The second moment theory of families of \(L\)-functions-the case of twisted Hecke \(L\)-functions. (English) Zbl 1519.11001
Memoirs of the American Mathematical Society 1394. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-5678-8/pbk; 978-1-4704-7350-1/ebook). v, 148 p. (2023).
After giving an overview of their book, the authors discuss some preliminary material in Chapter 2, such as Hecke \(L\)-functions, Rankin-Selberg \(L\)-functions, and the prime number theorem. Chapter 3 presents material on exponential and character sums (in particular Kloosterman sums) occurring in the book. In the next two chapters, estimates of the first and second twisted moment of a family of \(L\)-functions are proved; the rest of the book presents the proofs of the main theorems.
For stating them, let \(f\) be a primitive cusp form with respect to some congruence subgroup \(\Gamma_0(r)\) with trivial central character \(\chi_r\), and let \(\chi\) denote a primitive Dirichlet character defined modulo \(q\). Then (1) a positive proportion of central values \(L(f \otimes \chi, \frac12)\) are nonzero, and actually can be bounded from below, (2) there exist characters for which the central value is very large, and (3) the second moments estimates establish a special case of a conjecture of Mazur and Rubin on the distribution of modular symbols. These twisted moments estimates provide asymptotic expansions of \(\sum \varepsilon_\chi^k \chi(\ell) L(f \otimes \chi, \frac12)\), where \(\varepsilon_\chi\) is a normalized Gauss sum, and of \(\sum |L(f \otimes \chi, s)|^2\), where the sums are over all primitive Dirichlet characters \(\chi\) modulo \(q\).
For stating them, let \(f\) be a primitive cusp form with respect to some congruence subgroup \(\Gamma_0(r)\) with trivial central character \(\chi_r\), and let \(\chi\) denote a primitive Dirichlet character defined modulo \(q\). Then (1) a positive proportion of central values \(L(f \otimes \chi, \frac12)\) are nonzero, and actually can be bounded from below, (2) there exist characters for which the central value is very large, and (3) the second moments estimates establish a special case of a conjecture of Mazur and Rubin on the distribution of modular symbols. These twisted moments estimates provide asymptotic expansions of \(\sum \varepsilon_\chi^k \chi(\ell) L(f \otimes \chi, \frac12)\), where \(\varepsilon_\chi\) is a normalized Gauss sum, and of \(\sum |L(f \otimes \chi, s)|^2\), where the sums are over all primitive Dirichlet characters \(\chi\) modulo \(q\).
Reviewer: Franz Lemmermeyer (Jagstzell)
MSC:
| 11-02 | Research exposition (monographs, survey articles) pertaining to number theory |
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 11F11 | Holomorphic modular forms of integral weight |
| 11F12 | Automorphic forms, one variable |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11L05 | Gauss and Kloosterman sums; generalizations |
| 11L40 | Estimates on character sums |
| 11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |
| 11T23 | Exponential sums |
Keywords:
\(L\)-functions; modular forms; special values of \(L\)-functions; moments; mollification; analytic rank; shifted convolution sums; root number; Kloosterman sums; resonator methodReferences:
| [1] | Aistleitner, Christoph, Large values of L-functions from the Selberg class, J. Math. Anal. Appl., 345-364 (2017) · Zbl 1359.11075 · doi:10.1016/j.jmaa.2016.08.044 |
| [2] | Bagchi, B., Statistical behaviour and universality properties of the Riemann zeta function and other allied Dirichlet series (1981) |
