Search results for "Modular form"
showing 10 items of 11 documents
Modular quintics inℙ4
2003
We will examine the arithmetic of some of the members of a pencil of symmetric quintics in projective 4-space. We will give evidence for the modularity of some of the exceptional members (even the non-rigid ones) and give a proof in one rigid case. (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
A note on the admissibility of modular function spaces
2017
Abstract In this paper we prove the admissibility of modular function spaces E ρ considered and defined by Kozlowski in [17] . As an application we get that any compact and continuous mapping T : E ρ → E ρ has a fixed point. Moreover, we prove that the same holds true for any retract of E ρ .
Numerical evaluation of iterated integrals related to elliptic Feynman integrals
2021
We report on an implementation within GiNaC to evaluate iterated integrals related to elliptic Feynman integrals numerically to arbitrary precision within the region of convergence of the series expansion of the integrand. The implementation includes iterated integrals of modular forms as well as iterated integrals involving the Kronecker coefficient functions $g^{(k)}(z,\tau)$. For the Kronecker coefficient functions iterated integrals in $d\tau$ and $dz$ are implemented. This includes elliptic multiple polylogarithms.
Feynman integrals and iterated integrals of modular forms
2017
In this paper we show that certain Feynman integrals can be expressed as linear combinations of iterated integrals of modular forms to all orders in the dimensional regularisation parameter $\varepsilon$ . We discuss explicitly the equal mass sunrise integral and the kite integral. For both cases we give the alphabet of letters occurring in the iterated integrals. For the sunrise integral we present a compact formula, expressing this integral to all orders in $\varepsilon$ as iterated integrals of modular forms.
Hilbert modularity of some double octic Calabi--Yau threefolds
2018
We exhibit three double octic Calabi--Yau threefolds over the certain quadratic fields and prove their modularity. The non-rigid threefold has two conjugate Hilbert modular forms of weight [4,2] and [2,4] attached while the two rigid threefolds correspond to a Hilbert modular form of weight [4,4] and to the twist of the restriction of a classical modular form of weight 4.
The electron self-energy in QED at two loops revisited
2018
We reconsider the two-loop electron self-energy in quantum electrodynamics. We present a modern calculation, where all relevant two-loop integrals are expressed in terms of iterated integrals of modular forms. As boundary points of the iterated integrals we consider the four cases $p^2=0$, $p^2=m^2$, $p^2=9m^2$ and $p^2=\infty$. The iterated integrals have $q$-expansions, which can be used for the numerical evaluation. We show that a truncation of the $q$-series to order ${\mathcal O}(q^{30})$ gives numerically for the finite part of the self-energy a relative precision better than $10^{-20}$ for all real values $p^2/m^2$.
The unequal mass sunrise integral expressed through iterated integrals on M‾1,3
2020
Abstract We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter e. In order to do so, we transform the system of differential equations for the master integrals to an e-form. The sunrise integral with unequal masses depends on three kinematical variables. We perform a change of variables to standard coordinates on the moduli space M 1 , 3 of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on M ‾ 1 , 3 . On the hypersurface τ = const our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.
On a Class of Feynman Integrals Evaluating to Iterated Integrals of Modular Forms
2019
In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic curves and modular forms. Feynman integrals, which evaluate to iterated integrals of modular forms go beyond the class of multiple polylogarithms. Nevertheless, we may bring for all examples considered the associated system of differential equations by a non-algebraic transformation to an \(\varepsilon \)-form, which makes a solution in terms of iterated integrals immediate.
Modular Calabi-Yau threefolds of level eight
2005
In the studies on the modularity conjecture for rigid Calabi-Yau threefolds several examples with the unique level 8 cusp form were constructed. According to the Tate Conjecture correspondences inducing isomorphisms on the middle cohomologies should exist between these varieties. In the paper we construct several examples of such correspondences. In the constructions elliptic fibrations play a crucial role. In fact we show that all but three examples are in some sense built upon two modular curves from the Beauville list.
Frobenius polynomials for Calabi–Yau equations
2008
We describe a variation of Dwork’ s unit-root method to determine the degree 4 Frobenius polynomial for members of a 1-modulus Calabi–Yau family over P1 in terms of the holomorphic period near a point of maximal unipotent monodromy. The method is illustrated on a couple of examples from the list [3]. For singular points, we find that the Frobenius polynomial splits in a product of two linear factors and a quadratic part 1− apT + p3T 2. We identify weight 4 modular forms which reproduce the ap as Fourier coefficients.