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**Modularity of solvable Artin representations of \(\text{GO}(4)\)-type.**
*(English)*
Zbl 1002.11045

Let \(F\) be a number field and let \((\rho,V)\) be a continuous representation of \(\text{Gal}(\overline F/F)\) on a 4-dimensional complex vector space whose image is solvable and lies in the subgroup of orthogonal similitudes \(\text{GO}(V)\). Then the author shows that \(\rho\) is modular; that is, that the Artin \(L\)-function \(L(s,\rho)\) is equal to the \(L\)-function of an isobaric automorphic representation of \(\text{GL}(4,{\mathbf A}_F)\). Moreover, \(\pi\) is cuspidal if and only if \(\rho\) is irreducible.

There are many corollaries. For example, combining this with well-known information about Rankin-Selberg convolutions, the author concludes that if \(\rho\), \(\rho'\) are continuous Galois representations of solvable \(\text{GO}(4)\)-type, then the Artin conjecture holds for the (16-dimensional) representation \(\rho\otimes\rho'\). The author shows that for each \(F\), in fact there is a doubly infinite family of examples where the representations \(\rho\otimes\rho'\) are primitive (i.e. not induced from a representation of a proper subgroup). This is proved using a recent criterion of M. Aschbacher [J. Algebra 234, 627-640 (2000; Zbl 0991.20008)]. Thus the main result gives new instances where Artin’s conjecture is proved.

In addition, the result gives new examples of extensions \(N/F\) such that the strong Dedekind conjecture holds. (Recall that the Dedekind conjecture is the assertion that for any extension \(N/F\) of number fields the quotient of Dedekind zeta functions \(\zeta_N(s)/\zeta_F(s)\) is entire. The strong Dedekind conjecture asserts that this ratio is in fact the standard \(L\)-function of an isobaric automorphic representation on \(\text{GL}([N:F]-1,{\mathbf A}_F)\).) For these extensions \(N/F\) the author also shows that given \(\pi\) on \(\text{GL}(n,{\mathbf A}_F)\) the standard \(L\)-function of the admissible representation \(\pi_N\) which should give the base change of \(\pi\) to \(N\) has meromorphic continuation and functional equation and is divisible by \(L(s,\pi)\).

The proof makes use of the work of Langlands and Tunnell establishing the strong Artin conjecture for 2-dimensional Galois representations with solvable image. This result together with the Gelbart-Jacquet lift allows the author to reduce the main theorem to the following result: if \(\pi\) is a cuspidal automorphic representation of \(\text{GL}(2,{\mathbf A}_F)\) and \(K/F\) is a quadratic extension, then the Asai lift of \(\pi\) is automorphic, and cuspidal in precisely described instances. This lifting result, which is an instance of Langlands functoriality, is established using the converse theorem of J. W. Cogdell and I. I. Piatetski-Shapiro [Math. Res. Lett. 3, 67-76 (1996; Zbl 0864.22009)], together with the integral representations for certain Rankin-Selberg \(L\)-functions of I. I. Piatetski-Shapiro and S. Rallis [ Compos. Math. 64, 31-115 (1987; Zbl 0637.10023)] and additional information obtained by Ikeda. The author draws heavily upon the base change and descent techniques he developed in his paper [Ann. Math. (2) 152, 45-111 (2000; Zbl 0989.11023)].

There are many corollaries. For example, combining this with well-known information about Rankin-Selberg convolutions, the author concludes that if \(\rho\), \(\rho'\) are continuous Galois representations of solvable \(\text{GO}(4)\)-type, then the Artin conjecture holds for the (16-dimensional) representation \(\rho\otimes\rho'\). The author shows that for each \(F\), in fact there is a doubly infinite family of examples where the representations \(\rho\otimes\rho'\) are primitive (i.e. not induced from a representation of a proper subgroup). This is proved using a recent criterion of M. Aschbacher [J. Algebra 234, 627-640 (2000; Zbl 0991.20008)]. Thus the main result gives new instances where Artin’s conjecture is proved.

In addition, the result gives new examples of extensions \(N/F\) such that the strong Dedekind conjecture holds. (Recall that the Dedekind conjecture is the assertion that for any extension \(N/F\) of number fields the quotient of Dedekind zeta functions \(\zeta_N(s)/\zeta_F(s)\) is entire. The strong Dedekind conjecture asserts that this ratio is in fact the standard \(L\)-function of an isobaric automorphic representation on \(\text{GL}([N:F]-1,{\mathbf A}_F)\).) For these extensions \(N/F\) the author also shows that given \(\pi\) on \(\text{GL}(n,{\mathbf A}_F)\) the standard \(L\)-function of the admissible representation \(\pi_N\) which should give the base change of \(\pi\) to \(N\) has meromorphic continuation and functional equation and is divisible by \(L(s,\pi)\).

The proof makes use of the work of Langlands and Tunnell establishing the strong Artin conjecture for 2-dimensional Galois representations with solvable image. This result together with the Gelbart-Jacquet lift allows the author to reduce the main theorem to the following result: if \(\pi\) is a cuspidal automorphic representation of \(\text{GL}(2,{\mathbf A}_F)\) and \(K/F\) is a quadratic extension, then the Asai lift of \(\pi\) is automorphic, and cuspidal in precisely described instances. This lifting result, which is an instance of Langlands functoriality, is established using the converse theorem of J. W. Cogdell and I. I. Piatetski-Shapiro [Math. Res. Lett. 3, 67-76 (1996; Zbl 0864.22009)], together with the integral representations for certain Rankin-Selberg \(L\)-functions of I. I. Piatetski-Shapiro and S. Rallis [ Compos. Math. 64, 31-115 (1987; Zbl 0637.10023)] and additional information obtained by Ikeda. The author draws heavily upon the base change and descent techniques he developed in his paper [Ann. Math. (2) 152, 45-111 (2000; Zbl 0989.11023)].

Reviewer: Solomon Friedberg (Chestnut Hill)

### MSC:

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

11R42 | Zeta functions and \(L\)-functions of number fields |

11R39 | Langlands-Weil conjectures, nonabelian class field theory |