Burgess-like subconvex bounds for \(\mathrm{GL}_{2}\times \mathrm{GL}_{1}\). (English) Zbl 1376.11037
Summary: Let \(F\) be a number field, \(\pi\) an irreducible cuspidal representation of GL\(_2(\mathbb{A}_F)\) with unitary central character, and \({\chi}\) a Hecke character of analytic conductor \(Q\). Then \(L(1/2, \pi \otimes \chi) \ll Q^{\frac{1}{2} - \frac{1}{8}(1-2\theta)+\varepsilon}\), where \(0 \leq \theta \leq 1/2\) is any exponent towards the Ramanujan-Petersson conjecture. The proof is based on an idea of unipotent translation originated from
P. Sarnak [Commun. Pure Appl. Math. 38, 167–178 (1985; Zbl 0577.10026)] then developed by P. Michel and A. Venkatesh [Publ. Math., Inst. Hautes Étud. Sci. 111, 171–271 (2010; Zbl 1376.11040)], combined with a method of amplification.
MSC:
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 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 |
Keywords:
cuspidal representation; Hecke character; analytic conductor; \(L\)-function; Ramanujan-Petersson conjectureReferences:
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