## The circle method and bounds for $L$-functions—III: $t$-aspect subconvexity for $GL(3)$ $L$-functions

HTML articles powered by AMS MathViewer

- by Ritabrata Munshi PDF
- J. Amer. Math. Soc.
**28**(2015), 913-938 Request permission

## Abstract:

Let $\pi$ be a Hecke-Maass cusp form for $SL(3,\mathbb {Z})$. In this paper we will prove the following subconvex bound: \[ L\left (\tfrac {1}{2}+it,\pi \right )\ll _{\pi ,\varepsilon } (1+|t|)^{\frac {3}{4}-\frac {1}{16}+\varepsilon }. \]## References

- Valentin Blomer,
*Subconvexity for twisted $L$-functions on $\textrm {GL}(3)$*, Amer. J. Math.**134**(2012), no. 5, 1385–1421. MR**2975240**, DOI 10.1353/ajm.2012.0032 - Dorian Goldfeld,
*Automorphic forms and $L$-functions for the group $\textrm {GL}(n,\mathbf R)$*, Cambridge Studies in Advanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006. With an appendix by Kevin A. Broughan. MR**2254662**, DOI 10.1017/CBO9780511542923 - Anton Good,
*The square mean of Dirichlet series associated with cusp forms*, Mathematika**29**(1982), no. 2, 278–295 (1983). MR**696884**, DOI 10.1112/S0025579300012377 - M. N. Huxley,
*On stationary phase integrals*, Glasgow Math. J.**36**(1994), no. 3, 355–362. MR**1295511**, DOI 10.1017/S0017089500030962 - Henryk Iwaniec and Emmanuel Kowalski,
*Analytic number theory*, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR**2061214**, DOI 10.1090/coll/053 - Hervé Jacquet and Joseph Shalika,
*Rankin-Selberg convolutions: Archimedean theory*, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 125–207. MR**1159102** - Xiaoqing Li,
*Bounds for $\textrm {GL}(3)\times \textrm {GL}(2)$ $L$-functions and $\textrm {GL}(3)$ $L$-functions*, Ann. of Math. (2)**173**(2011), no. 1, 301–336. MR**2753605**, DOI 10.4007/annals.2011.173.1.8 - Stephen D. Miller and Wilfried Schmid,
*Automorphic distributions, $L$-functions, and Voronoi summation for $\textrm {GL}(3)$*, Ann. of Math. (2)**164**(2006), no. 2, 423–488. MR**2247965**, DOI 10.4007/annals.2006.164.423 - Ritabrata Munshi,
*Bounds for twisted symmetric square $L$-functions*, J. Reine Angew. Math.**682**(2013), 65–88. MR**3181499**, DOI 10.1515/crelle.2012.029 - R. Munshi,
*Bounds for twisted symmetric square $L$-functions—II*. Unpublished. - Ritabrata Munshi,
*Bounds for twisted symmetric square $L$-functions—III*, Adv. Math.**235**(2013), 74–91. MR**3010051**, DOI 10.1016/j.aim.2012.11.010 - Ritabrata Munshi,
*The circle method and bounds for $L$-functions—I*, Math. Ann.**358**(2014), no. 1-2, 389–401. MR**3158002**, DOI 10.1007/s00208-013-0968-4 - Ritabrata Munshi,
*The circle method and bounds for $L$-functions—II: Subconvexity for twists of $GL(3)$ $L$-functions*, Amer. J. Math.**137**(2015), no. 3, 791–812., DOI 10.1353/ajm.2015.0018 - B. R. Srinivasan,
*The lattice point problem of many dimensional hyperboloids. III*, Math. Ann.**160**(1965), 280–311. MR**181614**, DOI 10.1007/BF01371611 - H. Weyl,
*Zur Abschätzung von $\zeta (1+ti)$*, Math. Z.**10**(1921), 88–101., DOI 10.1007/BF02102307

## Additional Information

**Ritabrata Munshi**- Affiliation: School of Mathematics, Tata Institute of Fundamental Research, 1 Dr. Homi Bhabha Road, Colaba, Mumbai 400005, India
- MR Author ID: 817043
- Email: rmunshi@math.tifr.res.in
- Received by editor(s): March 31, 2014
- Published electronically: July 13, 2015
- © Copyright 2015 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**28**(2015), 913-938 - MSC (2010): Primary 11F66, 11M41; Secondary 11F55
- DOI: https://doi.org/10.1090/jams/843
- MathSciNet review: 3369905