2016/04/10 by Paul Monsky, Monsky, Paul
Computer Science · Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1604.02622
openalex publication_date 2016/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The "Nicolas-Serre code", (a,b) ↔ tn, is a bijection between N× N and those tn, n odd, in Z/2[t]. Suppose An, n odd, in Z/2[t] are defined by: A1= A5= 0, A3= t, A7= t5, and An+8= t8 An + t2 An+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 An. The unpublished highly technical proof has been simplified by Gerbelli-Gauthier.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 An+16= t16 An + t4 An+4 + t2 An+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) ↔ wn which is a bijection between N× N and those wn, n ≡ 1,3,7,9 \bmod20, in Z/2[w]. We then study the recurrence An+80= w80 An+ w20 An+20, n ≡ 1,3,7,9 \bmod20, 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.