# Equations satisfied by Igusa Invariants of split Jacobians

Nils Bruin and Kevin Doerksen

Below we list the equations satisfied by the Igusa Invariants of genus 2 curves with (n,n) split Jacobians. To fix notation:

• Let Y^2=f(X) be a model of a genus 2 curve
• Let [I2:I4:I6:I8:I10] be the Igusa invariants of this curve
• Let (i1,i2,i3)=(144*I4/I2^2,-1728*(I2*I4-3*I6)/I2^3,486*I10/I2^5) be the absolute invariants

If Jac(C) is optimally (n,n)-split then its absolute invariants satisfy the equations:

Note: All these equations are quite straightforward to find by interpolation if one has a universal model for a genus 2 curve with an appropriately split Jacobian. The case n=2 is classically known. For n=3, see Kuhn's 1988 paper. For n=4, see [Nils Bruin, Kevin Doerksen, The arithmetic of genus two curves with (4,4)-split Jacobians, Canad. J. Math. 63 (2011), 992-1021] or see ArXiv preprint arXiv:0902.3480, 2009).

Usage: The syntax used in the files above will work in most computer algebra packages. For example, in MAGMA you can read in these equations using:

``` R<i1,i2,i3>:=PolynomialRing(Rationals(),3); H4:=eval Read("split2x2.m"); H9:=eval Read("split3x3.m"); H16:=eval Read("split4x4.m"); ```

In the 1887 paper

• Oskar Bolza, Ueber die reduction hyperelliptischer integrale erster ordnung und erster gattung auf elliptische durch eine transformation vierten grade, Math. Ann. 28 (1887), no. 3, 447-456
Bolza gives a family of genus 2 curves with (4,4)-split Jacobians. His family can be related to the formulas above using the formulas below: