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2014 | 12 | 7 | 976-990
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On effective determination of symmetric-square lifts

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Let F be the symmetric-square lift with Laplace eigenvalue λ F (Δ) = 1+4µ2. Suppose that |µ| ≤ Λ. We show that F is uniquely determined by the central values of Rankin-Selberg L-functions L(s, F ⋇ h), where h runs over the set of holomorphic Hecke eigen cusp forms of weight κ ≡ 0 (mod 4) with κ≍ϱ+ɛ, t9 = max {4(1+4θ)/(1−18θ), 8(2−9θ)/3(1−18θ)} for any 0 ≤ θ < 1/18 and any ∈ > 0. Here θ is the exponent towards the Ramanujan conjecture for GL2 Maass forms.
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Bibliografia
  • [1] Chinta G., Diaconu A., Determination of a GL3 cuspform by twists of central L-values, Int. Math. Res. Not., 2005, 48, 2941–2967 http://dx.doi.org/10.1155/IMRN.2005.2941
  • [2] Ganguly S., Hoffstein J., Sengupta J., Determining modular forms on SL2(ℤ) by central values of convolution L-functions, Math. Ann., 2009, 345(4), 843–857 http://dx.doi.org/10.1007/s00208-009-0380-2
  • [3] Goldfeld D., Automorphic Forms and L-functions for the Group GL(n,ℝ), Cambridge Stud. Adv. Math., 99, Cambridge University Press, Cambridge, 2006
  • [4] Goldfeld D., Li X., Voronoi formulas on GL(n), Int. Math. Res. Not., 2006, #86295
  • [5] Hoffstein J., Lockhart P., Coefficients of Maass forms and the Siegel zero, Ann. Math., 1994, 140(1), 161–181 http://dx.doi.org/10.2307/2118543
  • [6] Iwaniec H., Topics in Classical Automorphic Forms, Grad. Stud. Math., 17, American Mathematical Society, Providence, 1997
  • [7] Iwaniec H., Kowalski E., Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., 53, American Mathematical Society, Providence, 2004
  • [8] Kim H.H., Sarnak P., Appendix 2 in Functoriality for the exterior square of GL4 and the symmetric fourth of GL2, J. Amer. Math. Soc., 2003, 16(1), 139–183 http://dx.doi.org/10.1090/S0894-0347-02-00410-1
  • [9] Li J., Determination of a GL2 automorphic cuspidal representation by twists of critical L-values, J. Number Theory, 2007, 123(2), 255–289 http://dx.doi.org/10.1016/j.jnt.2006.07.014
  • [10] Liu S.-C., Determination of GL(3) cusp forms by central values of GL(3)×GL(2) L-functions, Int. Math. Res. Not., 2010, 21, 4025–4041
  • [11] Liu S.-C., Determination of GL(3) cusp forms by central values of GL(3)×GL(2) L-functions, level aspect, J. Number Theory, 2011, 131(8), 1397–1408 http://dx.doi.org/10.1016/j.jnt.2011.01.014
  • [12] Luo W., Special L-values of Rankin-Selberg convolutons, Math. Ann., 1999, 314(3), 591–600 http://dx.doi.org/10.1007/s002080050308
  • [13] Luo W., Ramakrishnan D., Determination of modular forms by twists of critical L-values, Invent. Math., 1997, 130(2), 371–398 http://dx.doi.org/10.1007/s002220050189
  • [14] Luo W., Ramakrishnan D., Determination of modular elliptic curves by Heegner points, Pacific J. Math., 1997, 181(3), 251–258 http://dx.doi.org/10.2140/pjm.1997.181.251
  • [15] Munshi R., On effective determination of modular forms by twists of critical L-values, Math. Ann., 2010, 347(4), 963–978 http://dx.doi.org/10.1007/s00208-009-0465-y
  • [16] Pi Q., Determining cusp forms by central values of Rankin-Selberg L-functions, J. Number Theory, 2010, 130(10), 2283–2292 http://dx.doi.org/10.1016/j.jnt.2010.06.002
  • [17] Pi Q., Determination of cusp forms by central values of Rankin-Selberg L-functions, Lith. Math. J., 2011, 51(4), 543–561 http://dx.doi.org/10.1007/s10986-011-9147-z
  • [18] Ramakrishnan D., Wang S., On the exceptional zeros of Rankin-Selberg L-functions, Composotio Math., 2003, 135(2), 211–244 http://dx.doi.org/10.1023/A:1021761421232
  • [19] Sun Q., On determination of GL 3 cusp forms, Acta Arith., 2012, 151(1), 39–54 http://dx.doi.org/10.4064/aa151-1-4
  • [20] Zhang Y., Determining modular forms of general level by central values of convolution L-functions, Acta Arith., 2011, 150(1), 93–103 http://dx.doi.org/10.4064/aa150-1-5
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Bibliografia
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bwmeta1.element.doi-10_2478_s11533-014-0404-3
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