Modular Forms

Cohen, Henrí; Strömberg, Fredrik

doi:10.1090/gsm/179

Cited by 101 publications

(48 citation statements)

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“…We briefly recall some definitions and key facts about modular forms on the congruence subgroup Γ 0 (N ) as discussed in [CS17]. Let M k (Γ 0 (N ), χ) be the space holomorphic modular forms on Γ 0 (N ) of weight k and character χ.…”

Section: The Main Resultsmentioning

confidence: 99%

“…where the sum is over all pairs (c, d) ∈ Z × Z with (c, d) = 0, with the additional condition N 1 |c. If χ 1 is primitive then the series G k (χ 1 , χ 2 ) belongs to the space M k (Γ 0 (N ), χ) and has the Fourier expansion (see [CS17,Corollary 8.5.5])…”

Section: The Main Resultsmentioning

confidence: 99%

“…where | k is the usual slash operator of weight k acting on the space M k (Γ 0 (N ), χ). The Fricke involution of G k (χ 1 , χ 2 ) is given by (see [CS17,Proposition 8.5.3])…”

Section: The Main Resultsmentioning

confidence: 99%

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Eisenstein series whose Fourier coefficients are zeta functions of binary Hermitian forms

Flórez

Karabulut

2020

Proc. Amer. Math. Soc.

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Section: The Main Resultsmentioning

confidence: 99%

Section: The Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Eisenstein series whose Fourier coefficients are zeta functions of binary Hermitian forms

Flórez

Karabulut

2020

Proc. Amer. Math. Soc.

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“…For all of these modular forms one may construct now a finite dimensional basis representation [43] through Lambert-Eisenstein series [46,47] and products thereof [48]. They are of the form…”

Section: The Functional Structure Of Feynman Integrals In the Single mentioning

confidence: 99%

“…Due to the multiplication relation of the elliptic polylogarithm all of these terms can be represented in the form of elliptic polylogarithms, which are formal power series in q. However, there are still the factors 1/η k (τ) for which the closed form representation of its formal power series in q is yet unknown, unlike its infinite product representation [48]. This means that for the case that all k = 0, the integral-relation of the elliptic polylogarithms will yield elliptic polylogarithms again.…”

Section: The Functional Structure Of Feynman Integrals In the Single mentioning

confidence: 99%

Iterative and Iterative-Noniterative Integral Solutions in 3-Loop Massive QCD Calculations

Bluemlein¹

2018

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

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Various of the single scale quantities in massless and massive QCD up to 3-loop order can be expressed by iterative integrals over certain classes of alphabets, from the harmonic polylogarithms to root-valued alphabets. Examples are the anomalous dimensions to 3-loop order, the massless Wilson coefficients and also different massive operator matrix elements. Starting at 3-loop order, however, also other letters appear in the case of massive operator matrix elements, the so called iterative non-iterative integrals, which are related to solutions based on complete elliptic integrals or any other special function with an integral representation that is definite but not a Volterratype integral. After outlining the formalism leading to iterative non-iterative integrals,we present examples for both of these cases with the 3-loop anomalous dimension γ (2) qg and the structure of the principle solution in the iterative non-interative case of the 3-loop QCD corrections to the ρ-parameter.

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Iterative Non-iterative Integrals in Quantum Field Theory

Blümlein

2019

Texts &Amp; Monographs in Symbolic Computation

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Single scale Feynman integrals in quantum field theories obey difference or differential equations with respect to their discrete parameter N or continuous parameter x. The analysis of these equations reveals to which order they factorize, which can be different in both cases. The simplest systems are the ones which factorize to first order. For them complete solution algorithms exist. The next interesting level is formed by those cases in which also irreducible second order systems emerge. We give a survey on the latter case. The solutions can be obtained as general 2 F 1 solutions. The corresponding solutions of the associated inhomogeneous differential equations form so-called iterative non-iterative integrals. There are known conditions under which one may represent the solutions by complete elliptic integrals. In this case one may find representations in terms of meromorphic modular functions, out of which special cases allow representations in the framework of elliptic polylogarithms with generalized parameters. These are in general weighted by a power of 1/η(τ), where η(τ) is Dedekind's η-function. Single scale elliptic solutions emerge in the ρ-parameter, which we use as an illustrative example. They also occur in the 3-loop QCD corrections to massive operator matrix elements and the massive 3-loop form factors.

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Modular Forms

Cited by 101 publications

References 0 publications

Eisenstein series whose Fourier coefficients are zeta functions of binary Hermitian forms

Eisenstein series whose Fourier coefficients are zeta functions of binary Hermitian forms

Iterative and Iterative-Noniterative Integral Solutions in 3-Loop Massive QCD Calculations

Iterative Non-iterative Integrals in Quantum Field Theory

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