On automorphic \(L\)-functions for the Jacobi group of degree one and a relation with \(L\)-functions for Jacobi forms
The paper under review discusses automorphic \(L\)-functions attached to certain automorphic representations of the Jacobi group of degree one. Following the two well-known methods in the \(GL(2)\)-theory, the authors investigate the local \(L\)-factor given by the greatest common divisor of certain zeta integrals \(\zeta_p (W,s)\), \(W\) running over the Whittaker model, and compare this factor with the \(L\)-factor given by the zeta integral \(\zeta_p (W^0,s)\) of a normalized spherical Whittaker function \(W^0\). In contrast to the \(GL(2)\)-theory, these two methods lead in general to different local \(L\)-factors and the second one even depends on the character specifying the Whittaker model. The authors give an explanation of the above situation in the special case of an automorphic representation attached to a Hecke eigenform \(f\) of index one. They show (under a certain condition on \(f)\) that the first method leads to the local \(L\)-factor attached to \(f\) by \textit{V. Gritsenko} [Math. USSR, Sb. 47, 237-268 (1984); translation from Mat. Sb., Nov. Ser. 119(161), 248-277 (1982; Zbl 0507.10017); and Mathematica Gottingensis, SFB Göttingen, Heft 12] and \textit{T. Arakawa} [Comment. Math. Univ. St. Pauli 43, 181-216 (1994; Zbl 0837.11027)] using the Hecke eigenvalues of \(f\), whereas the zeta integral \(\zeta_p(W^0,s)\) is equal to the \(L\)-factor of the modular form of half-integral weight associated to \(f\) as defined by \textit{G. Shimura} [Ann. Math., II. Ser. 97, 440-481 (1973; Zbl 0266.10022)]. Moreover it is shown that a certain linear combination of these latter \(L\)-functions with local factors \(\zeta_p (W^0,s)\) is equal to the Mellin transform of \(f\) as defined by \textit{R. Berndt} [Manuscr. Math. 84, 101-112 (1994; Zbl 0827.11028)].
automorphic representations of the Jacobi group
scientific article; zbMATH DE number 1019709
scientific article