The four binary cubic forms, homogeneous cubics of the form $C_j(x,y) = a x^3 + b x^2 y + c x y^2+d y^3 = (a,b,c,d)$,

$C_1 = (1,3,-15,-23)$, $C_2 = (1,3,-51,-203)$, $C_3 = (1,3,-33,-113)$, $C_4 = (2,18,36,-15)$

have Hessian $(b^2-3ac, b c-9 a d, c^2-3 b d)$ quadratic covariant forms respectively

$Q_1 = 54(1,3,8)$, $Q_2 = 54(3,31,82)$, $Q_3 = 54(2,17,39)$, $Q_4 = Q_3$.

The substitution $\begin{pmatrix}{x}\\{y}\end{pmatrix} \mapsto \begin{pmatrix}{-5}&{-19}\\{1}&{4}\end{pmatrix} \begin{pmatrix}{x}\\{y}\end{pmatrix}$ transforms $Q_2$ into $Q_3$ and $C_2$ into $C_4$ so $C_2$ and $C_4$ belong to the same $GL_2(\mathbb{Z})$ class of binary cubic forms of discriminant 22356. The polynomials $C_j(x,1)$ generate three non-isomorphic cubic fields of the same discriminant, see Question 3448. $C_3$ and $C_4$ should belong to different $GL_2(\mathbb{Z})$ classes of forms but have the same Hessian.

Question: Why does this not agree with Eisenstein's result that sending $SL_2(\mathbb{Z})$ classes of binary cubic forms of positive discriminant to $SL_2(\mathbb{Z})$ classes of corresponding Hessian binary quadratic forms is injective? Have I overlooked something?

  • $\begingroup$ I can't find this result of Eisenstein's stated anywhere. Do you have a reference? $\endgroup$ – Qiaochu Yuan Nov 23 '13 at 2:36
  • $\begingroup$ Dickson's Volume 3 History of the theory of numbers, Page 253 and 254, archive.org/details/historyoftheoryo03dickuoft $\endgroup$ – Samuel Hambleton 2 Nov 23 '13 at 7:14
  • $\begingroup$ I thought I'd seen Eisenstein's result written more generally somewhere and stating that the map is always surjective, and injective whenever the disc of $C$ is positive. I've probably just misinterpreted the result. $\endgroup$ – Samuel Hambleton 2 Nov 23 '13 at 7:30
  • $\begingroup$ The closest thing I can find on pages 253 and 254 requires that the discriminant be squarefree, and that doesn't seem to be the case here. There also seems to be a second condition ("...which by triplication produces the principal class...") which I assume is about having order $3$ in some class group, and also the result is noted immediately afterwards to have an error. $\endgroup$ – Qiaochu Yuan Nov 23 '13 at 7:54
  • $\begingroup$ Thanks Qiaochu. I've just found www2.warwick.ac.uk/fac/sci/maths/people/staff/bouyer/… Page 28 which makes the map clear and states that the kernel is isomorphic to units of the quadratic field mod cubed units. $\endgroup$ – Samuel Hambleton 2 Nov 23 '13 at 8:09

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