|OFFICIAL Version (PDF)||The official published version.|
|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.|
Date: Tue, 30 May 2006 11:16:01 -0500 From: Scott Ahlgren Subject: Your conjecture Dear William and Frank, I've attached a new draft of the paper about congruences for forms of weights two and four. We did figure out a way to prove the statement given only a congruence modulo the maximal ideal in \zpbar (i.e. the hypothesis in your conjecture). The idea is to use the fact that the degrees of the fields generated by the coefficients of the forms in question are too small to allow wild ramification. Therefore we can take a trace to the unramified part of the extension without losing mod p information. The trace map does not preserve eigenforms of the Hecke operators. It does, however, preserve eigenforms of w_p (since the eigenvalue can be read off of the pth coefficient). And that is enough for the rest of the argument to work. Best, Scott Ahlgren ---- Frank responds: Dear Scott, Thanks for sending the latest version. I realized sometime this semester that the conjecture probably followed from a result of Breuil--Mezard; I checked this was so today and have attached a brief argument. However, your argument is more elementary and so certainly good to have. Best, Frank.