Abstract
We determine the asymptotic distribution of Manin’s iterated integrals of length at most 2. For all lengths, we compute all the asymptotic moments. We show that if the length is at least 3, these moments do in general not determine a unique distribution.
Funding source: Natur og Univers, Det Frie Forskningsråd
Award Identifier / Grant number: DFF-7014-00060B
Funding statement: The research of Nils Matthes was supported by a Walter Benjamin Fellowship of the DFG. The research of Morten S. Risager was supported by the Grant DFF-7014-00060B from Independent Research Fund Denmark.
Acknowledgements
We are grateful to Christian Berg for many useful comments on Section 5, and to Wadim Zudilin, Christian Krattenthaler and Guoniu Han for discussions and correspondence that led to (6.1). We are also grateful to the referee for useful comments.
References
[1] P. Banks, E. Panzer and B. Pym, Multiple zeta values in deformation quantization, Invent. Math. 222 (2020), no. 1, 79–159. 10.1007/s00222-020-00970-xSearch in Google Scholar
[2] C. Berg, The cube of a normal distribution is indeterminate, Ann. Probab. 16 (1988), no. 2, 910–913. 10.1214/aop/1176991795Search in Google Scholar
[3] C. Berg and M. Thill, Rotation invariant moment problems, Acta Math. 167 (1991), no. 3–4, 207–227. 10.1007/BF02392450Search in Google Scholar
[4] S. Bettin, High moments of the Estermann function, Algebra Number Theory 13 (2019), no. 2, 251–300. 10.2140/ant.2019.13.251Search in Google Scholar
[5] S. Bettin and S. Drappeau, Limit laws for rational continued fractions and value distribution of quantum modular forms, Proc. Lond. Math. Soc. (3) 125 (2022), no. 6, 1377–1425. 10.1112/plms.12485Search in Google Scholar
[6] P. Billingsley, Probability and measure, 3rd ed., Wiley Ser. Probab. Stat., John Wiley & Sons, New York 1995. Search in Google Scholar
[7] P. Billingsley, Convergence of probability measures, 2nd ed., Wiley Ser. Probab. Stat., John Wiley & Sons, New York 1999. 10.1002/9780470316962Search in Google Scholar
[8] B. J. Birch, Elliptic curves over 𝑄: A progress report, 1969 Number theory institute (Stony Brook 1969), Proc. Sympos. Pure Math. 20, American Mathematical Society, Providence (1971), 396–400. 10.1090/pspum/020/0314845Search in Google Scholar
[9] C. Bogner and F. Brown, Feynman integrals and iterated integrals on moduli spaces of curves of genus zero, Commun. Number Theory Phys. 9 (2015), no. 1, 189–238. 10.4310/CNTP.2015.v9.n1.a3Search in Google Scholar
[10] F. Brown, A multi-variable version of the completed Riemann zeta function and other 𝐿-functions, RIMS Kôkyûroku 2120 (2019), 1–27. Search in Google Scholar
[11] F. Brown, From the Deligne–Ihara conjecture to multiple modular values, RIMS Kôkyûroku 2120 (2019), 1–24. Search in Google Scholar
[12] F. Brown, Mixed Tate motives over ℤ, Ann. of Math. (2) 175 (2012), no. 2, 949–976. 10.4007/annals.2012.175.2.10Search in Google Scholar
[13] R. Bruggeman and Y. Choie, Multiple period integrals and cohomology, Algebra Number Theory 10 (2016), no. 3, 645–664. 10.2140/ant.2016.10.645Search in Google Scholar
[14] R. Bruggeman and N. Diamantis, Fourier coefficients of Eisenstein series formed with modular symbols and their spectral decomposition, J. Number Theory 167 (2016), 317–335. 10.1016/j.jnt.2016.03.009Search in Google Scholar
[15] D. Bump, Automorphic forms and representations, Cambridge Stud. Adv. Math. 55, Cambridge University, Cambridge 1997. 10.1017/CBO9780511609572Search in Google Scholar
[16] J. I. Burgos Gil and J. Fresán, Multiple zeta values: From numbers to motives, Clay Math. Proc., to appear. Search in Google Scholar
