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

HTML articles powered by AMS MathViewer

- 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 $p$.## References

- A. O. L. Atkin and J. Lehner,
*Hecke operators on $\Gamma _{0}(m)$*, Math. Ann.**185**(1970), 134–160. MR**268123**, DOI 10.1007/BF01359701 - Joël Bellaïche and Chandrashekhar Khare,
*Level 1 Hecke algebras of modular forms modulo $p$*, Compos. Math.**151**(2015), no. 3, 397–415. MR**3320566**, DOI 10.1112/S0010437X1400774X - Gaëtan Chenevier,
*The $p$-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings*, Automorphic forms and Galois representations. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 414, Cambridge Univ. Press, Cambridge, 2014, pp. 221–285. MR**3444227** - Shaunak V. Deo,
*Structure of Hecke algebras of modular forms modulo $p$*, Algebra Number Theory**11**(2017), no. 1, 1–38. MR**3602765**, DOI 10.2140/ant.2017.11.1 - Fred Diamond,
*Congruence primes for cusp forms of weight $k\ge 2$*, Astérisque**196-197**(1991), 6, 205–213 (1992) (English, with French summary). Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). MR**1141459** - Fred Diamond and John Im,
*Modular forms and modular curves*, Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994) CMS Conf. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 1995, pp. 39–133. MR**1357209** - Fred Diamond and Jerry Shurman,
*A first course in modular forms*, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005. MR**2112196** - Eyal Z. Goren,
*Lectures on Hilbert modular varieties and modular forms*, CRM Monograph Series, vol. 14, American Mathematical Society, Providence, RI, 2002. With the assistance of Marc-Hubert Nicole. MR**1863355**, DOI 10.1090/crmm/014 - Benedict H. Gross,
*A tameness criterion for Galois representations associated to modular forms (mod $p$)*, Duke Math. J.**61**(1990), no. 2, 445–517. MR**1074305**, DOI 10.1215/S0012-7094-90-06119-8 - Nicholas M. Katz and Barry Mazur,
*Arithmetic moduli of elliptic curves*, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR**772569**, DOI 10.1515/9781400881710 - William J. McGraw and Ken Ono,
*Modular form congruences and Selmer groups*, J. London Math. Soc. (2)**67**(2003), no. 2, 302–318. MR**1956137**, DOI 10.1112/S0024610702004064 - Anna Medvedovsky,
*Lower bounds on dimensions of mod-p Hecke algebras: The nilpotence method*, ProQuest LLC, Ann Arbor, MI, 2015. Thesis (Ph.D.)–Brandeis University. MR**3358260** - Anna Medvedovsky,
*Nilpotence order growth of recursion operators in characteristic $p$*, Algebra Number Theory**12**(2018), no. 3, 693–722. MR**3815310**, DOI 10.2140/ant.2018.12.693 - Toshitsune Miyake,
*Modular forms*, Reprint of the first 1989 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006. Translated from the 1976 Japanese original by Yoshitaka Maeda. MR**2194815** - Paul Monsky,
*A Hecke algebra attached to mod-2 modular forms of level 3*, available at http://arxiv.org/pdf/1508.07523.pdf, 2015. - Paul Monsky,
*A Hecke algebra attached to mod-2 modular forms of level 5*, available at http://arxiv.org/pdf/1610.07058.pdf, 2016. - Jean-Louis Nicolas and Jean-Pierre Serre,
*Formes modulaires modulo 2: l’ordre de nilpotence des opérateurs de Hecke*, C. R. Math. Acad. Sci. Paris**350**(2012), no. 7-8, 343–348 (English, with English and French summaries). MR**2922080**, DOI 10.1016/j.crma.2012.03.013 - Jean-Louis Nicolas and Jean-Pierre Serre,
*Formes modulaires modulo 2: structure de l’algèbre de Hecke*, C. R. Math. Acad. Sci. Paris**350**(2012), no. 9-10, 449–454 (French, with English and French summaries). MR**2929047**, DOI 10.1016/j.crma.2012.03.019 - Ken Ono and Nick Ramsey,
*A mod $\ell$ Atkin-Lehner theorem and applications*, Arch. Math. (Basel)**98**(2012), no. 1, 25–36. MR**2885530**, DOI 10.1007/s00013-011-0347-x - Kartik Prasanna,
*Arithmetic properties of the Shimura-Shintani-Waldspurger correspondence*, Invent. Math.**176**(2009), no. 3, 521–600. With an appendix by Brian Conrad. MR**2501296**, DOI 10.1007/s00222-008-0169-z - Kenneth A. Ribet,
*Congruence relations between modular forms*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 503–514. MR**804706** - Jean-Pierre Serre,
*Formes modulaires et fonctions zêta $p$-adiques*, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 191–268 (French). MR**0404145** - Jean-Pierre Serre,
*Congruences et formes modulaires [d’après H. P. F. Swinnerton-Dyer]*, Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. 319–338 (French). MR**0466020**

## 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