I am looking for a Siegel modular form of genus $2$ (living on the Siegel modular 3-fold $A\_2=\mathrm{Sp}(4,\mathbb{Z})\backslash \mathfrak H\_2$) which becomes "roughly" the product of two eta functions on the locus where the parametrized abelian surface is the product of two elliptic curves. Here "roughly" means up to multiplication by something like $e^{\pi i \tau/12}$. 


Does anyone know a reference for this sort of result? 



A Siegel modular form related to the product of two eta functions
I hesitate to answer my own question, but the answer below may be useful to some people. 


The Siegel cusp form $\chi\_{10}$ of weight $10$ of genus $2$ has the following asymptotic behaviour as $z$ goes to $0$
$$
\chi\_{10}(x,y,z) = \eta(x)^{24}\eta(y)^{24}(\pi z)^2+O(z^4),
$$
where $x,y,z$ are the coordinates of the Siegel upper-half plane with $z$ off diagonal (so the divisor $z=0$ is precisely the locus where the abelian surface is the product of two elliptic curves). 


