + x −1 + y + y −1 + t 1/2 or x 3 + y 3 + 1 − t 1/3 xy, depending on j, and A ∼ Q B means B = cA for some c ∈ Q. (Note that by the functional equation of L(E, s), one has |L ′ (E, 0)| = (2π) −2 N E L(E, 2), where N E is the conductor of E.) However, most of these conjectured formulas have remained unproved. Rodriguez Villegas [16] proved some of these formulas when the corresponding elliptic curves have complex multiplication by showing first that m 2 (t) and m 3 (t) can be expressed in terms of Eisenstein-Kronecker series. Afterward, Lalín and Rogers [10] verified that for every t ∈ C, if E is the elliptic curve defined by x + x −1 + y + y −1 + ...

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Some remarks on superspecial and ordinary curves of low genus

Hasegawa

2012

Mathematische Nachrichten

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In the first half of this paper, we investigate a superspecialty criterion with respect to $\lambda$ for the curve which is a generalization of a theorem for an elliptic curve by Max Deuring (1941). In the last half, we study a criterion with respect to the characteristic $p>0$ of the ground field such that the Picard curve is superspecial or ordinary. In the final section (appendix), we consider another viewpoint of the first result by using a fact that the first curve has a completely decomposable Jacobian.

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Ramanujan’s Theories of Elliptic Functions to Alternative Bases

Cited by 66 publications

References 2 publications

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

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

The elliptic trilogarithm and Mahler measures of K3 surfaces

Some remarks on superspecial and ordinary curves of low genus

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