Sup-norm bounds for Eisenstein series. (English) Zbl 1433.11042
Summary: The paper deals with establishing bounds for Eisenstein series on congruence quotients of the upper half plane, with control of both the spectral parameter and the level. The key observation in this work is that we exploit better the structure of the amplifier by just supporting on primes for the Eisenstein series, which can use both the analytic method as Young did to get a lower bound for the amplifier and the geometric method as Harcos-Templier did to obtain a more efficient treatment for the counting problem.
MSC:
| 11F12 | Automorphic forms, one variable |
References:
| [1] | C. B. Balogh, Asymptotic expansions of the modified Bessel function of the third kind of imaginary order, SIAM J. Appl. Math. 15 (1967), 1315-1323.; Balogh, C. B., Asymptotic expansions of the modified Bessel function of the third kind of imaginary order, SIAM J. Appl. Math., 15, 1315-1323 (1967) · Zbl 0157.12303 |
| [2] | V. Blomer, G. Harcos, P. Maga and D. Milićević, The sup-norm problem for \(\text{GL}(2)\) over number fields, preprint (2016), .; Blomer, V.; Harcos, G.; Maga, P.; Milićević, D., The sup-norm problem for \(\operatorname{GL} <mml:mo stretchy=''false``>(2 <mml:mo stretchy=''false``>)\) over number fields (2016) · Zbl 1442.11083 |
| [3] | V. Blomer and R. Holowinsky, Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (2010), no. 3, 645-681.; Blomer, V.; Holowinsky, R., Bounding sup-norms of cusp forms of large level, Invent. Math., 179, 3, 645-681 (2010) · Zbl 1243.11059 |
| [4] | J. B. Conrey and H. Iwaniec, The cubic moment of central values of automorphic -functions, Ann. of Math. (2)151 (2000), no. 3, 1175-1216.; 2020-01-09 09:30:06 Citation within text. · Zbl 0973.11056 |
| [5] | W. Duke, J. B. Friedlander and H. Iwaniec, The subconvexity problem for Artin L-functions, Invent. Math. 149 (2002), no. 3, 489-577.; Duke, W.; Friedlander, J. B.; Iwaniec, H., The subconvexity problem for Artin L-functions, Invent. Math., 149, 3, 489-577 (2002) · Zbl 1056.11072 |
| [6] | É. Fouvry, E. Kowalski and P. Michel, Algebraic trace functions over the primes, Duke Math. J. 163 (2014), no. 9, 1683-1736.; Fouvry, É.; Kowalski, E.; Michel, P., Algebraic trace functions over the primes, Duke Math. J., 163, 9, 1683-1736 (2014) · Zbl 1318.11103 |
| [7] | G. Harcos and N. Templier, On the sup-norm of Maass cusp forms of large level: II, Int. Math. Res. Not. IMRN 2012 (2012), no. 20, 4764-4774.; Harcos, G.; Templier, N., On the sup-norm of Maass cusp forms of large level: II, Int. Math. Res. Not. IMRN, 2012, 20, 4764-4774 (2012) · Zbl 1332.11048 |
| [8] | G. Harcos and N. Templier, On the sup-norm of Maass cusp forms of large level. III, Math. Ann. 356 (2013), no. 1, 209-216.; Harcos, G.; Templier, N., On the sup-norm of Maass cusp forms of large level. III, Math. Ann., 356, 1, 209-216 (2013) · Zbl 1332.11049 |
| [9] | H. Iwaniec, Spectral Methods of Automorphic Forms, 2nd ed., Grad. Stud. Math. 53, American Mathematical Society, Providence, 2002.; Iwaniec, H., Spectral Methods of Automorphic Forms (2002) · Zbl 1006.11024 |
| [10] | H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, American Mathematical Society, Providence, 2004.; Iwaniec, H.; Kowalski, E., Analytic Number Theory (2004) · Zbl 1059.11001 |
| [11] | H. Iwaniec and P. Sarnak, \(L^{\infty}\) norms of eigenfunctions of arithmetic surfaces, Ann. of Math. (2) 141 (1995), no. 2, 301-320.; Iwaniec, H.; Sarnak, P., \(L^{\infty}\) norms of eigenfunctions of arithmetic surfaces, Ann. of Math. (2), 141, 2, 301-320 (1995) · Zbl 0833.11019 |
| [12] | S. Ramanujan, Some formulæ in the analytic theory of numbers [Messenger Math. 45 (1916), 81-84], Collected Papers of Srinivasa Ramanujan, American Mathematical Society, Providence (2000), 133-135.; Ramanujan, S., Some formulæ in the analytic theory of numbers [Messenger Math. 45 (1916), 81-84], Collected Papers of Srinivasa Ramanujan, 133-135 (2000) · JFM 45.1250.01 |
| [13] | N. Templier, On the sup-norm of Maass cusp forms of large level, Selecta Math. (N.S.) 16 (2010), no. 3, 501-531.; Templier, N., On the sup-norm of Maass cusp forms of large level, Selecta Math. (N.S.), 16, 3, 501-531 (2010) · Zbl 1260.11032 |
| [14] | N. Templier, Hybrid sup-norm bounds for Hecke-Maass cusp forms, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 8, 2069-2082.; Templier, N., Hybrid sup-norm bounds for Hecke-Maass cusp forms, J. Eur. Math. Soc. (JEMS), 17, 8, 2069-2082 (2015) · Zbl 1376.11030 |
| [15] | M. Young, A note on the sup norm of Eisenstein series, preprint (2016), .; Young, M., A note on the sup norm of Eisenstein series (2016) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
