Iterated primitives of meromorphic quasimodular forms for $\operatorname {SL}_2(\mathbb Z)$
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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
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Additional Information
- Nils Matthes
- Affiliation: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
- Address at time of publication: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark
- MR Author ID: 1123080
- Email: nils.matthes@maths.ox.ac.uk
- Received by editor(s): January 27, 2021
- Received by editor(s) in revised form: February 15, 2021, and July 10, 2021
- Published electronically: December 2, 2021
- Additional Notes: This project had received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 724638).
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 1443-1460
- MSC (2020): Primary 11F37; Secondary 11F67
- DOI: https://doi.org/10.1090/tran/8538
- MathSciNet review: 4369252