Year: 2025
Type: article
Abstract: We study congruences for Eisenstein series on SL 2 ( ℤ ) {{\rm SL}_{2}(\mathbb{Z})} modulo p 2 {p^{2}} , where p ≥ 5 {p\geq 5} is prime. It is classically known that all Eisenstein series of weight at least 4 are determined modulo p 2 {p^{2}} by those of weight at most p 2 - p + 2 {p^{2}-p+2} . We prove that up to powers of E p - 1 {E_{p-1}} , each such Eisenstein series is in fact determined modulo p 2 {p^{2}} by a modular form of weight at most 2 p - 4 {2p-4} . We also determine E 2 {E_{2}} modulo p 2 {p^{2}} in terms of a modular form of weight p + 1 {p+1} .
Source: Forum Mathematicum
Authors: Scott Ahlgren, Michael Hanson, Martin Raum, Olav K. Richter
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funders: Simons Foundation, Vetenskapsrådet
OpenAlex ID: W4408920007