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  • Broadly I would like to know what is the connection between homogeneous binary quartic forms and elliptic curves.

    I see that the invariants on the first space under the action of $SL(2,\mathbb{C})$ are precisely the functions $g_2$ and $g_3$. Also given such a quartic form on $\mathbb{C}^2$, if one thinks its 0-set as the intersection of two conics in $\mathbb{C}^3$ then the determinant of the complex symmetric matrix representing the most general conic passing through this intersection is precisely the "RHS" of the elliptic curve equation.

  • What does this determinant mean geometrically?

More specifically consider the $SL(2,\mathbb{C})$ action on $\mathbb{C}^2$ (spanned by $x$ and $y$) and define $U = x^2$, $V = 2xy$ and $W=y^2$.

If the action is by the matrix $ \left [ \begin{array}{c c } a & b \\ c & d \\ \end{array}\right ]$ ($ad-bc=1$) of $SL(2,\mathbb{C})$ then the following transformation is affected on $\mathbb{C}^2$,

$$ \left [ \begin{array}{c} x\\ y\\ \end{array} \right ] \rightarrow \left [ \begin{array}{c} ax + by\\ cx + dy\\ \end{array} \right ] $$

This in turn implies the following transformation on $\mathbb{C}^3$ (defining the matrices, $X$ and $A$),

$$X = \left [ \begin{array}{c} U\\ V\\ W \end{array} \right ] \rightarrow A.X = \left [ \begin{array}{c c c} a^2 & ab & b^2\\ 2ac & (ad+bc) & 2bd \\ c^2 & cd & d^2\\ \end{array} \right ] \left [ \begin{array}{c} U\\ V\\ W \end{array} \right ] $$

$4UW-V^2$ being identically equal to $0$ is obviously kept invariant by the above transformation.

The invariance of $4UW-V^2$ can be thought of as $SL(2,\mathbb{C})$ preserving an inner-product on $\mathbb{C}^3$ with signature, $\left [ \begin{array}{c c c} & & 2 \\ & -1 & \\ 2 & & \\ \end{array} \right ]$

  • But $A$ does not seem to be an orthogonal matrix though it satisfies the property of $det(A)=1$. Then why is it being said in the books that the $SL(2,\mathbb{C})$ action on $\mathbb{C}^2$ (spanned by $x$ and $y$) induces an action of $SO(3)$ on $\mathbb{C}^3$ (spanned by $U$, $V$ and $W$) with respect to the inner product $ \left [ \begin{array}{c c c} & & 2 \\ & -1 & \\ 2 & & \\ \end{array}\right ]$ ?

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