| [3] | Blomer, Valentin, Shifted convolution sums and subconvexity bounds for automorphic \(L\)-functions, Int. Math. Res. Not., 3905-3926 (2004) · Zbl 1066.11019 · doi:10.1155/S1073792804142505 |
| [4] | Blomer, Valentin, On moments of twisted \(L\)-functions, Amer. J. Math., 707-768 (2017) · Zbl 1476.11081 · doi:10.1353/ajm.2017.0019 |
| [5] | Blomer, Valentin, Hybrid bounds for twisted \(L\)-functions, J. Reine Angew. Math., 53-79 (2008) · Zbl 1193.11044 · doi:10.1515/CRELLE.2008.058 |
| [6] | Blomer, Valentin, The second moment of twisted modular \(L\)-functions, Geom. Funct. Anal., 453-516 (2015) · Zbl 1400.11097 · doi:10.1007/s00039-015-0318-7 |
| [7] | Burgess, D. A., On character sums and \(L\)-series, Proc. London Math. Soc. (3), 193-206 (1962) · Zbl 0106.04004 · doi:10.1112/plms/s3-12.1.193 |
| [8] | Bushnell, C. J., An upper bound on conductors for pairs, J. Number Theory, 183-196 (1997) · Zbl 0884.11049 · doi:10.1006/jnth.1997.2142 |
| [9] | Bykovski\u{\i }, V. A., A trace formula for the scalar product of Hecke series and its applications, J. Math. Sci. (New York). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 14-36, 235-236 (1996) · Zbl 0898.11017 · doi:10.1007/BF02358528 |
| [10] | Chinta, Gautam, Analytic ranks of elliptic curves over cyclotomic fields, J. Reine Angew. Math., 13-24 (2002) · Zbl 1028.11040 · doi:10.1515/crll.2002.021 |
| [11] | Coates, J., Iwasawa theory for the symmetric square of an elliptic curve, J. Reine Angew. Math., 104-156 (1987) · Zbl 0609.14013 · doi:10.1515/crll.1987.375-376.104 |
| [12] | Conrey, J. B., Integral moments of \(L\)-functions, Proc. London Math. Soc. (3), 33-104 (2005) · Zbl 1075.11058 · doi:10.1112/S0024611504015175 |
| [13] | Conrey, J. B., Real zeros of quadratic Dirichlet \(L\)-functions, Invent. Math., 1-44 (2002) · Zbl 1042.11053 · doi:10.1007/s00222-002-0227-x |
| [14] | Deligne, Pierre, La conjecture de Weil. I, Inst. Hautes \'{E}tudes Sci. Publ. Math., 273-307 (1974) · Zbl 0287.14001 |
| [15] | Deligne, Pierre, La conjecture de Weil. II, Inst. Hautes \'{E}tudes Sci. Publ. Math., 137-252 (1980) · Zbl 0456.14014 |
| [16] | Duke, W., Bounds for automorphic \(L\)-functions, Invent. Math., 1-8 (1993) · Zbl 0765.11038 · doi:10.1007/BF01232422 |
| [17] | Fouvry, \'{E}tienne, Sur le probl\`eme des diviseurs de Titchmarsh, J. Reine Angew. Math., 51-76 (1985) · Zbl 0547.10039 · doi:10.1515/crll.1985.357.51 |
| [18] | Fouvry, \'{E}tienne, Algebraic trace functions over the primes, Duke Math. J., 1683-1736 (2014) · Zbl 1318.11103 · doi:10.1215/00127094-2690587 |
| [19] | Fouvry, \'{E}tienne, Colloquium De Giorgi 2013 and 2014. Trace functions over finite fields and their applications, Colloquia, 7-35 (2014), Ed. Norm., Pisa · Zbl 1408.11080 |
| [20] | Fouvry, \'{E}tienne, Algebraic twists of modular forms and Hecke orbits, Geom. Funct. Anal., 580-657 (2015) · Zbl 1344.11036 · doi:10.1007/s00039-015-0310-2 |
| [21] | Fouvry, \'{E}tienne, On the exponent of distribution of the ternary divisor function, Mathematika, 121-144 (2015) · Zbl 1317.11080 · doi:10.1112/S0025579314000096 |
| [22] | Fres\'{a}n, Javier, \'{E}quir\'{e}partition de Sommes exponentielles [travaux de Katz], Ast\'{e}risque, Exp. No. 1141, 205-250 (2019) · Zbl 1484.11219 · doi:10.24033/ast.1085 |
| [23] | Gao, Peng, The second moment of Dirichlet twists of Hecke \(L\)-functions, Acta Arith., 57-65 (2009) · Zbl 1242.11035 · doi:10.4064/aa140-1-4 |
| [24] | Gelbart, Stephen S., Automorphic forms on ad\`ele groups, x+267 pp. (1975), Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo · Zbl 0329.10018 |
