##
**Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic L-functions and their derivatives.**
*(English)*
Zbl 0699.10039

Let f be a cusp form on hyperbolic 3-space, transforming according to one of the two characters of \(PGL_ 2({\mathbb{Z}}[i])\) that are trivial on the principal congruence subgroup modulo \((1+i)\). Suppose that f is a Hecke eigenform. For the twisted automorphic L-functions \(L(s,f,\chi)\) it is proved:

1) If Re \(s\geq 1/2\) there are infinitely many quadratic characters \(\chi\) of \({\mathbb{Q}}(i)\) such that \(L(s,f,\chi)\neq 0.\)

2) If the character of \(PGL_ 2({\mathbb{Z}}[i])\) is non-trivial there are infinitely many quadratic characters \(\chi\) of \({\mathbb{Q}}(i)\) for which \(L(1/2,f,\chi)=0\) and \(\frac{\partial L}{\partial s}(1/2,f,\chi)\neq 0.\)

The proof is complicated, and contains many computations. The reviewer admires the insight of the authors in deciding what automorphic forms to use and what to compute.

The basis idea is to consider Eisenstein series on \(G=GSp_ 4({\mathbb{C}})\) for certain congruence subgroups \(\Gamma\), and to compute Fourier coefficients. Actually, these Eisenstein series live on the metaplectic group. That means here that they are functions on G transforming according to a certain character of \(\Gamma\), the Kubota symbol. The existence of this symbol and the necessary properties are proved. The Eisenstein series considered here are connected to a maximal parabolic subgroup, with \(GL_ 2\) as the Levi component. So they depend on one complex parameter s, and on an automorphic form on \(GL_ 2({\mathbb{C}})\), as which the cusp form f is chosen.

Whittaker-Fourier coefficients are now computed, not only for the Eisenstein series themselves, but for various left translates as well. A suitable parametrization of the flag variety for G is developed to describe explicitly the Dirichlet series that are the main factor of these coefficients.

It is shown that in the region of convergence certain linear combinations of these Dirichlet series happen to be of the form \(L(s-1,f,\chi)\) up to some explicitly known factors. One of these factors is the symmetric square L-series \(L(2s-2,f,V^ 2)\) for f. It occurs in the denominator, hence its nonvanishing is important. All Fourier coefficients in the linear combination have the same order, but several left translates of the Eisenstein series are involved. The character \(\chi\) is quadratic, and depends on a number \(n\in {\mathbb{Q}}(i)\) that occurs in the order of the corresponding Fourier coefficients.

The equality thus obtained stays valid after meromorphic continuation in s. It is multiplied by \((Nn)^{-u+s-3/2}\), Re u sufficiently large, and summed over n. The resulting function of u is studied by a method of M. E. Novodvorsky [Proc. Symp. Pure Math. 33, No.2, 87-95 (1979; Zbl 0408.12013)]. This gives the meromorphic continuation in u, and explicit knowledge concerning location and principal part of the singularities.

The presence of singularities in this function (and further analysis of the influence of other factors in the equality) gives the nonvanishing result on \(L(s,f,\chi)\). For the other result the explicit knowledge of the singularities is needed, but the central idea is the same.

1) If Re \(s\geq 1/2\) there are infinitely many quadratic characters \(\chi\) of \({\mathbb{Q}}(i)\) such that \(L(s,f,\chi)\neq 0.\)

2) If the character of \(PGL_ 2({\mathbb{Z}}[i])\) is non-trivial there are infinitely many quadratic characters \(\chi\) of \({\mathbb{Q}}(i)\) for which \(L(1/2,f,\chi)=0\) and \(\frac{\partial L}{\partial s}(1/2,f,\chi)\neq 0.\)

The proof is complicated, and contains many computations. The reviewer admires the insight of the authors in deciding what automorphic forms to use and what to compute.

The basis idea is to consider Eisenstein series on \(G=GSp_ 4({\mathbb{C}})\) for certain congruence subgroups \(\Gamma\), and to compute Fourier coefficients. Actually, these Eisenstein series live on the metaplectic group. That means here that they are functions on G transforming according to a certain character of \(\Gamma\), the Kubota symbol. The existence of this symbol and the necessary properties are proved. The Eisenstein series considered here are connected to a maximal parabolic subgroup, with \(GL_ 2\) as the Levi component. So they depend on one complex parameter s, and on an automorphic form on \(GL_ 2({\mathbb{C}})\), as which the cusp form f is chosen.

Whittaker-Fourier coefficients are now computed, not only for the Eisenstein series themselves, but for various left translates as well. A suitable parametrization of the flag variety for G is developed to describe explicitly the Dirichlet series that are the main factor of these coefficients.

It is shown that in the region of convergence certain linear combinations of these Dirichlet series happen to be of the form \(L(s-1,f,\chi)\) up to some explicitly known factors. One of these factors is the symmetric square L-series \(L(2s-2,f,V^ 2)\) for f. It occurs in the denominator, hence its nonvanishing is important. All Fourier coefficients in the linear combination have the same order, but several left translates of the Eisenstein series are involved. The character \(\chi\) is quadratic, and depends on a number \(n\in {\mathbb{Q}}(i)\) that occurs in the order of the corresponding Fourier coefficients.

The equality thus obtained stays valid after meromorphic continuation in s. It is multiplied by \((Nn)^{-u+s-3/2}\), Re u sufficiently large, and summed over n. The resulting function of u is studied by a method of M. E. Novodvorsky [Proc. Symp. Pure Math. 33, No.2, 87-95 (1979; Zbl 0408.12013)]. This gives the meromorphic continuation in u, and explicit knowledge concerning location and principal part of the singularities.

The presence of singularities in this function (and further analysis of the influence of other factors in the equality) gives the nonvanishing result on \(L(s,f,\chi)\). For the other result the explicit knowledge of the singularities is needed, but the central idea is the same.

Reviewer: R.W.Bruggeman

### MSC:

11F27 | Theta series; Weil representation; theta correspondences |

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

22E30 | Analysis on real and complex Lie groups |