Modularity of the Rankin-Selberg \(L\)-series, and multiplicity one for \(\mathrm{SL}(2)\).

*(English)*Zbl 0989.11023The main result of the paper is the following theorem. Let \(\pi\) and \(\pi'\) be two isobaric automorphic representations of \(\mathrm{GL}(2)\) over a number field \(F\). Then there exists an automorphic representation \(\Pi\) of \(\mathrm{GL}(4)\) over \(F\) whose \(L\)- and \(\varepsilon\)-factors are \(L(s,\pi_v \times \pi'_v)\) and \(\varepsilon(s,\pi_v \times \pi'_v)\) (the \(L\)-function of \(\Pi\) is the “convolution” of the \(L\)-functions of \(\pi\) and \(\pi'\)). The theorem is completed by a criterion for the cuspidality of \(\Pi\).

The result is derived from a converse theorem for \(\mathrm{GL}(4)\). In order to define candidates for \(\Pi_v\) at all places, the author uses base change to an infinite family of solvable extensions of \(F\). Then the analytic properties of \(L(s,\Pi\times\eta)\), with \(\eta\) cuspidal on \(\mathrm{GL}(1)\) or \(\mathrm{GL}(2)\), have to be proved, so that the converse theorem can be applied. The most difficult part is the required boundedness in vertical strips of these \(L\)-functions (for \(\eta\) on \(\mathrm{GL}(2)\)). Here the author uses the integral representation of the triple \(L\)-function defined by Piatetski-Shapiro and Rallis. That integral involves an Eisenstein series \(E(f_s)\) on \(\mathrm{GSp}(6)\), which has to be estimated. Arthur’s truncation functor is used to do that. Finally one descends to the original base field \(F\).

Applications of the theorem are, among others, multiplicity one for cusp forms on \(\mathrm{SL}(2)\) and a proof of the Tate conjecture for 4-fold products of modular curves.

The result is derived from a converse theorem for \(\mathrm{GL}(4)\). In order to define candidates for \(\Pi_v\) at all places, the author uses base change to an infinite family of solvable extensions of \(F\). Then the analytic properties of \(L(s,\Pi\times\eta)\), with \(\eta\) cuspidal on \(\mathrm{GL}(1)\) or \(\mathrm{GL}(2)\), have to be proved, so that the converse theorem can be applied. The most difficult part is the required boundedness in vertical strips of these \(L\)-functions (for \(\eta\) on \(\mathrm{GL}(2)\)). Here the author uses the integral representation of the triple \(L\)-function defined by Piatetski-Shapiro and Rallis. That integral involves an Eisenstein series \(E(f_s)\) on \(\mathrm{GSp}(6)\), which has to be estimated. Arthur’s truncation functor is used to do that. Finally one descends to the original base field \(F\).

Applications of the theorem are, among others, multiplicity one for cusp forms on \(\mathrm{SL}(2)\) and a proof of the Tate conjecture for 4-fold products of modular curves.

Reviewer: J. G. M. Mars (Utrecht)

##### MSC:

11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

11G18 | Arithmetic aspects of modular and Shimura varieties |

11F80 | Galois representations |

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |