1. On the Failure of the Gorenstein Property for Hecke Algebras of Prime Weight
- Author
-
Gabor Wiese and L. J. P. Kilford
- Subjects
Pure mathematics ,11F80 ,General Mathematics ,Mathematics::Number Theory ,Modular form ,Galois group ,Splitting of prime ideals in Galois extensions ,Embedding problem ,mod-$p$ modular forms ,FOS: Mathematics ,Multiplicities of Galois representations ,Number Theory (math.NT) ,Hecke algebras ,Mathematics ,Gorenstein property ,ddc:510 ,Discrete mathematics ,Mathematics - Number Theory ,Mathematics::Commutative Algebra ,11F80 (primary), 11F33, 11F25 (secondary) ,510 Mathematik ,11F33 ,Galois module ,Differential Galois theory ,Finite field ,11F25 ,Mathematics [G03] [Physical, chemical, mathematical & earth Sciences] ,Mathématiques [G03] [Physique, chimie, mathématiques & sciences de la terre] ,Hecke operator - Abstract
In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight p at maximal ideals of residue characteristic p such that the attached mod p Galois representation is unramified at p and the Frobenius at p acts by scalars. The results lead us to the ask the question whether the Gorenstein defect and the multplicity of the attached Galois representation are always equal to 2. We review the literature on the failure of the Gorenstein property and multiplicity one, discuss in some detail a very important practical improvement of the modular symbols algorithm over finite fields and include precise statements on the relationship between the Gorenstein defect and the multiplicity of Galois representations. Appendix A: Manual of Magma package HeckeAlgebra, Appendix B: Tables of Hecke algebras., Comment: 52 pages LaTeX, 2 appendices
- Published
- 2008