On special values of certain \(L\)-functions
\textit{P. Deligne} [Proc. Symp. Pure Math. 33, 313--346 (1979; Zbl 0449.10022)] has conjectured that the special values of motivic \(L\)-functions at certain critical points, when divided by suitable periods, are algebraic. This paper gives a new algebraicity result that is consistent with this expectation. Let \(h\) be a holomorphic newform on \(\Gamma_0(N)\) of weight \(k\) and Nebentypus \(\varepsilon\), and let \(\sigma\) be the irreducible unitary cuspidal automorphic representation associated to \(h\). Let \(V\) be a quadratic space over \(\mathbb{Q}\) of \(\mathbb{Q}\)-dimension \(n\geq 4\) such that \(V\otimes_{\mathbb{Q}}{\mathbb{R}}\) is anisotropic, and let \(\tau\) be an automorphic representation of \(\mathrm{SO}(V,A_{\mathbb{Q}})\) whose infinity component is the trivial representation. Suppose that \(k>2n\). Then the authors show that there exists a finite set \(S\) of places of \(\mathbb{Q}\) containing the Archimedean place such that the partial \(L\)-function \(L_S(s,\sigma\otimes\tau)\), divided by suitable powers of \(\pi\) and the Petersson inner product \(\langle h,h\rangle\), is algebraic at \(s=(k-n+1)/2\). Here the \(L\)-function is normalized to have functional equation under \(s\mapsto 1-s\). This location is expected to be the rightmost critical point in the sense of Deligne.NEWLINENEWLINEThe proof is based on an integral representation of the \(L\)-function due to \textit{D. Ginzburg} et al. [Mem. Am. Math. Soc. 611, 218 p. (1997; Zbl 0884.11022)]. This integral representation is based on taking the Bessel coefficient of a certain Eisenstein series. The authors show that at the special value in question, the Eisenstein series is a holomorphic Eisenstein series on a type IV domain and is arithmetic for a proper choice of data. Thus for this data its Bessel coefficient is algebraic. The authors then deduce their algebraicity result by an inductive argument on \(n\).NEWLINENEWLINEWhen \(n=4\) (in which case the integral representation is due to the first author [Am. J. Math. 115, No. 4, 823--860 (1993; Zbl 0792.11012)]), the authors apply their result to give a new algebraicity result concerning the complete (i.e.\ with \(S=\{\infty\}\)) \(\mathrm{GL}_2\) Rankin triple product \(L\)-function in an unbalanced case. When \(n=5\) they note that their result is consistent with a conjecture of \textit{H. Yoshida} [Am. J. Math. 123, No. 6, 1171--1197 (2001; Zbl 0998.11022)] concerning the special values of the tensor product \(L\)-function for \(\mathrm{GSp}_4\times \mathrm{GL}_2\).
scientific article; zbMATH DE number 6365643