[17] K.-T. Chen, Iterated path integrals and generalized paths, Bull. Amer. Math. Soc. 73 (1967), 935–938. 10.1090/S0002-9904-1967-11858-5Search in Google Scholar
[18] K.-T. Chen, Algebras of iterated path integrals and fundamental groups, Trans. Amer. Math. Soc. 156 (1971), 359–379. 10.1090/S0002-9947-1971-0275312-1Search in Google Scholar
[19] K.-T. Chen, Iterated integrals of differential forms and loop space homology, Ann. of Math. (2) 97 (1973), 217–246. 10.2307/1970846Search in Google Scholar
[20] K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), no. 5, 831–879. 10.1090/S0002-9904-1977-14320-6Search in Google Scholar
[21] G. Chinta, N. Diamantis and C. O’Sullivan, Second order modular forms, Acta Arith. 103 (2002), no. 3, 209–223. 10.4064/aa103-3-2Search in Google Scholar
[22] G. Chinta, I. Horozov and C. O’Sullivan, Noncommutative modular symbols and Eisenstein series, Automorphic forms and related topics, Contemp. Math. 732, American Mathematical Society, Providence (2019), 27–45. 10.1090/conm/732/14784Search in Google Scholar
[23] Y. Choie, Parabolic cohomology and multiple Hecke 𝐿-values, Ramanujan J. 41 (2016), no. 1–3, 543–561. 10.1007/s11139-016-9844-7Search in Google Scholar
[24] Y. Choie and N. Diamantis, Rankin–Cohen brackets on higher order modular forms, Multiple Dirichlet series, automorphic forms, and analytic number theory, Proc. Sympos. Pure Math. 75, American Mathematical Society, Providence (2006), 193–201. 10.1090/pspum/075/2279937Search in Google Scholar
[25] Y. Choie and K. Ihara, Iterated period integrals and multiple Hecke 𝐿-functions, Manuscripta Math. 142 (2013), no. 1–2, 245–255. 10.1007/s00229-013-0605-2Search in Google Scholar
[26] Y. Choie and K. Matsumoto, Functional equations for double series of Euler type with coefficients, Adv. Math. 292 (2016), 529–557. 10.1016/j.aim.2016.01.016Search in Google Scholar
[27]
P. Constantinescu,
Distribution of modular symbols in
[28] J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University, Cambridge 1997. Search in Google Scholar
[29] A. Deitmar and I. Horozov, Iterated integrals and higher order invariants, Canad. J. Math. 65 (2013), no. 3, 544–552. 10.4153/CJM-2012-020-8Search in Google Scholar
[30] P. Deligne and A. B. Goncharov, Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), no. 1, 1–56. 10.1016/j.ansens.2004.11.001Search in Google Scholar
[31] N. Diamantis and D. Sim, The classification of higher-order cusp forms, J. reine angew. Math. 622 (2008), 121–153. 10.1515/CRELLE.2008.067Search in Google Scholar
[32] S. Drappeau and A. C. Nordentoft, Central values of additive twists of Maass forms 𝐿-functions, preprint (2022), https://arxiv.org/abs/2208.14346. Search in Google Scholar
[33] W. Duke, J. B. Friedlander and H. Iwaniec, The subconvexity problem for Artin 𝐿-functions, Invent. Math. 149 (2002), no. 3, 489–577. 10.1007/s002220200223Search in Google Scholar
[34] J. D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. reine angew. Math. 293(294) (1977), 143–203. 10.1515/crll.1977.293-294.143Search in Google Scholar
[35] J. Françon and G. Viennot, Permutations selon leurs pics, creux, doubles montées et double descentes, nombres d’Euler et nombres de Genocchi, Discrete Math. 28 (1979), no. 1, 21–35. 10.1016/0012-365X(79)90182-1Search in Google Scholar
[36] D. Goldfeld, The distribution of modular symbols, Number theory in progress. Vol. 2 (Zakopane–Kościelisko 1997), De Gruyter, Berlin (1999), 849–865. 10.1515/9783110285581.849Search in Google Scholar
