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?

  • $\begingroup$ For "roughly", you could take a weight $1/2$ holomorphic (?!) Siegel-type Eisenstein series, and restrict to get a sum of weight-$1/2$ holomorphic (?!) Eisenstein series on the $SL_2\times SL_2$ plus a sum of (tensor) products over all weight-$1/2$ cuspforms... roughly. A somewhat different version of such a computation is in my old paper in the Taniguchi Symposium (at Katata) published about 1984 in a volume "Automorphic Forms in Several Variables". The work of Rallis-PiatetskiShapiro on "doubling" emphasizes obtaining integral representations of $L$-functions. $\endgroup$ – paul garrett Jun 1 '15 at 22:24
  • $\begingroup$ Have you looked at MR0669299, around page 335 (section 5)? $\endgroup$ – Lev Borisov Jun 2 '15 at 1:49

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).

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  • $\begingroup$ Dear Kanazawa, do you have a reference for the asymptotic identity given above. I would be quite interested to see how you proved it. $\endgroup$ – Hugo Chapdelaine Jun 30 '15 at 17:49
  • $\begingroup$ I don't know a good reference for it, but it follows from the expression of $\chi_{10}$ in terms of the Eisenstein series of weight $4,6$ and $12$. $\endgroup$ – Atsushi Kanazawa Jul 1 '15 at 20:01
  • $\begingroup$ Hi Atsushi, may I find this relation in Klingen's book on Siegel modular forms? $\endgroup$ – Hugo Chapdelaine Jul 2 '15 at 20:49
  • $\begingroup$ Hi Hugo. I don't know about the book, but the relation essentially appeared in Igusa's famous paper "On Siegel modular forms of genus two". Actually $\chi_{12}$ has a similar property. $\endgroup$ – Atsushi Kanazawa Jul 3 '15 at 2:11

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