Conjectures about Discriminants of Hecke Algebras of Prime Level

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(Γ0(p)), but there are almost always many such congruences when the weight is bigger than 2. 1 Basic Definitions We first recall some commutative algebra related to discriminants, then introduce Hecke algebras of spaces of cusp forms.

• Type: Conference
• Year: 2004
• Where: ANTS
• Authors: Frank Calegari, William A. Stein
• Keywords: Algorithms, ANTS 2004, Cusp Forms, Hecke Algebras, Precise Conjecture