Iterated primitives of meromorphic quasimodular forms for 𝑆𝐿₂(ℤ)

Matthes, Nils

doi:10.1090/tran/8538

Iterated primitives of meromorphic quasimodular forms for $\operatorname {SL}_2(\mathbb Z)$
HTML articles powered by AMS MathViewer

by Nils Matthes PDF

Trans. Amer. Math. Soc. 375 (2022), 1443-1460 Request permission

Abstract:

We introduce and study iterated primitives of meromorphic quasimodular forms for $\operatorname {SL}_2(\mathbb Z)$, generalizing work of Manin and Brown for holomorphic modular forms. We prove that the algebra of iterated primitives of meromorphic quasimodular forms is naturally isomorphic to a certain explicit shuffle algebra. We deduce from this an Ax–Lindemann–Weierstrass type algebraic independence criterion for primitives of meromorphic quasimodular forms which includes a recent result of Paşol–Zudilin as a special case. We also study spaces of meromorphic modular forms with restricted poles, generalizing results of Guerzhoy in the weakly holomorphic case.

References

G. Bol, Invarianten linearer differentialgleichungen, Abh. Math. Sem. Univ. Hamburg 16 (1949), no. nos. 3-4, 1–28 (German). MR 33411, DOI 10.1007/BF03343515
F. Brown, Multiple modular values and the relative completion of the fundamental group of $M_{1,1}$, arXiv:1407.5167v4, 2017.
Francis Brown and Richard Hain, Algebraic de Rham theory for weakly holomorphic modular forms of level one, Algebra Number Theory 12 (2018), no. 3, 723–750. MR 3815311, DOI 10.2140/ant.2018.12.723
Kuo Tsai Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), no. 5, 831–879. MR 454968, DOI 10.1090/S0002-9904-1977-14320-6
Alain Connes and Dirk Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000), no. 1, 249–273. MR 1748177, DOI 10.1007/s002200050779
Matthieu Deneufchâtel, Gérard H. E. Duchamp, Vincel Hoang Ngoc Minh, and Allan I. Solomon, Independence of hyperlogarithms over function fields via algebraic combinatorics, Algebraic informatics, Lecture Notes in Comput. Sci., vol. 6742, Springer, Heidelberg, 2011, pp. 127–139. MR 2846744, DOI 10.1007/978-3-642-21493-6_{8}
P. Guerzhoy, Hecke operators for weakly holomorphic modular forms and supersingular congruences, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3051–3059. MR 2407067, DOI 10.1090/S0002-9939-08-09277-0
Masanobu Kaneko and Don Zagier, A generalized Jacobi theta function and quasimodular forms, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 165–172. MR 1363056, DOI 10.1007/978-1-4612-4264-2_{6}
P. Yu. Kozlov, On the algebraic independence of functions of a certain class, Izv. Ross. Akad. Nauk Ser. Mat. 77 (2013), no. 1, 23–32 (Russian, with Russian summary); English transl., Izv. Math. 77 (2013), no. 1, 20–29. MR 3087085, DOI 10.1070/IM2013v077n01ABEH002627
K. Mahler, On algebraic differential equations satisfied by automorphic functions. , J. Austral. Math. Soc. 10 (1969), 445–450. MR 0262493
Dominique Manchon, Hopf algebras in renormalisation, Handbook of algebra. Vol. 5, Handb. Algebr., vol. 5, Elsevier/North-Holland, Amsterdam, 2008, pp. 365–427. MR 2523455, DOI 10.1016/S1570-7954(07)05007-3
Yuri I. Manin, Iterated integrals of modular forms and noncommutative modular symbols, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser Boston, Boston, MA, 2006, pp. 565–597. MR 2263200, DOI 10.1007/978-0-8176-4532-8_{1}0
Nils Matthes, On the algebraic structure of iterated integrals of quasimodular forms, Algebra Number Theory 11 (2017), no. 9, 2113–2130. MR 3735463, DOI 10.2140/ant.2017.11.2113
V. Paşol, W. Zudilin: Magnetic (quasi-)modular forms. arXiv:2009.14609, 2020.
David E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Algebra 58 (1979), no. 2, 432–454. MR 540649, DOI 10.1016/0021-8693(79)90171-6
Christophe Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New Series, vol. 7, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1231799
Emmanuel Royer, Quasimodular forms: an introduction, Ann. Math. Blaise Pascal 19 (2012), no. 2, 297–306 (English, with English and French summaries). MR 3025137