Multiplication and Division on Elliptic Curves, Torsion Points, and Roots of Modular Equations
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MULTIPLICATION AND DIVISION ON ELLIPTIC CURVES, TORSION POINTS, AND ROOTS OF MODULAR EQUATIONS S. Adlaj∗
UDC 512.772.7, 511.381, 512.622
Having expressed the ratio of the length of the lemniscate of Bernoulli to the length√of its cocentered superscribing circle as the reciprocal of the arithmetic-geometric mean of 1 and 2, Gauss wrote in his diary, on May 30, 1799, that thereby “an entirely new field of analysis” emerged. 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 not only enables rederiving classical results eloquently, but allows one to undertake new calculations, which seemed 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-torsion points on a corresponding elliptic curve, and we shall rebring to light Galois’ exceptional, never really surpassable, and far from fully appreciated impact. Bibliography: 19 titles.
Introduction: an integral, tightly cohesive subject of elliptic functions and elliptic curves Given a parameter β ∈ C\ {−1, 0, 1}, we introduce, as in [1–4, 8, 9], the Galois essential elliptic function, which is a (meromorphic) function R = Rβ = Rβ ( · ) = R( · , β) possessing a (double) pole at the origin and satisfying the differential equation R2 = 4R (R + β) (R + 1/β) .
(1)
Denote the lattice of the function Rβ by Λβ , and call the parameter β the elliptic modulus. The map z → 1, Rβ (z), Rβ (z) extends, with 0 → (0, 0, 1), to a map from the period parallelogram C/Λβ to the complex projective space PC2 . The (extended) map induces on its image Eβ , which we shall call the associated elliptic curve,1 an isomorphism of Riemann surfaces, as well as an isomorphism of groups.2 This map, further, enables an identification (exploiting the j-invariant) of isomorphism classes for projective complex elliptic curves with the homothety classes of lattices L/C× , which might, in turn, be identified with the fundamental domain Γ\H for the action of the modular group Γ := PSL(2, Z) on the upper half plane H, as is well explained in [17]. From now on, we exploit the identification of the points on the torus C/Λβ , which might be viewed as the domain of Rβ , with the points on the elliptic curve Eβ , which might be viewed as the image of the functional pair (Rβ , Rβ ). Keeping in mind that the value of the function Rβ at every point determines, up to sign, via Eq. (1), the value of its derivative Rβ at this point, we can further identify a pair of (not necessarily distinct) points on Eβ sharing the first ∗
Federal Research Center “Informatics and Control” of the Russian Academy of Sciences, Moscow, Russia, e-mail
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