Munshi, Ritabrata The circle method and bounds for \(L\)-functions. IV: Subconvexity for twists of \(\mathrm{GL}(3)\) \(L\)-functions. (English) Zbl 1333.11046 Ann. Math. (2) 182, No. 2, 617-672 (2015). 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). Reviewer: Zhengyu Mao (Newark) Cited in 25 Documents MSC: 11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols Keywords:subconvexity bound; Hecke-Maass cusp form Citations:Zbl 1312.11037; Zbl 1344.11042; Zbl 1354.11036 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] V. Blomer, ”Subconvexity for twisted \(L\)-functions on \({ GL}(3)\),” Amer. J. Math., vol. 134, iss. 5, pp. 1385-1421, 2012. · Zbl 1297.11046 · doi:10.1353/ajm.2012.0032 [2] D. Bump, Automorphic Forms on \({ GL}(3,{\mathbf R})\), New York: Springer-Verlag, 1984, vol. 1083. · Zbl 0543.22005 [3] D. A. Burgess, ”On character sums and primitive roots,” Proc. London Math. Soc., vol. 12, pp. 179-192, 1962. · Zbl 0106.04003 · doi:10.1112/plms/s3-12.1.179 [4] W. Duke, J. Friedlander, and H. Iwaniec, ”Bounds for automorphic \(L\)-functions,” Invent. Math., vol. 112, iss. 1, pp. 1-8, 1993. · Zbl 0765.11038 · doi:10.1007/BF01232422 [5] D. Goldfeld, Automorphic Forms and \(L\)-Functions for the Group \({ GL}(n,\mathbb R)\), Cambridge: Cambridge Univ. Press, 2006, vol. 99. · Zbl 1108.11039 · doi:10.1017/CBO9780511542923 [6] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53. · Zbl 1059.11001 [7] H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, ”Rankin-Selberg convolutions,” Amer. J. Math., vol. 105, iss. 2, pp. 367-464, 1983. · Zbl 0525.22018 · doi:10.2307/2374264 [8] X. Li, ”Bounds for \({ GL}(3)\times { GL}(2)\) \(L\)-functions and \({ GL}(3)\) \(L\)-functions,” Ann. of Math., vol. 173, iss. 1, pp. 301-336, 2011. · Zbl 1320.11046 · doi:10.4007/annals.2011.173.1.8 [9] S. D. Miller, ”Cancellation in additively twisted sums on \({ GL}(n)\),” Amer. J. Math., vol. 128, iss. 3, pp. 699-729, 2006. · Zbl 1142.11033 · doi:10.1353/ajm.2006.0027 [10] S. D. Miller and W. Schmid, ”Automorphic distributions, \(L\)-functions, and Voronoi summation for \({ GL}(3)\),” Ann. of Math., vol. 164, iss. 2, pp. 423-488, 2006. · Zbl 1162.11341 · doi:10.4007/annals.2006.164.423 [11] R. Munshi, ”Bounds for twisted symmetric square \(L\)-functions,” J. Reine Angew. Math., vol. 682, pp. 65-88, 2013. · Zbl 1330.11032 · doi:10.1515/crelle-2012-0038 [12] R. Munshi, Bounds for twisted symmetric square \(L\)-functions - II. · Zbl 1330.11032 · doi:10.1515/crelle-2012-0038 [13] R. Munshi, ”Bounds for twisted symmetric square \(L\)-functions - III,” Adv. Math., vol. 235, pp. 74-91, 2013. · Zbl 1271.11055 · doi:10.1016/j.aim.2012.11.010 [14] R. Munshi, ”The circle method and bounds for \(L\)-functions - I,” Math. Ann., vol. 358, iss. 1-2, pp. 389-401, 2014. · Zbl 1312.11037 · doi:10.1007/s00208-013-0968-4 [15] R. Munshi, ”The circle method and bounds for \(L\)-functions, II: Subconvexity and twists of GL(3) \(L\)-functions,” Amer. J. Math., vol. 137, pp. 791-812, 2015. · Zbl 1344.11042 · doi:10.1353/ajm.2015.0018 [16] R. Munshi, The circle method and bounds for \(L\)-functions-III. \(t\)-aspect subconvexity for GL(3) \(L\)-functions, 2013. · Zbl 1354.11036 · doi:10.1090/jams/843 [17] R. Munshi, Hybrid subconvexity for Rankin-Selberg \(L\)-functions. · Zbl 1305.11036 [18] R. Holowinsky, R. Munshi, and Z. Qi, Hybrid subconvexity bounds for \(L(\tfrac12,\mathrm{Sym}^2 f\otimes g)\), 2014. · Zbl 1408.11045 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.