Poisson equations for elliptic modular graph functions

Elliptic Functions and Equations of Modular Curves

Let p ≥ 5 be a prime. We show that the space of weight one Eisenstein series defines an embedding into P(p−3)/2 of the modular curve X1(p) for the congruence group Γ1(p) that is scheme-theoretically cut out by explicit quadratic equations.

Trigonometric functions, elliptic functions, elliptic modular forms

0 Elliptic Functions and Equations of Modular Curves

Let p ≥ 5 be a prime. We show that the space of weight one Eisenstein series defines an embedding into P of the modular curve X1(p) for the congruence group Γ1(p) that is scheme-theoretically cut out by explicit quadratic equations.

Introducing elliptic, an R package for elliptic and modular functions

This paper introduces the elliptic package of R routines, for numerical calculation of elliptic and related functions. Elliptic functions furnish interesting and instructive examples of many ideas of complex analysis, and package elliptic illustrates these numerically and visually. A statistical application in fluid mechanics is presented. An earlier version of this vignette was published as Ha...

Generalized Elliptic Integrals and Modular Equations

In geometric function theory, generalized elliptic integrals and functions arise from the Schwarz-Christoffel transformation of the upper half-plane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic number theory in the study of Ramanujan’s modular equations and approximations to π. The authors stu...