Abstract
Let $\pi$ be an $\mathrm{SL}(3,\mathbb Z)$ Hecke-Maass cusp form satisfying the Ramanujan conjecture and the Selberg-Ramanujan conjecture, and let $\chi$ be a primitive Dirichlet character modulo $M$, which we assume to be prime for simplicity. We will prove that there is a computable absolute constant $\delta>0$ such that $$ L\left(\tfrac{1}{2},\pi\otimes\chi\right)\ll_{\pi} M^{\frac{3}{4}-\delta}. $$
-
[B]
V. Blomer, "Subconvexity for twisted $L$-functions on ${ GL}(3)$," Amer. J. Math., vol. 134, iss. 5, pp. 1385-1421, 2012.
@article{B, mrkey = {2975240},
author = {Blomer, Valentin},
title = {Subconvexity for twisted {$L$}-functions on {${\rm GL}(3)$}},
journal = {Amer. J. Math.},
fjournal = {American Journal of Mathematics},
volume = {134},
year = {2012},
number = {5},
pages = {1385--1421},
issn = {0002-9327},
coden = {AJMAAN},
mrclass = {11F67 (11F70)},
mrnumber = {2975240},
mrreviewer = {Jannis A. Antoniadis},
doi = {10.1353/ajm.2012.0032},
zblnumber = {1297.11046},
} -
[DB] D. Bump, Automorphic Forms on ${ GL}(3,{\bf R})$, New York: Springer-Verlag, 1984, vol. 1083.
@book{DB, mrkey = {0765698},
author = {Bump, Daniel},
title = {Automorphic Forms on {${\rm GL}(3,{\bf R})$}},
series = {Lecture Notes in Math.},
volume = {1083},
publisher = {Springer-Verlag},
year = {1984},
pages = {xi+184},
isbn = {3-540-13864-1},
mrclass = {11F55 (11F70)},
mrnumber = {0765698},
mrreviewer = {Stephen Gelbart},
address = {New York},
zblnumber = {0543.22005},
} -
[Bu]
D. A. Burgess, "On character sums and primitive roots," Proc. London Math. Soc., vol. 12, pp. 179-192, 1962.
@article{Bu, mrkey = {0132732},
author = {Burgess, D. A.},
title = {On character sums and primitive roots},
journal = {Proc. London Math. Soc.},
fjournal = {Proceedings of the London Mathematical Society. Third Series},
volume = {12},
year = {1962},
pages = {179--192},
issn = {0024-6115},
mrclass = {10.41},
mrnumber = {0132732},
mrreviewer = {L. Carlitz},
doi = {10.1112/plms/s3-12.1.179},
zblnumber = {0106.04003},
} -
[DFI-1]
W. Duke, J. Friedlander, and H. Iwaniec, "Bounds for automorphic $L$-functions," Invent. Math., vol. 112, iss. 1, pp. 1-8, 1993.
@article{DFI-1, mrkey = {1207474},
author = {Duke, W. and Friedlander, J. and Iwaniec, H.},
title = {Bounds for automorphic \hbox{{$L$}-functions}},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {112},
year = {1993},
number = {1},
pages = {1--8},
issn = {0020-9910},
coden = {INVMBH},
mrclass = {11F66 (11F11)},
mrnumber = {1207474},
mrreviewer = {George Gilbert},
doi = {10.1007/BF01232422},
zblnumber = {0765.11038},
} -
[G]
D. Goldfeld, Automorphic Forms and $L$-Functions for the Group ${ GL}(n,\Bbb R)$, Cambridge: Cambridge Univ. Press, 2006, vol. 99.
