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Let $k$ be a field of characteristic not 2 or 3. Let $V$ be the $k$-space ofbinary cubic polynomials. The natural symplectic structure on $k^2$ promotes toa symplectic structure $\omega$ on $V$ and from the natural symplectic actionof $\textrm{Sl}(2,k)$ one obtains the symplectic module $(V,\omega)$. We give acomplete analysis of this symplectic module from the point of view of theassociated moment map, its norm square $Q$ (essentially the classicaldiscriminant) and the symplectic gradient of $Q$. Among the results are asymplectic derivation of the Cardano-Tartaglia formulas for the roots of acubic, detailed parameters for all $\textrm{Sl}(2,k)$ and$\textrm{Gl}(2,k)$-orbits, in particular identifying a group structure on theset of $\textrm{Sl}(2,k)$-orbits of fixed nonzero discriminant, and a purelysymplectic generalization of the classical Eisenstein syzygy for the covariantsof a binary cubic. Such fine symplectic analysis is due to the specialsymplectic nature inherited from the ambient exceptional Lie algebra $\mathfrakG_2$.

The Special Symplectic Structure of Binary Cubics