The elliptic polylogarithm

Beilinson, Alexander; Levin, A.

doi:10.1090/pspum/055.2/1265553

Cited by 73 publications

(133 citation statements)

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“…Beilinson and Deligne [6], Huber and Wildeshaus [35]. Geometric versions of polylogarithms have been formulated (Goncharov [28,29], Cartier [12]), as well as an elliptic curve generalization of the polylogarithm (Beilinson and Levin [7]). …”

Section: Prior Workmentioning

confidence: 99%

The Lerch Zeta function III. Polylogarithms and special values

Lagarias

2016

Res Math Sci

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This paper studies algebraic and analytic structures associated with the Lerch zeta function, complex variables viewpoint taken in part II. The Lerch transcendent (s, z, c) := ∞ n=0 z n (n+c) s is obtained from the Lerch zeta function ζ (s, a, c) by the change of variable z = e 2πia . We show that it analytically continues to a maximal domain of holomorphy in three complex variables (s, z, c), as a multivalued function defined over the base manifold C × (P 1 (C) {0, 1, ∞}) × (C Z) and compute the monodromy functions describing the multivaluedness. For positive integer values s = m and c = 1 this function is closely related to the classical m-th order polylogarithm Li m (z). We study its behavior as a function of two variables (z, c) for "special values" where s = m is an integer. For m ≥ 1 we show that it is a one-parameter deformation of Li m (z), which satisfies a linear ODE, depending on c ∈ C, of order m + 1 of Fuchsian type on the Riemann sphere. We determine the associated (m + 1)-dimensional monodromy representation, which is a non-algebraic deformation of the monodromy of Li m (z).

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Section: Prior Workmentioning

confidence: 99%

The Lerch Zeta function III. Polylogarithms and special values

Lagarias

2016

Res Math Sci

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“…In the mathematical literature there exist various slightly different definitions of elliptic polylogarithms [39,[58][59][60][61][62][63][64][65]. In order to express the sunrise and the kite integral to all orders in ε we introduce the functions…”

Section: Beyond Multiple Polylogarithms: Single Scale Integralsmentioning

confidence: 99%

Differential equations for Feynman integrals beyond multiple polylogarithms

Adams¹,

Bogner²,

Chaubey³

et al. 2018

Proceedings of 13th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology) —

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“…which generalize the famous symbol on K2(X) of Beilinson and Deligne [3,14]. (For simplicity we formulate results for curves over Q.)…”

Section: Hi(x/rr(n -I))mentioning

confidence: 99%

“…The weight 3 motivic complex. By this we will refer to the complex F(X, 3) introduced in [9,10] for an arbitrary regular scheme X. If X = Spec(F), where F is an arbitrary field, it looks as follows: In [9] we have constructed homomorphisms of groups ~,~: g,,(F) | Q ~ n~r(Spec(r); 3) | Q.…”

Section: Motivicmentioning

confidence: 99%