Elliptic modular functions

Trigonometric functions, elliptic functions, elliptic modular forms

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...

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.

The p-Adic L-Functions of Modular Elliptic Curves

The arithmetic theory of elliptic curves enters the new century with some of its major secrets intact. Most notably, the Birch and Swinnerton-Dyer conjecture, which relates the arithmetic of an elliptic curve to the analytic behaviour of its associated L-series, is still unproved in spite of important advances in the last decades, some of which are recalled in Chapter 1. In the 1960’s the pione...

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.