The fourth power moment of automorphic 𝐿-functions for 𝐺𝐿(2) over a short interval

Ye, Yangbo

doi:10.1090/S0002-9947-05-03831-6

The fourth power moment of automorphic $L$-functions for $GL(2)$ over a short interval
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Trans. Amer. Math. Soc. 358 (2006), 2259-2268 Request permission

Abstract:

In this paper we will prove bounds for the fourth power moment in the $t$ aspect over a short interval of automorphic $L$-functions $L(s,g)$ for $GL(2)$ on the central critical line Re$\ s=1/2$. Here $g$ is a fixed holomorphic or Maass Hecke eigenform for the modular group $SL_{2}(\mathbb {Z})$, or in certain cases, for the Hecke congruence subgroup $\Gamma _{0}({\mathcal {N}})$ with $\mathcal {N}>1$. The short interval is from a large $K$ to $K+K^{103/135+\varepsilon }$. The proof is based on an estimate in the proof of subconvexity bounds for Rankin-Selberg $L$-function for Maass forms by Jianya Liu and Yangbo Ye (2002) and Yuk-Kam Lau, Jianya Liu, and Yangbo Ye (2004), which in turn relies on the Kuznetsov formula (1981) and bounds for shifted convolution sums of Fourier coefficients of a cusp form proved by Sarnak (2001) and by Lau, Liu, and Ye (2004).

References

A. O. L. Atkin and Wen Ch’ing Winnie Li, Twists of newforms and pseudo-eigenvalues of $W$-operators, Invent. Math. 48 (1978), no. 3, 221–243. MR 508986, DOI 10.1007/BF01390245
J. B. Conrey and H. Iwaniec, The cubic moment of central values of automorphic $L$-functions, Ann. of Math. (2) 151 (2000), no. 3, 1175–1216. MR 1779567, DOI 10.2307/121132
W. Duke, J. Friedlander, and H. Iwaniec, Bounds for automorphic $L$-functions, Invent. Math. 112 (1993), no. 1, 1–8. MR 1207474, DOI 10.1007/BF01232422
W. Duke, J. B. Friedlander, and H. Iwaniec, Bounds for automorphic $L$-functions. II, Invent. Math. 115 (1994), no. 2, 219–239. MR 1258904, DOI 10.1007/BF01231759
W. Duke, J. B. Friedlander, and H. Iwaniec, Bounds for automorphic $L$-functions. III, Invent. Math. 143 (2001), no. 2, 221–248. MR 1835388, DOI 10.1007/s002220000104
A. Good, Beiträge zur Theorie der Dirichletreihen, die Spitzenformen zugeordnet sind, J. Number Theory 13 (1981), no. 1, 18–65 (German, with English summary). MR 602447, DOI 10.1016/0022-314X(81)90028-7
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
A. Good, The convolution method for Dirichlet series, The Selberg trace formula and related topics (Brunswick, Maine, 1984) Contemp. Math., vol. 53, Amer. Math. Soc., Providence, RI, 1986, pp. 207–214. MR 853560, DOI 10.1090/conm/053/853560
D. R. Heath-Brown, The fourth power moment of the Riemann zeta function, Proc. London Math. Soc. (3) 38 (1979), no. 3, 385–422. MR 532980, DOI 10.1112/plms/s3-38.3.385
Aleksandar Ivić, The Riemann zeta-function, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985. The theory of the Riemann zeta-function with applications. MR 792089
Henryk Iwaniec, Fourier coefficients of cusp forms and the Riemann zeta-function, Seminar on Number Theory, 1979–1980 (French), Univ. Bordeaux I, Talence, 1980, pp. Exp. No. 18, 36. MR 604215
Henry H. Kim, Functoriality for the exterior square of $\textrm {GL}_4$ and the symmetric fourth of $\textrm {GL}_2$, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR 1937203, DOI 10.1090/S0894-0347-02-00410-1
Henry H. Kim, Functoriality for the exterior square of $\textrm {GL}_4$ and the symmetric fourth of $\textrm {GL}_2$, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR 1937203, DOI 10.1090/S0894-0347-02-00410-1
Henry H. Kim and Freydoon Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), no. 1, 177–197. MR 1890650, DOI 10.1215/S0012-9074-02-11215-0
E. Kowalski, P. Michel, and J. VanderKam, Mollification of the fourth moment of automorphic $L$-functions and arithmetic applications, Invent. Math. 142 (2000), no. 1, 95–151. MR 1784797, DOI 10.1007/s002220000086
N. V. Kuznecov, The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums, Mat. Sb. (N.S.) 111(153) (1980), no. 3, 334–383, 479 (Russian). MR 568983
Yuk-Kam Lau, Jianya Liu, and Yangbo Ye, Subconvexity bounds for Rankin-Selberg $L$-functions for congruence subgroups, to appear in J. Number Theory.
Jianya Liu and Yangbo Ye, Subconvexity for Rankin-Selberg $L$-functions of Maass forms, Geom. Funct. Anal. 12 (2002), no. 6, 1296–1323. MR 1952930, DOI 10.1007/s00039-002-1296-0
Jianya Liu and Yangbo Ye, Petersson and Kuznetsov trace formulas, in Lie Groups and Automorphic Forms, Amer. Math. Soc. and International Press, 2005.
Wenzhi Luo, Zeév Rudnick, and Peter Sarnak, On the generalized Ramanujan conjecture for $\textrm {GL}(n)$, Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996) Proc. Sympos. Pure Math., vol. 66, Amer. Math. Soc., Providence, RI, 1999, pp. 301–310. MR 1703764
Tom Meurman, On the order of the Maass $L$-function on the critical line, Number theory, Vol. I (Budapest, 1987) Colloq. Math. Soc. János Bolyai, vol. 51, North-Holland, Amsterdam, 1990, pp. 325–354. MR 1058223
Peter Sarnak, Estimates for Rankin-Selberg $L$-functions and quantum unique ergodicity, J. Funct. Anal. 184 (2001), no. 2, 419–453. MR 1851004, DOI 10.1006/jfan.2001.3783
Freydoon Shahidi, On the Ramanujan conjecture and finiteness of poles for certain $L$-functions, Ann. of Math. (2) 127 (1988), no. 3, 547–584. MR 942520, DOI 10.2307/2007005