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Cycle integrals of modular functions and their applications

Murakami Yuya 東北大学

2022.03.25

概要

Singular moduli are values of modular functions at imaginary quadratic points. According to the theory of complex multiplication, they generate class fields of imaginary quadratic fields. For real quadratic cases, cycle integrals of modular functions can be regarded as the “values” of modular functions at the real quadratic numbers, and they are expected to play a role in the real quadratic analogue of singular moduli.

In this thesis, we study properties of cycle integrals of modular functions as “functions” on real quadratic numbers. Firstly, we reveal the continuity of cycle integrals with respect to continued fraction expansions, and it turns out that it is different from the continuity with respect to Euclidean topology. Secondly, we extend the definition of cycle integrals of modular functions from real quadratic numbers to badly approximable numbers. In the case when a modular function is constant, our extended-cycle integrals are an extension of the fundamental units of real quadratic fields. Thirdly, we give explicit representations of values of extended-cycle integrals for some cases. The first case deals with badly approximable numbers with infinitely long cyclic parts in their continued fractions. The second case deals with badly approximable numbers whose continued fraction expansions are Thue-Morse words, typical words that do not have cyclic parts.

We also study cycle integrals from the viewpoint of symbolic dynamics. We interpret cycle integrals by the notation of symbolc dynamics and study its topology. We also define dynamical zeta functions of Ruelle type and express them as Fredholm determinants of transfer operators to give their analytic extension.

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参考文献

[Aig97] M. Aigner. Markov’s theorem and 100 years of the uniqueness conjecture. Springer, 1997.

[Bea83] A. F. Beardon. The geometry of discrete groups, volume 91 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1983.

[BI19] P. Bengoechea and Ö. Imamoglu. Cycle integrals of modular functions, Markov geodesics and a conjecture of Kaneko. Algebra Number Theory, 13(4):943––962, 2019.

[BI20] P. Bengoechea and Ö. Imamoglu. Values of modular functions at real quadratics and conjectures of Kaneko. Math. Ann., 377(1-2):249–266, 2020.

[CK97] Christian Choffrut and Juhani Karhumäki. Combinatorics of words. In Handbook of formal languages, Vol. 1, pages 329–438. Springer, Berlin, 1997.

[Dal11] F. Dal’Bo. Geodesic and horocyclic trajectories. Universitext. Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011. Translated from the 2007 French original.

[DIT11] W. Duke, Ö. Imamoḡlu, and Á. Tóth. Cycle integrals of the j-function and mock modular forms. Ann. of Math. (2), 173(2):947–981, 2011.

[FJ03] Aihua Fan and Yunping Jiang. Spectral theory of transfer operators. In Complex dynamics and related topics: lectures from the Morningside Center of Mathematics, volume 5 of New Stud. Adv. Math., pages 63–128. Int. Press, Somerville, MA, [2003].

[FZ99] Dominique Foata and Doron Zeilberger. A combinatorial proof of Bass’s evaluations of the Ihara-Selberg zeta function for graphs. Trans. Amer. Math. Soc., 351(6):2257–2274, 1999.

[Gro55] Alexandre Grothendieck. Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc., 16:Chapter 1: 196 pp.; Chapter 2: 140, 1955.

[Kan09] M. Kaneko. Observations on the ‘values’ of the elliptic modular function j(τ ) at real quadratics. Kyushu Journal of Mathematics, 63(2):353–364, 2009.

[Lot02] M. Lothaire. Algebraic combinatorics on words, volume 90 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2002. A collective work by Jean Berstel, Dominique Perrin, Patrice Seebold, Julien Cassaigne, Aldo De Luca, Steffano Varricchio, Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon, Veronique Bruyere, Christiane Frougny, Filippo Mignosi, Antonio Restivo, Christophe Reutenauer, Dominique Foata, Guo-Niu Han, Jacques Desarmenien, Volker Diekert, Tero Harju, Juhani Karhumaki and Wojciech Plandowski, With a preface by Berstel and Perrin.

[Man04] Yu. I. Manin. Real multiplication and noncommutative geometry (ein Alterstraum). In The legacy of Niels Henrik Abel, pages 685–727. Springer, Berlin, 2004.

[Mat20] T. Matsusaka. A hyperbolic analogue of the rademacher symbol. 2020. arXiv:2003.12354.

[Mur21a] Y. Murakami. A continuity of cycle integrals of modular functions. Ramanujan J., 55(3):1177–1187, 2021.

[Mur21b] Y. Murakami. Extended-cycle integrals of modular functions for badly approximable numbers. 2021. arXiv:2110.08452.

[Neu99] Jürgen Neukirch. Algebraic number theory, volume 322 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder.

[P1̈6] S. Päpcke. Values of the j-function and its relation to Markov numbers. Master’s thesis, The Department of Mathematics at Eidgenössische Technische Hochschule Zürich, 2016.

[Pea08] John Clifford Pearson. The Noncommutative Geometry of ultrametric Cantor sets. ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–Georgia Institute of Technology.

[Rue94] David Ruelle. Dynamical zeta functions for piecewise monotone maps of the interval, volume 4 of CRM Monograph Series. American Mathematical Society, Providence, RI, 1994.

[Rue02] David Ruelle. Dynamical zeta functions and transfer operators. Notices Amer. Math. Soc., 49(8):887–895, 2002.

[Sem82] Zbigniew Semadeni. Schauder bases in Banach spaces of continuous functions, volume 918 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1982.

[Zag81] D. B. Zagier. Zetafunktionen und quadratische Körper. Hochschultext [University Textbooks]. Springer-Verlag, Berlin-New York, 1981. Eine Einführung in die höhere Zahlentheorie. [An introduction to higher number theory].

[Zag02] D. Zagier. Traces of singular moduli. In Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), volume 3 of Int. Press Lect. Ser., pages 211–244. Int. Press, Somerville, MA, 2002.

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