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Equations satisfied by Igusa invariants of split Jacobians
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Polynomials
$n$
$\phi_n$
INPUT{polynomials.yaml} (not shown in preview)
Definition
This table contains equations satisfied by the Igusa Invariants of genus $2$ curves with $(n,n)$ split Jacobians. More precisely, let $C$ be a genus $2$ curve, $Y^2 = f(X)$ a model thereof, $[I_2:I_4:I_6:I_8:I_{10}]$ be the Igusa invariants of this curve, and $(i_1,i_2,i_3)=(144 I_4/I_2^2,-1728(I_2 I_4-3 I_6)/I_2^3,486 I_{10}/I_2^5)$ the absolute invariants. If $\text{Jac}(C)$ is optimally $(n,n)$-split then its absolute invariants satisfy the equation $\phi_n=0$ with $\phi_n$ in this table.
Parameters
$n$
—   integer ($n \geq 2$)
(1)
More complete data is available at [4].
(2)
The case $n=2$ is classical, $n=3$ due to Kuhn [3], and $n=4$ due to Bruin and Doerksen [2].
References
[1]
Oskar Bolza, "Ueber die Reduction hyperelliptischer Integrale erster Ordnung und erster Gattung auf elliptische durch eine Transformation vierten Grades", Math. Ann. 28 (1887), no. 3, 447-456.
[2]
Nils Bruin, Kevin Doerksen, "The arithmetic of genus two curves with (4,4)-split Jacobians", Canad. J. Math. 63 (2011), 992-1021. (arXiv)
[3]
R. M. Kuhn, "Curves of genus 2 with split Jacobian". Trans. Amer. Math. Soc. 307(1988), no. 1, 41-49.