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Transactions of the American Mathematical Society Series B

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Newforms mod $ p$ in squarefree level with applications to Monsky's Hecke-stable filtration


Authors: Shaunak V. Deo and Anna Medvedovsky; with an appendix by Alexandru Ghitza
Journal: Trans. Amer. Math. Soc. Ser. B 6 (2019), 245-273
MSC (2010): Primary 11F33; Secondary 11F30, 11F11, 11F25, 11F23
DOI: https://doi.org/10.1090/btran/35
Published electronically: October 2, 2019
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Abstract: We propose an algebraic definition of the space of $ \ell $-new mod-$ p$ modular forms for $ \Gamma _0(N\ell )$ in the case that $ \ell $ is prime to $ N$, which naturally generalizes to a notion of newforms modulo $ p$ in squarefree level. We use this notion of newforms to interpret the Hecke algebras on the graded pieces of the space of mod-$ 2$ level-$ 3$ modular forms described by Paul Monsky. Along the way, we describe a renormalized version of the Atkin-Lehner involution: no longer an involution, it is an automorphism of the algebra of modular forms, even in characteristic $ p$.


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Additional Information

Shaunak V. Deo
Affiliation: Mathematics Research Unit, University of Luxembourg, Maison du nombre, 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg
Address at time of publication: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
Email: deoshaunak@gmail.com

Anna Medvedovsky
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
Email: medved@gmail.com

Alexandru Ghitza
Affiliation: School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010, Australia
Email: aghitza@alum.mit.edu

DOI: https://doi.org/10.1090/btran/35
Keywords: Modular forms modulo $p$, newforms mod $p$, oldforms mod $p$, mod-$p$ Hecke algebras, Atkin-Lehner operators mod $p$, Igusa curves
Received by editor(s): July 26, 2018
Received by editor(s) in revised form: August 13, 2018, and November 26, 2018
Published electronically: October 2, 2019
Additional Notes: The first author was partially supported by the University of Luxembourg Internal Research Project AMFOR
The second author was supported by an NSF postdoctoral fellowship (grant DMS-1703834).
Article copyright: © Copyright 2019 by the author under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)