| [25] | Gelbart, Stephen, A relation between automorphic representations of \(\operatorname{GL}(2)\) and \(\operatorname{GL}(3)\), Ann. Sci. \'{E}cole Norm. Sup. (4), 471-542 (1978) · Zbl 0406.10022 |
| [26] | Goldston, Daniel A., Primes in tuples. I, Ann. of Math. (2), 819-862 (2009) · Zbl 1207.11096 · doi:10.4007/annals.2009.170.819 |
| [27] | Gradshteyn, I. S., Table of integrals, series, and products, xlviii+1171 pp. (2007), Elsevier/Academic Press, Amsterdam · Zbl 1208.65001 |
| [28] | Harcos, Gergely, The subconvexity problem for Rankin-Selberg \(L\)-functions and equidistribution of Heegner points. II, Invent. Math., 581-655 (2006) · Zbl 1111.11027 · doi:10.1007/s00222-005-0468-6 |
| [29] | Harper, A. J., Sharp conditional bounds for moments of the Riemann zeta function, Preprint (2013) |
| [30] | Harris, Michael, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, viii+276 pp. (2001), Princeton University Press, Princeton, NJ · Zbl 1036.11027 |
| [31] | Heath-Brown, D. R., Exponential decay in the frequency of analytic ranks of automorphic \(L\)-functions, Duke Math. J., 475-484 (2000) · Zbl 1166.11326 · doi:10.1215/S0012-7094-00-10235-9 |
| [32] | Henniart, Guy, Correspondance de Langlands et fonctions \(L\) des carr\'{e}s ext\'{e}rieur et sym\'{e}trique, Int. Math. Res. Not. IMRN, 633-673 (2010) · Zbl 1184.22009 · doi:10.1093/imrn/rnp150 |
| [33] | Hoffstein, J., Second moments and simultaneous non-vanishing of \(\GL (2)\) automorphic \(L\)-series, Preprint (2013) |
| [34] | Hough, Bob, The angle of large values of \(L\)-functions, J. Number Theory, 353-393 (2016) · Zbl 1412.11089 · doi:10.1016/j.jnt.2016.03.008 |
| [35] | Iwaniec, Henryk, Topics in classical automorphic forms, Graduate Studies in Mathematics, xii+259 pp. (1997), American Mathematical Society, Providence, RI · Zbl 0905.11023 · doi:10.1090/gsm/017 |
| [36] | Iwaniec, Henryk, Analytic number theory, American Mathematical Society Colloquium Publications, xii+615 pp. (2004), American Mathematical Society, Providence, RI · Zbl 1059.11001 · doi:10.1090/coll/053 |
| [37] | Iwaniec, H., Number theory in progress, Vol. 2. Dirichlet \(L\)-functions at the central point, 941-952 (1997), de Gruyter, Berlin · Zbl 0929.11025 |
| [38] | Iwaniec, H., Perspectives on the analytic theory of \(L\)-functions, Geom. Funct. Anal., 705-741 (2000) · Zbl 0996.11036 · doi:10.1007/978-3-0346-0425-3\_6 |
| [39] | Jacquet, H., Rankin-Selberg convolutions, Amer. J. Math., 367-464 (1983) · Zbl 0525.22018 · doi:10.2307/2374264 |
| [40] | Katz, Nicholas M., Convolution and equidistribution, Annals of Mathematics Studies, viii+203 pp. (2012), Princeton University Press, Princeton, NJ · Zbl 1261.11084 |
| [41] | Katz, Nicholas M., Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.), 1-26 (1999) · Zbl 0921.11047 · doi:10.1090/S0273-0979-99-00766-1 |
| [42] | Keating, J. P., Random matrix theory and \(\zeta (1/2+it)\), Comm. Math. Phys., 57-89 (2000) · Zbl 1051.11048 · doi:10.1007/s002200000261 |
| [43] | Kim, Henry H., Functoriality for the exterior square of \(\text{GL}_4\) and the symmetric fourth of \(\text{GL}_2\), J. Amer. Math. Soc., 139-183 (2003) · Zbl 1018.11024 · doi:10.1090/S0894-0347-02-00410-1 |
| [44] | Kim, Henry H., Functorial products for \(\text{GL}_2\times \text{GL}_3\) and the symmetric cube for \(\text{GL}_2\), Ann. of Math. (2), 837-893 (2002) · Zbl 1040.11036 · doi:10.2307/3062134 |
| [45] | Kim, Henry H., Cuspidality of symmetric powers with applications, Duke Math. J., 177-197 (2002) · Zbl 1074.11027 · doi:10.1215/S0012-9074-02-11215-0 |
