On the algebraicity of special \(L\)-values of Hermitian modular forms. (Q273555)

In this paper, the author studies Hermitian modular forms and adelic Hermitian modular forms. He describes some results on the algebraicity of special \(L\)-values attached to them. The paper is essentially based on the techniques of \textit{G. Shimura} from the book [Arithmeticity in the theory of automorphic forms. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0967.11001)]. The main results claim essentially, that certain special \(L\)-values over the Petersson inner product (of the modular forms involved) is, up to a certain power of \(\pi\), an algebraic number. We briefly describe the content of the paper. After introducing Hermitian modular and adelic Hermitian modular forms in the second section, in the third section the author recalls the definitions of (Siegel type) Eisenstein and theta series (mainly following Shimura's book). Then, he reviews some results on the coefficients of the Fourier expansions of these Eisenstein series (normalized) and recalls for which values of (a complex parameter \(s\)) these Eisenstein series are nearly holomorphic. In this section, the Galois action on these Eisenstein series is also discussed. In the fourth section, the definitions of the standard \(L\)-functions attached to a non-zero adelic Hermitian modular forms and a related Dirichlet series are given. Then, the author gives an integral expression (Rankin-Selberg method) for these Dirichlet series. In the fifth section, the author defines some Archimedean periods which become useful for the normalization of the special values of the \(L\)-functions. The main algebraicity results for these special values are given in the sixth section.NEWLINENEWLINEIn general, the author follows the ideas of Shimura, but is able to be more precise about the field of definition of the ratios mentioned above (due to some results of Klosin). Also, at the end, the author discusses the overlap and the differences between these results and similar results of \textit{M. Harris} [Ann. Sci. Éc. Norm. Supér. (4) 14, 77--120 (1981; Zbl 0465.10022); Ann. Math. (2) 119, 59--94 (1984; Zbl 0589.10030); J. Reine Angew. Math. 483, 75--161 (1997; Zbl 0859.11032)] (but Harris uses the doubling method, unlike the author).