Kevin, what sheer malice - 1999

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

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, eat. 5, pp. 1385-1421, 2012.
  • [DB] D. Bump, Automorphic Forms on $ {GL} (3, {\ bf R}) $, New York: Springer-Verlag, 1984, vol. 1083
  • [Bu] D. A. Burgess, "On character sums and primitive roots," Proc. London Math. Soc., vol. 12, pp. 179-192, 1962.
  • [DFI-1] W. Duke, J. Friedlander, and H. Iwaniec, "Bounds for automorphic $ L $ -functions," Invent. Math., vol. 112, eat. 1, pp. 1-8, 1993.
  • [G] D. Goldfeld, Automorphic Forms and $ L $ -Functions for the Group $ {GL} (n, \ Bbb R) $, Cambridge: Cambridge Univ. Press, 2006, vol. 99
  • [IK] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53.
  • [JPSS] H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, "Rankin-Selberg convolutions," Amer. J. Math., vol. 105, eat. 2, pp. 367-464, 1983.
  • [L] X. Li, "Bounds for $ {GL} (3) \ times {GL} (2) $ $ L $ -functions and $ {GL} (3) $ $ L $ -functions," Ann. of Math., vol. 173, eat. 1, pp. 301-336, 2011.
  • [M] S. D. Miller, "Cancellation in additively twisted sums on $ {GL} (n) $," Amer. J. Math., vol. 128, eat. 3, pp. 699-729, 2006.
  • [MS] S. D. Miller and W. Schmid, "Automorphic distributions, $ L $ -functions, and Voronoi summation for $ {GL} (3) $," Ann. of Math., vol. 164, eat. 2, pp. 423-488, 2006.
  • [Mu1] R. Munshi, "Bounds for twisted symmetric square $ L $ -functions," J. Reine Angew. Math., vol. 682, pp. 65-88, 2013.
  • [Mu2] R. Munshi, Bounds for twisted symmetric square $ L $ -functions - II.
  • [Mu3] R. Munshi, "Bounds for twisted symmetric square $ L $ -functions - III," Adv. Math., vol. 235, pp. 74-91, 2013.
  • [Mu] R. Munshi, "The circle method and bounds for $ L $ -functions - I," Math. Ann., vol. 358, eat. 1-2, pp. 389-401, 2014.
  • [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.
  • [Mu0] R. Munshi, The circle method and bounds for $ L $ functions — III. $ t $ -aspect subconvexity for GL (3) $ L $ -functions, 2013.
  • [Mu5] R. Munshi, Hybrid subconvexity for Rankin-Selberg $ L $ functions.
  • [HMQ] R. Holowinsky, R. Munshi, and Z. Qi, Hybrid subconvexity bounds for $ {L} (\ tfrac {1} {2}, \ mathrm {{S} ym} ^ 2 f \ otimes g) $, 2014.

Authors

Ritabrata Munshi Tata Institute of Fundamental Research, Colaba, Mumbai, India