Conjectures about Discriminants of Hecke Algebras of Prime Level

@inproceedings{Calegari2004ConjecturesAD,
  title={Conjectures about Discriminants of Hecke Algebras of Prime Level},
  author={Frank Calegari and W. Stein},
  booktitle={ANTS},
  year={2004}
}
In this paper, we study p-divisibility of discriminants of Hecke algebras associated to spaces of cusp forms of prime level. By considering cusp forms of weight bigger than 2, we are are led to make a precise conjecture about indexes of Hecke algebras in their normalisation which implies (if true) the surprising conjecture that there are no mod p congruences between non-conjugate newforms in S 2(Γ0(p)), but there are almost always many such congruences when the weight is bigger than 2. 
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