Variations on a Lemma of Nicolas and Serre

Paul Monsky

paper

Variations on a Lemma of Nicolas and Serre

arXiv:1604.02622

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 The unpublished highly technical proof has been simplified by Gerbelli-Gauthier.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 $w^{n}$ , $n \equiv 1, 3, 7, 9 mod 20$ , in $Z /2 [w]$ . We then study the recurrence $A_{n + 80} = w^{80} A_{n} + w^{20} A_{n + 20}$ , $n \equiv 1, 3, 7, 9 mod 20$ , 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.

References added to the applications of Lemma 5.5 to Hecke algebras. 14 pages

Variations on a Lemma of Nicolas and Serre

Abstract

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