[37] D. Goldfeld, Zeta functions formed with modular symbols, Automorphic forms, automorphic representations, and arithmetic (Fort Worth 1996), Proc. Sympos. Pure Math. 66, American Mathematical Society, Providence (1999), 111–121. 10.1090/pspum/066.1/1703748Search in Google Scholar
[38] R. M. Hain, The Hodge de Rham theory of relative Malcev completion, Ann. Sci. Éc. Norm. Supér. (4) 31 (1998), no. 1, 47–92. 10.1016/S0012-9593(98)80018-9Search in Google Scholar
[39]
K. Ihara and T. Takamuki,
The quantum
[40] H. Iwaniec, Spectral methods of automorphic forms, 2nd ed., Grad. Stud. Math. 53, American Mathematical Society, Providence 2002. 10.1090/gsm/053/05Search in Google Scholar
[41] H. Iwaniec and E. Kowalski, Analytic number theory, Amer. Math. Soc. Colloq. Publ. 53, American Mathematical Society, Providence 2004. 10.1090/coll/053Search in Google Scholar
[42] J. Jorgenson and C. O’Sullivan, Unipotent vector bundles and higher-order non-holomorphic Eisenstein series, J. Théor. Nombres Bordeaux 20 (2008), no. 1, 131–163. 10.5802/jtnb.619Search in Google Scholar
[43] P. Kleban and D. Zagier, Crossing probabilities and modular forms, J. Stat. Phys. 113 (2003), no. 3–4, 431–454. 10.1023/A:1026012600583Search in Google Scholar
[44] C. Kleiber and J. Stoyanov, Multivariate distributions and the moment problem, J. Multivariate Anal. 113 (2013), 7–18. 10.1016/j.jmva.2011.06.001Search in Google Scholar
[45] M. Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), 35–72. 10.1023/A:1007555725247Search in Google Scholar
[46] R. S. Kulkarni, An arithmetic-geometric method in the study of the subgroups of the modular group, Amer. J. Math. 113 (1991), no. 6, 1053–1133. 10.2307/2374900Search in Google Scholar
[47]
C. A. Kurth and L. Long,
Computations with finite index subgroups of
[48] T. Q. T. Le and J. Murakami, Kontsevich’s integral for the Homfly polynomial and relations between values of multiple zeta functions, Topology Appl. 62 (1995), no. 2, 193–206. 10.1016/0166-8641(94)00054-7Search in Google Scholar
[49] T. T. Q. Le and J. Murakami, Kontsevich’s integral for the Kauffman polynomial, Nagoya Math. J. 142 (1996), 39–65. 10.1017/S0027763000005638Search in Google Scholar
[50] J. Lee and H.-S. Sun, Dynamics of continued fractions and distribution of modular symbols, preprint (2019), https://arxiv.org/abs/1902.06277. Search in Google Scholar
[51] M. Loève, Probability theory. I, 4th ed., Grad. Texts in Math. 45, Springer, New York 1977. 10.1007/978-1-4757-6288-4Search in Google Scholar
[52] H. Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949), 141–183. 10.1007/BF01329622Search in Google Scholar
[53] J. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66. 10.1070/IM1972v006n01ABEH001867Search in Google Scholar
[54] Y. I. Manin, Iterated Shimura integrals, Mosc. Math. J. 5 (2005), no. 4, 869–881, 973. 10.17323/1609-4514-2005-5-4-869-881Search in Google Scholar
[55] Y. I. Manin, Iterated integrals of modular forms and noncommutative modular symbols, Algebraic geometry and number theory, Progr. Math. 253, Birkhäuser, Boston (2006), 565–597. 10.1007/978-0-8176-4532-8_10Search in Google Scholar
[56] Y. I. Manin, Lectures on modular symbols, Arithmetic geometry, Clay Math. Proc. 8, American Mathematical Society, Providence (2009), 137–152. Search in Google Scholar
[57] B. Mazur, Courbes elliptiques et symboles modulaires, Séminaire Bourbaki, 24ème année (1971/1972), Lecture Notes in Math. 317, Springer, Berlin (1973), 277–294, Exp. No. 414. 10.1007/BFb0069287Search in Google Scholar
[58] B. Mazur and K. Rubin, Arithmetic conjectures suggested by the statistical behavior of modular symbols, Exp. Math. 32 (2023), no. 4, 657–672. 10.1080/10586458.2021.1982424Search in Google Scholar
