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Counting integer points on quadrics with arithmetic weights. (English) Zbl 1465.11196

Summary: Let \(F \in \mathbf{Z}[\mathbb{x}]\) be a diagonal, non-singular quadratic form in four variables. Let \(\lambda (n)\) be the normalised Fourier coefficients of a holomorphic Hecke form of full level. We give an upper bound for the problem of counting integer zeros of \(F\) with \(\vert\mathbf{x}\vert \leqslant X\), weighted by \(\lambda (x_1)\).

MSC:

11P55 Applications of the Hardy-Littlewood method
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11F30 Fourier coefficients of automorphic forms
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[1] Baier, S.; Browning, T. D., Inhomogeneous quadratic congruences, Funct. Approx. Comment. Math., 47, part 2, 267-286 (2012) · Zbl 1351.11063 · doi:10.7169/facm/2012.47.2.9
[2] Blomer, Valentin, Sums of Hecke eigenvalues over values of quadratic polynomials, Int. Math. Res. Not. IMRN, 16, Art. ID rnn059. 29 pp. (2008) · Zbl 1232.11053 · doi:10.1093/imrn/rnn059
[3] Blomer, Valentin; Harcos, Gergely, Hybrid bounds for twisted \(L\)-functions, J. Reine Angew. Math., 621, 53-79 (2008) · Zbl 1193.11044 · doi:10.1515/CRELLE.2008.058
[4] Booker, Andrew R.; Milinovich, Micah B.; Ng, Nathan, Subconvexity for modular form \(L\)-functions in the \(t\) aspect, Adv. Math., 341, 299-335 (2019) · Zbl 1469.11280 · doi:10.1016/j.aim.2018.10.037
[5] de la Bret\`eche, R.; Browning, T. D., Le probl\`eme des diviseurs pour des formes binaires de degr\'{e} 4, J. Reine Angew. Math., 646, 1-44 (2010) · Zbl 1204.11158 · doi:10.1515/CRELLE.2010.064
[6] de la Bret\`eche, R.; Browning, T. D., Binary forms as sums of two squares and Ch\^atelet surfaces, Israel J. Math., 191, 2, 973-1012 (2012) · Zbl 1293.11058 · doi:10.1007/s11856-012-0019-y
[7] de la Bret\`eche, R\'{e}gis; Tenenbaum, G\'{e}rald, Moyennes de fonctions arithm\'{e}tiques de formes binaires, Mathematika, 58, 2, 290-304 (2012) · Zbl 1284.11126 · doi:10.1112/S0025579311002154
[8] Browning, Tim, The divisor problem for binary cubic forms, J. Th\'{e}or. Nombres Bordeaux, 23, 3, 579-602 (2011) · Zbl 1271.11091 · doi:10.5802/jtnb.778
[9] Daniel, Stephan, On the divisor-sum problem for binary forms, J. Reine Angew. Math., 507, 107-129 (1999) · Zbl 0913.11041 · doi:10.1515/crll.1999.010
[10] Duke, W.; Friedlander, J.; Iwaniec, H., Bounds for automorphic \(L\)-functions, Invent. Math., 112, 1, 1-8 (1993) · Zbl 0765.11038 · doi:10.1007/BF01232422
[11] Fouvry, \'{E}tienne; Ganguly, Satadal; Kowalski, Emmanuel; Michel, Philippe, Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions, Comment. Math. Helv., 89, 4, 979-1014 (2014) · Zbl 1306.11079 · doi:10.4171/CMH/342
[12] Friedlander, John; Iwaniec, Henryk, The polynomial \(X^2+Y^4\) captures its primes, Ann. of Math. (2), 148, 3, 945-1040 (1998) · Zbl 0926.11068 · doi:10.2307/121034
[13] Getz, Jayce R., Secondary terms in asymptotics for the number of zeros of quadratic forms over number fields, J. Lond. Math. Soc. (2), 98, 2, 275-305 (2018) · Zbl 1411.11032 · doi:10.1112/jlms.12130
[14] Greaves, G., On the divisor-sum problem for binary cubic forms, Acta Arith., 17, 1-28 (1970) · Zbl 0198.37903 · doi:10.4064/aa-17-1-1-28
[15] Heath-Brown, D. R., A new form of the circle method, and its application to quadratic forms, J. Reine Angew. Math., 481, 149-206 (1996) · Zbl 0857.11049 · doi:10.1515/crll.1996.481.149
[16] Heath-Brown, D. R., Primes represented by \(x^3+2y^3\), Acta Math., 186, 1, 1-84 (2001) · Zbl 1007.11055 · doi:10.1007/BF02392715
[17] Heath-Brown, D. R., Linear relations amongst sums of two squares. Number theory and algebraic geometry, London Math. Soc. Lecture Note Ser. 303, 133-176 (2003), Cambridge Univ. Press, Cambridge · Zbl 1161.11387
[18] Heath-Brown, D. R.; Pierce, L. B., Simultaneous integer values of pairs of quadratic forms, J. Reine Angew. Math., 727, 85-143 (2017) · Zbl 1396.11121 · doi:10.1515/crelle-2014-0112
[19] Hooley, Christopher, On the number of divisors of a quadratic polynomial, Acta Math., 110, 97-114 (1963) · Zbl 0116.03802 · doi:10.1007/BF02391856
[20] Iwaniec, Henryk; Kowalski, Emmanuel, Analytic number theory, American Mathematical Society Colloquium Publications 53, xii+615 pp. (2004), American Mathematical Society, Providence, RI · Zbl 1059.11001 · doi:10.1090/coll/053
[21] Kim, Henry H., Functoriality for the exterior square of \(\text{GL}_4\) and the symmetric fourth of \(\text{GL}_2\), J. Amer. Math. Soc., 16, 1, 139-183 (2003) · Zbl 1018.11024 · doi:10.1090/S0894-0347-02-00410-1
[22] Munshi, Ritabrata, The circle method and bounds for \(L\)-functions-I, Math. Ann., 358, 1-2, 389-401 (2014) · Zbl 1312.11037 · doi:10.1007/s00208-013-0968-4
[23] Stein, Elias M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, xiv+695 pp. (1993), Princeton University Press, Princeton, NJ · Zbl 0821.42001
[24] Templier, Nicolas; Tsimerman, Jacob, Non-split sums of coefficients of \(GL(2)\)-automorphic forms, Israel J. Math., 195, 2, 677-723 (2013) · Zbl 1334.11035 · doi:10.1007/s11856-012-0112-2
[25] Tsang, Kai-Man; Zhao, Lilu, On Lagrange’s four squares theorem with almost prime variables, J. Reine Angew. Math., 726, 129-171 (2017) · Zbl 1372.11096 · doi:10.1515/crelle-2014-0094
[26] Watson, G. N., A treatise on the theory of Bessel functions, Cambridge Mathematical Library, viii+804 pp. (1995), Cambridge University Press, Cambridge · Zbl 0849.33001
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