# zbMATH — the first resource for mathematics

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)].

##### MSC:
 11F33 Congruences for modular and $$p$$-adic modular forms 11F80 Galois representations
Full Text: