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A local analysis of congruences in the \((p,p)\) case. II. (English) Zbl 0971.11028
Fix an odd rational prime \(p\). Let \(\rho:G_\mathbb{Q} \to\text{GL}_2 (\overline \mathbb{F}_p)\) be a continuous, irreducible representation of the absolute Galois group \(G_\mathbb{Q}= \text{Gal} (\overline\mathbb{Q}/ \mathbb{Q})\). We say that \(\rho\) arises from a newform \(f\in S_k(\Gamma_1 (M))\) \((k\geq 2)\) if \(\rho\) is the reduction modulo the maximal ideal of an integral model of the irreducible \(p\)-adic representation \(\rho_f: G_\mathbb{Q} \to\text{GL}_2(K)\) attached to \(f\) by Eichler, Shimura and Deligne.
The author studies the local components at \(p\) of newforms \(f\) that give rise to \(\rho\) (Theorems 1 to 5). His proofs of these theorems (given in sections 3, 4 and 5) are along the lines outlined in the introduction of his paper [Compos. Math. 112, 363-376 (1998; Zbl 1072.11506)].

11F33 Congruences for modular and \(p\)-adic modular forms
11F80 Galois representations
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