G2-invariants of a genus 2 curve (reviewed)

The G2-invariants of a genus 2 curve are absolute invariants associated to the geometric isomorphism class of the curve. They are denoted $g_1,g_2,g_3$.

These invariants were defined by Cardona, Quer, Nart and Pujolàs. Like the Igusa invariants, they reflect the behavior of the curve in question at a prime; for example, if one of the invariants contains a prime in its denominator, then the curve will have bad reduction at that prime over any extension of the base field. However, the G2-invariants give less information than the Igusa invariants do about the particular type of reduction at a given prime.

The exact definition of the G2-invariants is as follows. Let $X$ be a curve of genus $2$ with Igusa invariants $J_2, J_4, J_6, J_8, J_{10}$. Then \[ (g_1,g_2,g_3) = (J_2^5 / J_{10},\, J_2^3 J_4 / J_{10},\, J_2^2 J_6 / J_{10}), \] so long as $J_2$ is nonzero ($J_{10}$ is always nonzero because it is a multiple of the discriminant of $f$). When $J_2$ is zero we instead have \[ (g_1,g_2,g_3) = (0,\, J_4^5 / J_{10}^2,\, J_4 J_6 / J_{10}),\qquad\ \ \] except when $J_4$ is also zero, in which case \[ (g_1,g_2,g_3) = (0,\,0,\,J_6^5 / J_{10}^3).\qquad\qquad\quad \]