an:01553216
Zbl 0971.11028
Khare, Chandrashekhar
A local analysis of congruences in the \((p,p)\) case. II
EN
Invent. Math. 143, No. 1, 129-155 (2001).
00071880
2001
j
11F33 11F80
\(p\)-adic Galois representation; modular curve; newform; Jacquet-Langlands correspondence; Steinberg lift
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)].
A.D??browski (Szczecin)
Zbl 1072.11506