Formes modulaires modulo 2 : L'ordre de nilpotence des opérateurs de Hecke
Abstract
Let Δ =∑m=0∞q (2 m + 1) 2 ∈F2 [ [ q ] ] be the reduction mod 2 of the Δ series. A modular form f modulo 2 of level 1 is a polynomial in Δ. If p is an odd prime, then the Hecke operator Tp transforms f in a modular form Tp (f) which is a polynomial in Δ whose degree is smaller than the degree of f, so that Tp is nilpotent.
The order of nilpotence of f is defined as the smallest integer g = g (f) such that, for every family of g odd primes p1 ,p2 , … ,pg, the relation Tp1Tp2 …Tpg (f) = 0 holds. We show how one can compute explicitly g (f); if f is a polynomial of degree d in Δ, one finds that g (f) ≪d 1 / 2.- Publication:
-
Comptes Rendus Mathematique
- Pub Date:
- April 2012
- DOI:
- 10.1016/j.crma.2012.03.013
- arXiv:
- arXiv:1204.1036
- Bibcode:
- 2012CRMat.350..343N
- Keywords:
-
- Mathematics - Number Theory;
- 11F33;
- 11F25
- E-Print:
- C. R. Acad. Sci. Paris, Ser. I 350 (2012)