Igusa invariants of a genus 2 curve (reviewed)

The Igusa invariants of a genus 2 curve are weighted projective invariants associated to the geometric isomorphism class of the curve. They are denoted $J_2, J_4, J_6, J_8, J_{10}$, where the subscript indicates the weight.

For genus 2 curves over $\Q$ we may assume that the Igusa invariants are integral. We can then normalize them by removing a factor of $d^k$ from $I_{2k}$, where $d$ is the largest positive integer for which the invariants remain integral after this normalization. We can also make the first nonzero $I_{2k}$ with $k$ odd nonnegative by multiplying each $I_{2k}$ by $(-i)^k$.

In the LMFDB, Igusa invariants for genus 2 curves over $\Q$ are always normalized in this way, which means that their values may differ from those returned by the corresponding functions in Sage or Magma; however, they are equivalent as $\overline\mathbb{Q}$-points in weighted projective space and identify the same geometric isomorphism class.

These invariants were first defined by Igusa to enable him to study the moduli space of genus 2 curves over $\Z$. In particular, they reflect the behavior of the corresponding curve when reducing modulo a prime. For instance, if in our normalized representation of the Igusa invariants a prime divides $I_{10}$, then the curve does not have good reduction at that prime, though it may still have potentially good reduction.

The exact definition of the Igusa invariants is as follows. Let $X$ be a curve of genus $2$, which we may assume to be defined by an equation $y^2 = f(x)$ with $f$ a sextic in $x$. Let $I_2, I_4, I_6, I_{10}$ be obtained from $f$ as in the definition of the Igusa-Clebsch invariants of $f$. Then the Igusa invariants are given by $J_2 = I_2 / 8,\qquad\qquad\qquad\quad\ \ \ \\ J_4 = (4 J_2^2 - I_4) / 96,\qquad\qquad\ \\ J_6 = (8 J_2^3 - 160 J_2 J_4 - I_6) / 576, \\ J_8 = (J_2 J_6 - J_4^2) / 4,\qquad\qquad\ \\ J_{10} = I_{10} / 4096.\qquad\qquad\qquad\ \,$ These invariants will be integral whenever $f$ is integral. The invariant $J_8$ may appear redundant, but is crucial when reducing to a field of even characteristic.