Eisenstein ideal and reducible $\lambda$-adic Representations Unramified Outside a Finite Number of Primes.

Ciavarella, Miriam

Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 3, pp. 711-721

Résumé

The object of this note is to study certain 2-dimensional $\lambda$-adic representations of $\operatorname{Gal}(\bar{Q}/Q)$; fixed $p_{1}, \ldots, p_{n}$ distinct primes, we will consider representations $\rho \colon G \to GL_{2}(A)$, given by the matrix $\rho = \left(\begin{smallmatrix} a b \\ c d \end{smallmatrix}\right)$ which are unramified outside $p_{1}, \ldots, p_{n}, \infty$ and the residue characteristic of $\lambda$, which are a product of $m$ representations over finite extensions of the ring of Witt vectors of the residue field and which are reducible modulo $\lambda$. In analogy with the theory of the modular representations, we will introduce the analogue of Mazur's Hecke algebra $T$, together with an ideal $I$ of $T$ which we will call the Eisenstein ideal. Following the Ribet and Papier's method [3], under the hypotheses: $\bullet$$p_{i} \not\equiv 1 \mod \ell$, for any $i = 1, \ldots, n$, $\bullet$ the semisimplification of $\bar{\rho}$ is described by two characters $\alpha$, $\beta$ which are distinct if restricted to $Z^{\times}_{\ell}$, we obtain the following results: PROPOSITION 0.3--The Eisenstein ideal $I$ is equal to $BC$, where $B$ is the $T$-submodule of $A$ generated by all $b(g)$ with $g \in G$ and similary $C$ is defined using the $c(g)$'s. Moreover, $I$ is the ideal of $T$ generated by the quantities $a(h) - 1$ for $h \in \operatorname{Gal}(K/Q^{ab} \cap K)$. PROPOSITION 0.4 -- Suppose that Vandiver's conjecture is true for $\ell$ and that $I$ is non-zero. Then, after replacement of $\rho$ by a conjugate, the representation $\rho$ takes values in $GL_{2}(T)$ and its matrix coefficients satisfy: \begin{equation*}a \equiv \varphi, \quad d \equiv \psi, \quad c \equiv 0 \pmod I\end{equation*}$\varphi \equiv a \mod \mathcal{M}$ and $\psi \equiv \beta \mod \mathcal{M}$, for $\mathcal{M} = T \cap (\lambda)$. . In particular there is one and only one surjective ring homomorphism from the universal deformation ring $\mathcal{R}(\bar{\rho})$ to $T$, inducing the identity isomorphism on residue fields.