Eisenstein series modulo p2

Scott Ahlgren; Michael Hanson; Martin Raum; Olav K. Richter

doi:10.1515/forum-2024-0470

Eisenstein series modulo p²

Scott Ahlgren, Michael Hanson, Martin Raum, Olav K. Richter

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study congruences for Eisenstein series on SL₂ (ℤ) modulo p² p ≥ 5 is prime. It is classically known that all Eisenstein series of weight at least 4 are determined modulo p² by those of weight at most p² - p + 2. We prove that up to powers of E_p-1, each such Eisenstein series is in fact determined modulo p² by a modular form of weight at most 2p - 4. We also determine E₂ modulo p² in terms of a modular form of weight p + 1.

Original language	English (US)
Journal	Forum Mathematicum
Early online date	Mar 28 2025
DOIs	https://doi.org/10.1515/forum-2024-0470
State	E-pub ahead of print - Mar 28 2025

Keywords

Eichler-Shimura relations
Eisenstein series
modular form congruences

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

Online availability

10.1515/forum-2024-0470

Library availability

Discover UIUC Full Text

Cite this

@article{e389009a17c94f7da00eeea4cdea7ddf,

title = "Eisenstein series modulo p2",

abstract = "We study congruences for Eisenstein series on SL2 (ℤ) modulo p2 p ≥ 5 is prime. It is classically known that all Eisenstein series of weight at least 4 are determined modulo p2 by those of weight at most p2 - p + 2. We prove that up to powers of Ep-1, each such Eisenstein series is in fact determined modulo p2 by a modular form of weight at most 2p - 4. We also determine E2 modulo p2 in terms of a modular form of weight p + 1.",

keywords = "Eichler-Shimura relations, Eisenstein series, modular form congruences",

author = "Scott Ahlgren and Michael Hanson and Martin Raum and Richter, {Olav K.}",

note = "Publisher Copyright: {\textcopyright} 2025 Walter de Gruyter GmbH, Berlin/Boston 2025.",

year = "2025",

month = mar,

day = "28",

doi = "10.1515/forum-2024-0470",

language = "English (US)",

journal = "Forum Mathematicum",

issn = "0933-7741",

publisher = "Walter de Gruyter GmbH",

}

TY - JOUR

T1 - Eisenstein series modulo p2

AU - Ahlgren, Scott

AU - Hanson, Michael

AU - Raum, Martin

AU - Richter, Olav K.

PY - 2025/3/28

Y1 - 2025/3/28

N2 - We study congruences for Eisenstein series on SL2 (ℤ) modulo p2 p ≥ 5 is prime. It is classically known that all Eisenstein series of weight at least 4 are determined modulo p2 by those of weight at most p2 - p + 2. We prove that up to powers of Ep-1, each such Eisenstein series is in fact determined modulo p2 by a modular form of weight at most 2p - 4. We also determine E2 modulo p2 in terms of a modular form of weight p + 1.

AB - We study congruences for Eisenstein series on SL2 (ℤ) modulo p2 p ≥ 5 is prime. It is classically known that all Eisenstein series of weight at least 4 are determined modulo p2 by those of weight at most p2 - p + 2. We prove that up to powers of Ep-1, each such Eisenstein series is in fact determined modulo p2 by a modular form of weight at most 2p - 4. We also determine E2 modulo p2 in terms of a modular form of weight p + 1.

KW - Eichler-Shimura relations

KW - Eisenstein series

KW - modular form congruences

UR - http://www.scopus.com/inward/record.url?scp=105001531015&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=105001531015&partnerID=8YFLogxK

U2 - 10.1515/forum-2024-0470

DO - 10.1515/forum-2024-0470

M3 - Article

AN - SCOPUS:105001531015

SN - 0933-7741

JO - Forum Mathematicum

JF - Forum Mathematicum

ER -