Testing Equivalence of Ternary Cubics

Let C be a smooth plane cubic curve with Jacobian E. We give a formula for the action of the 3-torsion of E on C, and explain how it is useful in studying the 3-Selmer group of an elliptic curve dened over a number eld. We work over a eld K of characteristic zero, with algebraic closure K. 1 The Invariants of a Ternary Cubic Let X3 = A10 be the space of all ternary cubics U(X, Y, Z) = aX3 + bY 3 + cZ3 + a2X2 Y + a3X2 Z + b1XY 2 + b3Y 2 Z + c1XZ2 + c2Y Z2 + mXY Z . The co-ordinate ring of X3 is the polynomial ring K[X3] = K[a, b, c, a2, a3, b1, b3, c1, c2, m] . There is a natural action of GL3 on X3 given by (gU)(X, Y, Z) = U(g11X + g21Y + g31Z, . . . , g13X + g23Y + g33Z) . The ring of invariants is K[X3]SL3 = {F K[X3] : F g = F for all g SL3(K)} . A homogeneous invariant F satises F g = (g)F (1) for all g GL3(K), for some rational character : GL3 Gm. But the only rational characters of GL3 are of the form (g) = (det g)k for k an integer. We say that F is an invariant of weight ...

• Type: Conference
• Year: 2006
• Where: ANTS
• Authors: Tom Fisher
• Keywords: Algorithms, ANTS 2006, Cubic Let X3, Plane Cubic Curve, Ternary Cubics