| [46] | Sun, Hae-Sang, Generation of cyclotomic Hecke fields by modular \(L\)-values with cyclotomic twists, Amer. J. Math., 907-940 (2019) · Zbl 1428.11092 · doi:10.1353/ajm.2019.0024 |
| [47] | Kowalski, E., Actes de la Conf\'{e}rence “Th\'{e}orie des Nombres et Applications”. Families of cusp forms, Publ. Math. Besan\c{c}on Alg\`ebre Th\'{e}orie Nr., 5-40 (2013), Presses Univ. Franche-Comt\'{e}, Besan\c{c}on · Zbl 1312.11030 |
| [48] | Kowalski, E., Exploring the Riemann zeta function. Bagchi’s theorem for families of automorphic forms, 181-199 (2017), Springer, Cham · Zbl 1497.11233 |
| [49] | Kowalski, E., The analytic rank of \(J_0(q)\) and zeros of automorphic \(L\)-functions, Duke Math. J., 503-542 (1999) · Zbl 1161.11359 · doi:10.1215/S0012-7094-99-10017-2 |
| [50] | Kowalski, E., Non-vanishing of high derivatives of automorphic \(L\)-functions at the center of the critical strip, J. Reine Angew. Math., 1-34 (2000) · Zbl 1020.11033 · doi:10.1515/crll.2000.074 |
| [51] | Kowalski, E., Rankin-Selberg \(L\)-functions in the level aspect, Duke Math. J., 123-191 (2002) · Zbl 1035.11018 · doi:10.1215/S0012-7094-02-11416-1 |
| [52] | Kowalski, Emmanuel, Bilinear forms with Kloosterman sums and applications, Ann. of Math. (2), 413-500 (2017) · Zbl 1441.11194 · doi:10.4007/annals.2017.186.2.2 |
| [53] | Kowalski, E., Mod-Gaussian convergence and the value distribution of \(\zeta (\frac 12+it)\) and related quantities, J. Lond. Math. Soc. (2), 291-319 (2012) · Zbl 1292.11103 · doi:10.1112/jlms/jds003 |
| [54] | Li, Wen Ch’ing Winnie, \(L\)-series of Rankin type and their functional equations, Math. Ann., 135-166 (1979) · Zbl 0396.10017 · doi:10.1007/BF01420487 |
| [55] | Liu, Jianya, Perron’s formula and the prime number theorem for automorphic \(L\)-functions, Pure Appl. Math. Q., 481-497 (2007) · Zbl 1246.11152 · doi:10.4310/PAMQ.2007.v3.n2.a4 |
| [56] | L\"{u}, Guangshi, The sixth and eighth moments of Fourier coefficients of cusp forms, J. Number Theory, 2790-2800 (2009) · Zbl 1195.11060 · doi:10.1016/j.jnt.2009.01.019 |
| [57] | Matz, Jasmin, Families of automorphic forms and the trace formula. Distribution of Hecke eigenvalues for \(\operatorname{GL}(n)\), Simons Symp., 327-350 (2016), Springer, [Cham] · Zbl 1408.11043 |
| [58] | Mazur, B., Arithmetic conjectures suggested by the statistical behavior of modular symbols (2019) |
| [59] | Mazur, B., On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math., 1-48 (1986) · Zbl 0699.14028 · doi:10.1007/BF01388731 |
| [60] | Mestre, Jean-Fran\c{c}ois, Formules explicites et minorations de conducteurs de vari\'{e}t\'{e}s alg\'{e}briques, Compositio Math., 209-232 (1986) · Zbl 0607.14012 |
| [61] | Miyake, Toshitsune, Modular forms, Springer Monographs in Mathematics, x+335 pp. (2006), Springer-Verlag, Berlin · Zbl 1159.11014 |
| [62] | Moreno, Carlos J., Analytic proof of the strong multiplicity one theorem, Amer. J. Math., 163-206 (1985) · Zbl 0564.10035 · doi:10.2307/2374461 |
| [63] | Petridis, Y., Arithmetic statistics of modular symbols, Invent. math., 1-57 (2018) |
| [64] | Pollack, Robert, Computations with modular forms. Overconvergent modular symbols, Contrib. Math. Comput. Sci., 69-105 (2014), Springer, Cham · Zbl 1359.11057 · doi:10.1007/978-3-319-03847-6\_3 |
| [65] | Radziwi\l \l , Maksym, Moments and distribution of central \(L\)-values of quadratic twists of elliptic curves, Invent. Math., 1029-1068 (2015) · Zbl 1396.11098 · doi:10.1007/s00222-015-0582-z |