@book{G, mrkey = {2254662},
author = {Goldfeld, Dorian},
title = {Automorphic Forms and {$L$}-Functions for the Group {${\rm GL}(n,\bold R)$}},
series = {Cambridge Stud. Adv. Math.},
volume = {99},
note = {with an appendix by Kevin A. Broughan},
publisher = {Cambridge Univ. Press},
address = {Cambridge},
year = {2006},
pages = {xiv+493},
isbn = {978-0-521-83771-2; 0-521-83771-5},
mrclass = {11F55 (11F66 11F70 11R39)},
mrnumber = {2254662},
mrreviewer = {Emmanuel P. Royer},
doi = {10.1017/CBO9780511542923},
zblnumber = {1108.11039},
} -
[IK] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53.
@book{IK, mrkey = {2061214},
author = {Iwaniec, Henryk and Kowalski, Emmanuel},
title = {Analytic Number Theory},
series = {Amer. Math. Soc. Colloq. Publ.},
volume = {53},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {2004},
pages = {xii+615},
isbn = {0-8218-3633-1},
mrclass = {11-02 (11Fxx 11Lxx 11Mxx 11Nxx)},
mrnumber = {2061214},
mrreviewer = {K. Soundararajan},
zblnumber = {1059.11001},
} -
[JPSS]
H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, "Rankin-Selberg convolutions," Amer. J. Math., vol. 105, iss. 2, pp. 367-464, 1983.
@article{JPSS, mrkey = {0701565},
author = {Jacquet, H. and Piatetskii-Shapiro, I. I. and Shalika, J. A.},
title = {Rankin-{S}elberg convolutions},
journal = {Amer. J. Math.},
fjournal = {American Journal of Mathematics},
volume = {105},
year = {1983},
number = {2},
pages = {367--464},
issn = {0002-9327},
coden = {AJMAAN},
mrclass = {11F67 (11F70 11R39 22E55)},
mrnumber = {0701565},
mrreviewer = {Freydoon Shahidi},
doi = {10.2307/2374264},
zblnumber = {0525.22018},
} -
[L]
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.
@article{L, mrkey = {2753605},
author = {Li, Xiaoqing},
title = {Bounds for {${\rm GL}(3)\times {\rm GL}(2)$} {$L$}-functions and {${\rm GL}(3)$} {$L$}-functions},
journal = {Ann. of Math.},
fjournal = {Annals of Mathematics. Second Series},
volume = {173},
year = {2011},
number = {1},
pages = {301--336},
issn = {0003-486X},
coden = {ANMAAH},
mrclass = {11F67 (11F37 11F66)},
mrnumber = {2753605},
mrreviewer = {Wen-Wei Li},
doi = {10.4007/annals.2011.173.1.8},
zblnumber = {05960661},
} -
[M]
S. D. Miller, "Cancellation in additively twisted sums on ${ GL}(n)$," Amer. J. Math., vol. 128, iss. 3, pp. 699-729, 2006.
@article{M, mrkey = {2230922},
author = {Miller, Stephen D.},
title = {Cancellation in additively twisted sums on {${\rm GL}(n)$}},
journal = {Amer. J. Math.},
fjournal = {American Journal of Mathematics},
volume = {128},
year = {2006},
number = {3},
pages = {699--729},
issn = {0002-9327},
coden = {AJMAAN},
mrclass = {11F67 (11F70)},
mrnumber = {2230922},
mrreviewer = {Emmanuel P. Royer},
zblnumber = {1142.11033},
doi = {10.1353/ajm.2006.0027},
} -
[MS]
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.
@article{MS, mrkey = {2247965},
author = {Miller, Stephen D. and Schmid, Wilfried},
title = {Automorphic distributions, {$L$}-functions, and {V}oronoi summation for {${\rm GL}(3)$}},
journal = {Ann. of Math.},
fjournal = {Annals of Mathematics. Second Series},
volume = {164},
year = {2006},
number = {2},
pages = {423--488},
issn = {0003-486X},
coden = {ANMAAH},
mrclass = {11F66 (11F70 11M41)},
mrnumber = {2247965},
mrreviewer = {Andre Reznikov},
doi = {10.4007/annals.2006.164.423},
zblnumber = {1162.11341},
} -
[Mu1] R. Munshi, "Bounds for twisted symmetric square $L$-functions," J. Reine Angew. Math., vol. 682, pp. 65-88, 2013.
