Congruences between cusp forms: The \((p,p)\) case
Let \(p\) be a prime. Let \(f\) be a newform of weight 2 for the group \(\Gamma_0 (N) \cap \Gamma_1(p^t)\) such that the associated \(\text{mod }p\) representation is absolutely irreducible. The main result of this paper (Theorem 5) determines all the various \(p\)-power levels and nebentypes with which \(f\) occurs \(\text{mod }p\). This provides an exhaustive analysis of congruences in the \((p,p)\) case (i.e. when \(p\) divides \(N)\). Let us stress that one cannot use the theorem of Langlands and Carayol to guess the congruences in this case -- in the non-\((p,p)\) case it is possible [see: \textit{H. Carayol}, Duke Math. J. 59, 785-801 (1989; Zbl 0703.11027)]. The results on the congruences between forms of weight 2 in the \((p,p)\) case are applied to study the ring of modular deformations of an absolutely irreducible \(\text{mod }p\) representation which is modular.
scientific article; zbMATH DE number 871840
scientific article