Eisenstein series modulo <i>p</i> <sup>2</sup>
Abstract We study congruences for Eisenstein series on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{\rm SL}_{2}(\mathbb{Z})} modulo <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {p^{2}} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>5</m:mn> </m:mrow> </m:math> {p\geq 5} is prime. It is classically known that all Eisenstein series of weight at least 4 are determined modulo <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {p^{2}} by those of weight at most <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>-</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {p^{2}-p+2} . We prove that up to powers of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>E</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> {E_{p-1}} , each such Eisenstein series is in fact determined modulo <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {p^{2}} by a modular form of weight at most <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mn>4</m:mn> </m:mrow> </m:math> {2p-4} . We also determine <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>E</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> {E_{2}} modulo <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {p^{2}} in terms of a modular form of weight <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {p+1} .
Authors
- Martin Raum (ORCID: https://orcid.org/0000-0002-3485-8596)
- Olav K. Richter (ORCID: https://orcid.org/0000-0003-3886-0893)
- Scott Ahlgren (ORCID: https://orcid.org/0000-0002-1465-2080)
- Michael Hanson (ORCID: https://orcid.org/0009-0005-4641-6575)
Institutions
- University of North Texas (US)
- University of Illinois Urbana-Champaign (US)
- Chalmers University of Technology (SE)
Publication Details
- Journal
- Forum Mathematicum
- Published
- 2025-03-28
- DOI
- https://doi.org/10.1515/forum-2024-0470
- Primary Topic
- Algebraic Geometry and Number Theory
- Type
- article
- Field-Weighted Citation Impact
- 0.00
Funders
- Simons Foundation
- Vetenskapsrådet