@article{Mu1, mrkey = {3181499},
author = {Munshi, Ritabrata},
title = {Bounds for twisted symmetric square {$L$}-functions},
journal = {J. Reine Angew. Math.},
fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
volume = {682},
year = {2013},
pages = {65--88},
issn = {0075-4102},
mrclass = {11F66 (11F67 11M41)},
mrnumber = {3181499},
mrreviewer = {A. Perelli},
zblnumber = {6221004},
} -
[Mu2] R. Munshi, Bounds for twisted symmetric square $L$-functions – II.
@misc{Mu2,
author = {Munshi, Ritabrata},
title = {Bounds for twisted symmetric square {$L$}-functions - {II}},
note = {unpublished},
} -
[Mu3]
R. Munshi, "Bounds for twisted symmetric square $L$-functions – III," Adv. Math., vol. 235, pp. 74-91, 2013.
@article{Mu3, mrkey = {3010051},
author = {Munshi, Ritabrata},
title = {Bounds for twisted symmetric square {$L$}-functions - {III}},
journal = {Adv. Math.},
fjournal = {Advances in Mathematics},
volume = {235},
year = {2013},
pages = {74--91},
issn = {0001-8708},
mrclass = {11F66 (11M41)},
mrnumber = {3010051},
mrreviewer = {Kazuyuki Hatada},
doi = {10.1016/j.aim.2012.11.010},
zblnumber = {1271.11055},
} -
[Mu]
R. Munshi, "The circle method and bounds for $L$-functions – I," Math. Ann., vol. 358, iss. 1-2, pp. 389-401, 2014.
@article{Mu, mrkey = {3158002},
author = {Munshi, Ritabrata},
title = {The circle method and bounds for {$L$}-functions - {I}},
journal = {Math. Ann.},
fjournal = {Mathematische Annalen},
volume = {358},
year = {2014},
number = {1-2},
pages = {389--401},
issn = {0025-5831},
mrclass = {11F66 (11M41)},
mrnumber = {3158002},
mrreviewer = {A. Perelli},
doi = {10.1007/s00208-013-0968-4},
zblnumber = {06269821},
} -
[Mu4]
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.
@article{Mu4,
author = {Munshi, Ritabrata},
title = {The circle method and bounds for {$L$}-functions, {II}: {S}ubconvexity and twists of {GL}(3) {$L$}-functions},
journal = {Amer. J. Math.},
fjournal = {American Journal of Mathematics},
volume = {137},
year = {2015},
pages = {791--812},
doi = {10.1353/ajm.2015.0018},
} -
[Mu0] R. Munshi, The circle method and bounds for $L$-functions—III. $t$-aspect subconvexity for GL(3) $L$-functions, 2013.
@misc{Mu0,
author = {Munshi, Ritabrata},
title = {The circle method and bounds for {$L$}-functions---{III}. $t$-aspect subconvexity for {GL}(3) {$L$}-functions},
arxiv = {1301.1007},
year = {2013},
} -
[Mu5] R. Munshi, Hybrid subconvexity for Rankin-Selberg $L$-functions.
@misc{Mu5,
author = {Munshi, Ritabrata},
title = {Hybrid subconvexity for {R}ankin-{S}elberg {$L$}-functions},
note = {preprint},
SORTYEAR={2015},
} -
[HMQ] R. Holowinsky, R. Munshi, and Z. Qi, Hybrid subconvexity bounds for ${L}(\tfrac{1}{2},\mathrm{{S}ym}^2 f\otimes g)$, 2014.
@misc{HMQ,
author={Holowinsky, R. and Munshi, R. and Qi, Z.},
TITLE={Hybrid subconvexity bounds for ${L}(\tfrac{1}{2},
\mathrm{{S}ym}^2 f\otimes g)$},
NOTE={preprint},
YEAR={2014},
ARXIV = {1401.6695},
}