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Quadratic twists of central values for \(\mathrm{GL}(3)\). (English) Zbl 1514.11031

It is of fundamental interest as to how much information one needs from the twists of an automorphic representation to determine the original automorphic representation. The fundamental work of W. Luo and D. Ramakrishnan [Invent. Math. 130, No. 2, 371–398 (1997; Zbl 0905.11024)] showed that holomorphic modular forms with trivial central character are distinguished by the central values of the \(L\)-functions of their quadratic twists. This result has concrete applications, such as the determination of a modular form from its \(p\)-adic \(L\)-function, and the determination of a half-integral weight modular form from its Fourier coefficients. Since then, there have been several generalizations, with different types of automorphic representations and twists.
In the current paper, the authors prove that an automorphic representation of \(\mathrm{GL}(3)\) over a number field with trivial central character, satisfying a technical hypothesis, is determined by the central values of the \(L\)-functions of its quadratic twists, under the assumption that there is a twist with nonvanishing central value. This extends the work of G. Chinta and A. Diaconu [Int. Math. Res. Not. 2005, No. 48, 2941–2967 (2005; Zbl 1085.11026)] which proved a similar result, without nonvanishing hypothesis, for Gelbart-Jacquet lifts.
The idea of the proof is, as in the previous works on a similar problem, to relate the residue at \((s,w)=(\frac{1}{2},1)\) of the double Dirichlet series\[Z(s,w;\pi,\chi_{N})=\sum_{D}\frac{L(s,\pi\otimes\chi_{D})}{|D|^{w}}\chi_{N}(D)\]and the \(N\)-th Fourier coefficient of \(\pi\), for a large enough \(N\). To execute this strategy, one needs to meromorphically continue the double Dirichlet series to a region around \((\frac{1}{2},1)\), which is the main technical innovation of the paper. The authors do this by first meromorphically continuing a related double Dirichlet series and deducing analytic properties of \(Z(s,w;\pi)\) by using a quadratic large sieve inequality.

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
11F30 Fourier coefficients of automorphic forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
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