Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

THE EXCEPTIONAL POINTS OF A CUBIC CURVE WHICH IS SYMMETRIC IN THE HOMOGENEOUS VARIABLES

ERNST S. SELMER
Mathematica Scandinavica
Vol. 2, No. 2 (February 24, 1955), pp. 227-236
Published by: Mathematica Scandinavica
Stable URL: http://www.jstor.org/stable/24489035
Page Count: 10
  • Read Online (Free)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
THE EXCEPTIONAL POINTS OF A CUBIC CURVE WHICH IS SYMMETRIC IN THE HOMOGENEOUS VARIABLES
Preview not available

Abstract

It is shown that the symmetric ternary cubic equation a (x+y+z)3 + b (xy+xz+yz)(x+y+z) + c xyz = 0 will represent (be equivalent to) any cubic curve with three rational inflections. The exceptional points, in number n = 3n1, are studied in homogeneous coordinates, by a method based on the forming of tangentials. The case b = 0 will cover all curves with n > 6, and a simple parametric representation is then given for a and c when 9 | n. The case a = 0 covers the general curve with 6 | n, and corresponding representations for b and c are then given when there is a cyclic or a non-cyclic exceptional subgroup of order 12. It is assumed throughout that the coefficients and variables are absolutely rational. Because of the choice of normal form, the parametric representations are much simpler than those previously obtained for the Weierstrass curve y2 = x3 - Ax - B for n = 6, 9 and 12.

Page Thumbnails

  • Thumbnail: Page 
[227]
    [227]
  • Thumbnail: Page 
228
    228
  • Thumbnail: Page 
229
    229
  • Thumbnail: Page 
230
    230
  • Thumbnail: Page 
231
    231
  • Thumbnail: Page 
232
    232
  • Thumbnail: Page 
233
    233
  • Thumbnail: Page 
234
    234
  • Thumbnail: Page 
235
    235
  • Thumbnail: Page 
236
    236