Multiplication and division on elliptic curves, torsion points and roots of modular equations

Having expressed the ratio of the length of the Lemniscate of Bernoulli to the length of its cocentred superscribing circle as the reciprocal of the arithmetic-geometric mean of $1$ and $\sqrt{2}$, Gauss wrote in his diary, on May 30, 1799, that thereby “an entirely new field of analysis” emerges. Yet, up to these days, the study of elliptic functions (and curves) has been based on two traditional approaches (namely, that of Jacobi and that of Weiestrass), rather than a single unifying approach. Replacing artificial dichotomy by a, methodologically justified, single unifying approach does not only enable re-deriving classical results eloquently but it allows for undertaking new calculations, which did seem either unfeasible or too cumbersome to be explicitly performed. Here, we shall derive readily verifiable explicit formulas for carrying out highly efficient arithmetic on complex projective elliptic curves. We shall explicitly relate calculating the roots of the modular equation of level $p$ to calculating the $p$-torsin points on a corresponding elliptic curve, and we shall re-bring to light Galois exceptional, never nearly surpassable and far from fully appreciated, impact.