Some non-Gorenstein Hecke algebras attached to spaces of modular forms

In this paper we exhibit some examples of non-Gorenstein Hecke algebras, and hence some modular forms for which mod 2 multiplicity one does not hold. Define S2(Γ0(N)) to be the space of classical cuspidal modular forms of weight 2, level N , and trivial character. The Hecke algebra TN is defined to be the subring of End(S2(Γ0(N)) generated by the Hecke operators {Tp : p 6 |N} and {Uq : q|N}. Let m be a maximal ideal of TN , and let l denote the characteristic of the finite field TN/m. By work of Shimura, one can associate to m a semi-simple Galois representation ρm : Gal(Q/Q) → GL2(TN/m) satisfying tr(ρ(Frobp)) ≡ Tp modm for all primes p ∤ Nl. We say that m is non-Eisenstein if ρm is absolutely irreducible. As an example, if E is a (modular) elliptic curve over Q of conductor N , let f := ∑ n≥1 anq n be the modular form in S2(Γ0(N)) associated to E. Associated to f is a minimal prime ideal of TN ; we say that m is associated to f , or to E, if m contains this minimal prime ideal. In this case, the representation associated to m is isomorphic to the semisimplification of E[l], where l is the characteristic of TN/m. The localisation (TN )m of TN at a maximal ideal m is frequently a Gorenstein ring, and such a phenomenon is related to the study of the m-torsion in the Jacobian J0(N) of X0(N). For example, if m is non-Eisenstein then by the main result of [2], the m-torsion in J0(N) is isomorphic to a direct sum of d ≥ 1 copies of ρm. If d = 1 then one says that the ideal m satisfies “mod l multiplicity one”, or just “multiplicity one”. In this case, the localisation (TN )m is known to be Gorenstein. Multiplicity one is a common phenomenon for maximal ideals m of TN . Let us restrict for the rest of this paper to the case of non-Eisenstein maximal ideals m. The first serious study of these mod l multiplicity one questions is that of Mazur [10], who proves that if N = q is prime and the characteristic of the