Conjectures About Discriminants of Hecke Algebras of Prime Level

by Frank Calegari and William Stein

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Abstract

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 S2(Gamma0(p)), but there are almost always many such congruences when the weight is bigger than 2.

 


Note from Scott Ahlgren (2005-06-02): Mugurel Barcau and I have just finished writing out a proof of Conjecture 4 (under the hypotheses which I asked about, which should be the same as those in your reply, since the coefficients of f and g must all be in some fixed number field). It relies mostly on the Serre/Swinnerton-Dyer theory.