Nonvanishing of the central critical value of the third symmetric power \(L\)-functions. (English) Zbl 1034.11033
Let \(\pi\) be an irreducible cuspidal automorphic representation of \(GL_2(\mathbb A)\) where \(\mathbb A\) is the ring of adeles of a number field \(F\). In this paper the authors study the behaviour of the \(L\)-function \(L(s,\pi,\text{Sym}^3)\) at the central point \(s = \frac 12\) of the critical strip. H. Kim and F. Shahidi [Ann. Math. (2) 150, 645–662 (2002; Zbl 0957.11026)] have shown that when \(\pi\) is not monomial then this \(L\)-function is holomorphic in the entire plane except for a simple pole at \(s=1\). On the other hand, D. Bump, D. Ginzburg and J. Hoffstein [Invent. Math. 125, 413–449 (1996; Zbl 0871.11038)] have shown that the \(L\)-function has a representation as a Rankin-Selberg-Shimura integral.
In this paper the authors make use of a split group of type \(G_2\) over \(F\) when \(F\) contains the third roots of unity. The group \(G_2\) is used first because of Langlands’ representation of \(L(s,\pi,\text{Sym}^3)\) and secondly because the cubic metaplectic cover of \(G_2\) splits over a certain copy of \(GL_2\). Using this the authors can show that \(L(\frac 12,\pi,\text{Sym}^3)=0\) precisely when a certain “period”, an integral involving an automorpic form of \(\pi\) and two exceptional theta functions vanishes.
In this paper the authors make use of a split group of type \(G_2\) over \(F\) when \(F\) contains the third roots of unity. The group \(G_2\) is used first because of Langlands’ representation of \(L(s,\pi,\text{Sym}^3)\) and secondly because the cubic metaplectic cover of \(G_2\) splits over a certain copy of \(GL_2\). Using this the authors can show that \(L(\frac 12,\pi,\text{Sym}^3)=0\) precisely when a certain “period”, an integral involving an automorpic form of \(\pi\) and two exceptional theta functions vanishes.
Reviewer: Samuel James Patterson (Göttingen)
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E46 | Semisimple Lie groups and their representations |
| 11F55 | Other groups and their modular and automorphic forms (several variables) |
Keywords:
symmetric power L-functions; metaplectic theta functions; the execeptional group \(G_2\); Eisenstein seriesReferences:
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