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Conjectures about discriminants of Hecke algebras of prime level. (English) Zbl 1125.11320
Buell, Duncan (ed.), Algorithmic number theory. 6th international symposium, ANTS-VI, Burlington, VT, USA, June 13–18, 2004. Proceedings. Berlin: Springer (ISBN 3-540-22156-5/pbk). Lecture Notes in Computer Science 3076, 140-152 (2004).
Summary: 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 $$S_{2}(\Gamma_{0}(p)$$), but there are almost always many such congruences when the weight is bigger than $$2$$.
For the entire collection see [Zbl 1052.11002].

##### MSC:
 11F25 Hecke-Petersson operators, differential operators (one variable) 11F11 Holomorphic modular forms of integral weight
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