Conjectures About Discriminants of Hecke Algebras of Prime Level

by Frank Calegari and William Stein

OFFICIAL Version (PDF) The official published version.
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LaTeX Sources calegari-stein-ANTS6-final-submission.tex


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 -- one of our conjectures has been proved (see below).
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.
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.