| [66] | Radziwi\l \l , Maksym, Selberg’s central limit theorem for \(\log{|\zeta (1/2+it)|} \), Enseign. Math., 1-19 (2017) · Zbl 1432.11117 · doi:10.4171/LEM/63-1/2-1 |
| [67] | \maks , M., Value distribution of \(L\)-functions, Oberwolfach report |
| [68] | Ramakrishnan, Dinakar, Modularity of the Rankin-Selberg \(L\)-series, and multiplicity one for \(\operatorname{SL}(2)\), Ann. of Math. (2), 45-111 (2000) · Zbl 0989.11023 · doi:10.2307/2661379 |
| [69] | Ramakrishnan, Dinakar, SCHOLAR-a scientific celebration highlighting open lines of arithmetic research. Recovering cusp forms on \(\operatorname{GL}(2)\) from symmetric cubes, Contemp. Math., 181-189 (2015), Amer. Math. Soc., Providence, RI · Zbl 1394.11074 · doi:10.1090/conm/655/13205 |
| [70] | Ricotta, G., Real zeros and size of Rankin-Selberg \(L\)-functions in the level aspect, Duke Math. J., 291-350 (2006) · Zbl 1122.11059 · doi:10.1215/S0012-7094-06-13124-1 |
| [71] | Rudnick, Ze\'{e}v, Zeros of principal \(L\)-functions and random matrix theory, Duke Math. J., 269-322 (1996) · Zbl 0866.11050 · doi:10.1215/S0012-7094-96-08115-6 |
| [72] | Rudnick, Z., Lower bounds for moments of \(L\)-functions, Proc. Natl. Acad. Sci. USA, 6837-6838 (2005) · Zbl 1159.11317 · doi:10.1073/pnas.0501723102 |
| [73] | Sarnak, Peter, Contributions to automorphic forms, geometry, and number theory. Nonvanishing of \(L\)-functions on \(\mathfrak{R}(s)=1, 719-732 (2004)\), Johns Hopkins Univ. Press, Baltimore, MD · Zbl 1088.11067 |
| [74] | Sarnak, Peter, Families of automorphic forms and the trace formula. Families of \(L\)-functions and their symmetry, Simons Symp., 531-578 (2016), Springer, [Cham] · Zbl 1417.11078 |
| [75] | W. Sawin, Quantitative sheaf theory · Zbl 07682633 |
| [76] | Selberg, Atle, On the zeros of Riemann’s zeta-function, Skr. Norske Vid.-Akad. Oslo I, 59 pp. (1942) · Zbl 0028.34903 |
| [77] | Selberg, Atle, Contributions to the theory of the Riemann zeta-function, Arch. Math. Naturvid., 89-155 (1946) · Zbl 0061.08402 |
| [78] | Shin, Sug Woo, Sato-Tate theorem for families and low-lying zeros of automorphic \(L\)-functions, Invent. Math., 1-177 (2016) · Zbl 1408.11042 · doi:10.1007/s00222-015-0583-y |
| [79] | Soundararajan, Kannan, Moments of the Riemann zeta function, Ann. of Math. (2), 981-993 (2009) · Zbl 1251.11058 · doi:10.4007/annals.2009.170.981 |
| [80] | Soundararajan, K., Extreme values of zeta and \(L\)-functions, Math. Ann., 467-486 (2008) · Zbl 1186.11049 · doi:10.1007/s00208-008-0243-2 |
| [81] | Stefanicki, Tomasz, Non-vanishing of \(L\)-functions attached to automorphic representations of \(\text{GL}(2)\) over \(\mathbf{Q} \), J. Reine Angew. Math., 1-24 (1996) · Zbl 0848.11023 · doi:10.1515/crll.1996.474.1 |
| [82] | Voronin, S. M., A theorem on the “universality” of the Riemann zeta-function, Izv. Akad. Nauk SSSR Ser. Mat., 475-486, 703 (1975) · Zbl 0315.10037 |
| [83] | Xi, Ping, Large sieve inequalities for algebraic trace functions, Int. Math. Res. Not. IMRN, 4840-4881 (2017) · Zbl 1405.11125 · doi:10.1093/imrn/rnw074 |
| [84] | Young, Matthew P., The fourth moment of Dirichlet \(L\)-functions, Ann. of Math. (2), 1-50 (2011) · Zbl 1296.11112 · doi:10.4007/annals.2011.173.1.1 |
| [85] | Zacharias, Rapha\"{e}l, Mollification of the fourth moment of Dirichlet \(L\)-functions, Acta Arith., 201-257 (2019) · Zbl 1454.11159 · doi:10.4064/aa170901-16-10 |
| [86] | Zacharias, Rapha\"{e}l, Simultaneous non-vanishing for Dirichlet \(L\)-functions, Ann. Inst. Fourier (Grenoble), 1459-1524 (2019) · Zbl 1454.11160 |
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