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Transactions of the American Mathematical Society Series B

Published by the American Mathematical Society, the Transactions of the American Mathematical Society Series B (BTRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 2330-0000

The 2020 MCQ for Transactions of the American Mathematical Society Series B is 1.79.

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Newforms mod $p$ in squarefree level with applications to Monsky’s Hecke-stable filtration
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by Shaunak V. Deo and Anna Medvedovsky; with an appendix by Alexandru Ghitza HTML | PDF
Trans. Amer. Math. Soc. Ser. B 6 (2019), 245-273

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
  • MR Author ID: 1195938
  • Email: deoshaunak@gmail.com
  • Anna Medvedovsky
  • Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
  • MR Author ID: 1280121
  • Email: medved@gmail.com
  • Alexandru Ghitza
  • Affiliation: School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010, Australia
  • MR Author ID: 713726
  • Email: aghitza@alum.mit.edu
  • 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).
  • © Copyright 2019 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
  • 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
  • MathSciNet review: 4014301