Abstract
We construct Siegel Poincaré series of weight three and genus two outside their natural domain of convergency by the method of analytic continuation. In contrast to genus two Poincaré series of higher weights greater three, the Poincaré series thus obtained do not define holomorphic cuspforms of weight three. In fact, additional nonholomorphic phantom parts show up and we are able to describe them explicitly in terms of holomorphic forms of weight one. As a consequence of this result on Poincaré series, in the case under consideration we can also show that Sturm’s operator not only projects to the holomorphic discrete series of weight three, as one would expect, but also gives rise to an additional phantom projection. For genus two the weight three case is distinguished, giving the first example for this new kind of phenomenon.
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Notes
The integral is \(\int _{\mathcal Y}b(T,Y)\exp (-4\pi {{\mathrm{tr}}}(TY))\det (TY)^{k-1/2+s}\frac{dY}{\det (Y)^{3/2}}\), which equals \((4\pi )^2a(T)\det (T)\int _{\mathcal Y}\Big (k(k-1/2)(4\pi )^{-2}\det (TY)^{-1}-(k-1/2)(4\pi )^{-1}\det (TY)^{-1}{{\mathrm{tr}}}(TY)+1\Big )\times \exp (-4\pi {{\mathrm{tr}}}(TY))\det (TY)^{k-1/2+s}\frac{dY}{\det (Y)^{3/2}}\). For the change of variables \(Y\mapsto 4\pi T^{\frac{1}{2}}Y T^{\frac{1}{2}}\) this equals \((4\pi )^{-2(k+1/2+s-1)}a(T)\det (T)\sqrt{\pi }s(s-1/2)\Gamma (s+k-1/2)\Gamma (s+k-1)\).
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This work was partially supported by the European Social Fund (Kathrin Maurischat).
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Maurischat, K., Weissauer, R. Phantom holomorphic projections arising from Sturm’s formula. Ramanujan J 47, 21–46 (2018). https://doi.org/10.1007/s11139-018-0033-8
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DOI: https://doi.org/10.1007/s11139-018-0033-8