Abstract
Plane quartic curves given by equations of the form y 2=P(x) with polynomials P of degree 4 represent singular models of elliptic curves which are directly related to elliptic integrals in the form studied by Euler and for which he developed his famous addition formulas. For cubic curves, the well-known secant and tangent construction establishes an immediate connection of addition formulas for the corresponding elliptic integrals with the structure of an algebraic group. The situation for quartic curves is considerably more complicated due to the presence of the singularity. We present a geometric construction, similar in spirit to the secant method for cubic curves, which defines an addition law on a quartic elliptic curve given by rational functions. Furthermore, we show how this addition on the curve itself corresponds to the addition in the (generalized) Jacobian variety of the curve, and we show how any addition formula for elliptic integrals of the form \(\int (1/\sqrt{P(x)})\,\mathrm{d}x\) with a quartic polynomial P can be derived directly from this addition law.
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References
Alling, N.L.: Real Elliptic Curves. North Holland, Amsterdam (1981)
Brieskorn, E., Knörrer, H.: Plane Algebraic Curves. Birkhäuser, Basel (1986)
Cooke, R.: Abel’s Theorem. In: Rowe, D.E., McCleary, J. (eds.) The History of Modern Mathematics, vol. I, pp. 389–421. Academic Press, San Diego (1989)
Fischer, G.: Ebene algebraische Kurven. Vieweg, Wiesbaden (1994)
Gray, J.J.: Algebraic Geometry in the late Nineteenth Century. In: Rowe, D.E., McCleary, J. (eds.) The History of Modern Mathematics, vol. I, pp. 361–385. Academic Press, San Diego (1989)
Hartshorne, R.: Algebraic Geometry. Springer, Berlin (1977)
Husemoller, D.: Elliptic Curves. Springer, Berlin (1987)
Klein, F.: Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Springer, Berlin (1926) (reprint Wissenschaftliche Buchgesellschaft, 1986)
Liu, Q.: Algebraic Geometry and Arithmetic Curves. Oxford University Press, London (2002)
Mordell, L.J.: On the rational solutions of the indeterminate equations of the 3rd and 4th degrees. Math. Proc. Camb. Philos. Soc. 21, 179–192 (1922)
Norio, A.: Elliptic curves from Fermat to Weil. Hist. Sci. 9(1), 27–35 (1999)
Poincaré, H.: Sur les propriétés arithmétiques des courbes algébriques. J. Math. Pures Appl., sér. 5 7(2), 161–233 (1901)
Rosenlicht, M.: Generalized Jacobian varieties. Ann. Math. 59, 505–530 (1954)
Schappacher, N.: Développement de la loi de groupe sur une cubique. Séminaire de théorie des nombres Paris 1988–1989. Prog. Math. 91, 159–184 (1990) Birkhäuser
Shafarevich, I.R.: Basic Algebraic Geometry. Springer, Berlin (1977)
Siegel, C.L.: Topics in Complex Function Theory, vol. 1. Wiley, New York (1969)
Silverman, J.H.: The Arithmetic of Elliptic Curves. Springer, Berlin (1986)
Weil, A.: L’arithmétique sur les courbes algébriques, Acta Mathematica 52 (1928)
Weil, A.: Zahlentheorie – Ein Gang durch die Geschichte von Hammurapi bis Legendre. Birkhäuser, Basel (1992)
Zagier, D.: Elliptische Kurven: Fortschritte und Anwendungen. Jahresberichte der Deutschen Mathematiker-Vereinigung 92, 58–76 (1990)
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Selder, E., Spindler, K. A geometric approach to Euler’s addition formula. Math Semesterber 58, 185–214 (2011). https://doi.org/10.1007/s00591-011-0090-1
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DOI: https://doi.org/10.1007/s00591-011-0090-1