Abstract
In this work we obtain algebraicity results on special L-values attached to Siegel–Jacobi modular forms. Our method relies on a generalization of the doubling method to the Jacobi group obtained in our previous work, and on introducing a notion of near holomorphy for Siegel–Jacobi modular forms. Some of our results involve also holomorphic projection, which we obtain by using Siegel–Jacobi Poincaré series of exponential type.
Highlights
This paper should be seen as a continuation of our earlier paper [2] on properties of the standard L-function attached to a Siegel–Jacobi modular form
In [2] we have established various analytic properties (Euler product decomposition, analytic continuation and detection of poles) of the standard L-function attached to Siegel–Jacobi modular forms, and in this paper we turn our attention to algebraicity properties of some special L-values
Siegel–Jacobi modular forms and—in particular—the algebraicity results obtained in this paper do not fit in this framework: since the Jacobi group is not reductive, it does not satisfy the necessary properties to be associated with a Shimura variety
Summary
This paper should be seen as a continuation of our earlier paper [2] on properties of the standard L-function attached to a Siegel–Jacobi modular form. In our previous work [2] we extended their results to a very general setting: non-trivial level, character and a totally real algebraic number field For this purpose we applied the doubling method to the Jacobi group, and related Siegel-type Jacobi Eisenstein series to the standard Lfunction. Results of the above form for the standard L-functions of automorphic forms associated to Shimura varieties, such as Siegel and Hermitian modular forms, were obtained by many researchers, most profoundly by Shimura (see for example [23]) These results can be understood in the general framework of Deligne’s Period Conjectures for critical values of motives [7]. Siegel–Jacobi modular forms and—in particular—the algebraicity results obtained in this paper do not fit in this framework: since the Jacobi group is not reductive, it does not satisfy the necessary properties to be associated with a Shimura variety. [2] contains our results towards the analytic properties of the standard L function, whereas this paper focuses on the algebraic properties
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