[59] P. Michel, Analytic number theory and families of automorphic 𝐿-functions, Automorphic forms and applications, IAS/Park City Math. Ser. 12, American Mathematical Society, Providence (2007), 181–295. 10.1090/pcms/012/05Search in Google Scholar
[60] S. Nadarajah, The Kotz-type distribution with applications, Statistics 37 (2003), no. 4, 341–358. 10.1080/0233188031000078060Search in Google Scholar
[61] A. C. Nordentoft, Central values of additive twists of modular 𝐿-functions, preprint (2018), https://arxiv.org/abs/1812.08378. Search in Google Scholar
[62] A. C. Nordentoft, Central values of additive twists of cuspidal 𝐿-functions, J. reine angew. Math. 776 (2021), 255–293. 10.1515/crelle-2021-0013Search in Google Scholar
[63] C. O’Sullivan, Properties of Eisenstein series formed with modular symbols, J. reine angew. Math. 518 (2000), 163–186. 10.1515/crll.2000.003Search in Google Scholar
[64] Y. N. Petridis, Spectral deformations and Eisenstein series associated with modular symbols, Int. Math. Res. Not. IMRN 2002 (2002), no. 19, 991–1006. 10.1155/S1073792802111159Search in Google Scholar
[65] Y. N. Petridis and M. S. Risager, Modular symbols have a normal distribution, Geom. Funct. Anal. 14 (2004), no. 5, 1013–1043. 10.1007/s00039-004-0481-8Search in Google Scholar
[66] Y. N. Petridis and M. S. Risager, Arithmetic statistics of modular symbols, Invent. Math. 212 (2018), no. 3, 997–1053. 10.1007/s00222-017-0784-7Search in Google Scholar
[67] C. Reutenauer, Free Lie algebras, London Math. Soc. Monogr. (N. S.) 7, Oxford University, New York 1993. 10.1093/oso/9780198536796.001.0001Search in Google Scholar
[68] W. Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, II, Math. Ann. 167 (1966), 292–337; ibid. 168 (1966), 261–324. 10.1007/BF01364540Search in Google Scholar
[69] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N. S.) 20 (1956), 47–87. Search in Google Scholar
[70] A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math. 8, American Mathematical Society, Providence 1965. 10.1090/pspum/008/0182610Search in Google Scholar
[71] M. Waldschmidt, Multiple polylogarithms: An introduction, Number theory and discrete mathematics (Chandigarh 2000), Trends Math., Birkhäuser, Basel (2002), 1–12. 10.1007/978-93-86279-10-1_1Search in Google Scholar
[72] M. P. Young, Explicit calculations with Eisenstein series, J. Number Theory 199 (2019), 1–48. 10.1016/j.jnt.2018.11.007Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The theory of F-rational signature
- Jacobian determinants for nonlinear gradient of planar ∞-harmonic functions and applications
- Fano varieties with torsion in the third cohomology group
- The distribution of Manin’s iterated integrals of modular forms
- On some 𝑝-adic and mod 𝑝 representations of quaternion algebra over ℚ𝑝
- Nonexistence of isoperimetric sets in spaces of positive curvature
- Simple 𝑝-adic Lie groups with abelian Lie algebras
- Hyperbolic lattice point counting in unbounded rank
- The (almost) integral Chow ring of ℳ̅3
Articles in the same Issue
- Frontmatter
- The theory of F-rational signature
- Jacobian determinants for nonlinear gradient of planar ∞-harmonic functions and applications
- Fano varieties with torsion in the third cohomology group
- The distribution of Manin’s iterated integrals of modular forms
- On some 𝑝-adic and mod 𝑝 representations of quaternion algebra over ℚ𝑝
- Nonexistence of isoperimetric sets in spaces of positive curvature
- Simple 𝑝-adic Lie groups with abelian Lie algebras
- Hyperbolic lattice point counting in unbounded rank
- The (almost) integral Chow ring of ℳ̅3