Variations on a Lemma of Nicolas and Serre

TL;DR

This paper generalizes and simplifies previous lemmas related to the Nicolas-Serre code and its variants, providing new recurrence relations and bijections to analyze Hecke algebras at levels 1, 3, and 5.

Contribution

It introduces generalized recurrence relations and bijections for codes related to modular forms, simplifying proofs and extending results to analyze Hecke algebras at multiple levels.

Findings

Simplified proof of a key lemma used in modular forms analysis.

Generalized recurrence relations for code bijections.

Application of new results to Hecke algebras at levels 1, 3, and 5.

Abstract

The "Nicolas-Serre code", $(a,b) \leftrightarrow t^{n}$, is a bijection between $N\times N$ and those $t^{n}$, $n$ odd, in $Z/2[t]$. Suppose $A_{n}$, $n$ odd, in $Z/2[t]$ are defined by: $A_{1}= A_{5}= 0$, $A_{3}= t$, $A_{7}= t^{5}$, and $A_{n+8}= t^{8} A_{n} + t^{2} A_{n+2}$. A lemma, Proposition 4.3 of [6], used to study the Hecke algebra attached to the space of mod $2$ level $1$ modular forms, gives information about the codes $(a,b)$ attached to the monomials appearing in $A_{n}$. The unpublished highly technical proof has been simplified by Gerbelli-Gauthier. Our Theorem 3.7 generalizes Proposition 4.3. The proof, in sections 1-3, is a further simplification of Gerbelli-Gauthier's argument. We build up to the theorem with variants involving the same recurrence, but having different sorts of initial conditions. Section 4 treats the recurrence $A_{n+16}= t^{16} A_{n} + t^{4} A_{n+4} + t^{2} A_{n+2}$. Theorem 4.1, the analog to Theorem 3.7 for this recurrence, is used in [2] and [3] to analyze level 3 Hecke algebras. Finally we introduce a variant code, $(a,b) \leftrightarrow w^{n}$ which is a bijection between $N\times N$ and those $w^{n}$, $n \equiv 1,3,7,9 \bmod{20}$, in $Z/2[w]$. We then study the recurrence $A_{n+80}= w^{80} A_{n}+ w^{20} A_{n+20}$, $n \equiv 1,3,7,9 \bmod{20}$, with appropriate initial conditions. Lemma 5.5, derived from the results of sections 1-3, is the precise analog of Proposition 4.3 for this code, this recurrence, and these initial conditions. It is used in [4] and [5] to analyze level 5 Hecke algebras.