Near holomorphy of some automorphic forms at critical points
In this long and subtle article some results of \textit{G. Shimura} [Math. Ann. 278, 1-28 (1987; Zbl 0636.10023)] on near holomorphy of a distinguished automorphic form f on \({\mathcal Z}\times {\mathbb{C}}\) are extended. Here \({\mathcal Z}\) is a (bounded) symmetric domain arising from symmetric bilinear forms S over totally real number fields F. The form f in question always has an integral representation involving a Hilbert modular form of a common subfield E of the F's, an Eisenstein series and a certain theta series associated to the S's which allows to get a meromorphic continuation to the whole plane and guarantees real analyticity in \({\mathcal Z}\). The following questions are at the core of investigations: (1) finiteness and holomorphy of \(w\mapsto f(w,s_ 0).\) (2) more generally, near holomorphy of \(w\mapsto f(w,s_ 0)\) which amounts to say to be annihilated by a power of a certain non holomorphic differential operator. (3) nearly holomorphy of the residue in case of a simple pole. As to the importance of the questions above ``an affirmative answer is essential to the study of the arithmetic properties of \(f(w,s_ 0)''\). All these questions are tackled by Shimura (loc.cit.) in the case \(s=0\). His method is extended to nonpositive integers in the right half plane \(s\geq -b/2\) for a certain (half) integer b associated to f. Furthermore under additional assumptions on the common subfield E, the involved Hilbert modular form and the theta series better results are obtained by the author for a product of f by a certain L-series.
scientific article; zbMATH DE number 4162992