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The circle method and bounds for \(L\)-functions. IV: Subconvexity for twists of \(\mathrm{GL}(3)\) \(L\)-functions. (English) Zbl 1333.11046

Let \(\pi\) be a Hecke-Maass cusp form for \(\mathrm {SL}(3,Z)\) and \(\lambda(m,n)\) be its normalized Fourier coefficients. Let \(\chi\) be a primitive Dirichlet character modulo \(M\). The author considers the value of the \(L\)-function \(L(s,\pi\otimes\chi)=\sum_{n=1}^{\infty}\lambda(1,n)\chi(n)n^{-s}\) at \(s=\frac12\).
A convexity bound of \(L(\frac12,\pi\otimes\chi)\) (in terms of \(M\)) is given by \(M^{3/4+\varepsilon}\). The paper breaks this convexity bound and proves that \(L(\frac12,\pi\otimes\chi)\ll_{\pi} M^{\frac34-\delta}\) where \(\delta>0\) can be chosen to be \(1/1612\) (in the paper, it is not tried to optimize this \(\delta\)). Previously, such subconvexity bound was only obtained for special cases (for example when \(\pi\) is a symmetric square lift of a \(\mathrm{GL}(2)\) form).
Previous parts were published in I: Math. Ann. 358, No. 1-2, 389–401 (2014; Zbl 1312.11037); II: Am. J. Math. 137, No. 3, 791–812 (2015; Zbl 1344.11042); III: J. Am. Math. Soc. 28, No. 4, 913–938 (2015; Zbl 1354.11036).

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols

References:

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