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

Bollettino dell'Unione Matematica Italiana (2006)

- Volume: 9-B, Issue: 3, page 711-721
- ISSN: 0392-4041

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topCiavarella, Miriam. "Eisenstein ideal and reducible $\lambda$-adic Representations Unramified Outside a Finite Number of Primes.." Bollettino dell'Unione Matematica Italiana 9-B.3 (2006): 711-721. <http://eudml.org/doc/289601>.

@article{Ciavarella2006,

abstract = {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.},

author = {Ciavarella, Miriam},

journal = {Bollettino dell'Unione Matematica Italiana},

language = {eng},

month = {10},

number = {3},

pages = {711-721},

publisher = {Unione Matematica Italiana},

title = {Eisenstein ideal and reducible $\lambda$-adic Representations Unramified Outside a Finite Number of Primes.},

url = {http://eudml.org/doc/289601},

volume = {9-B},

year = {2006},

}

TY - JOUR

AU - Ciavarella, Miriam

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

JO - Bollettino dell'Unione Matematica Italiana

DA - 2006/10//

PB - Unione Matematica Italiana

VL - 9-B

IS - 3

SP - 711

EP - 721

AB - 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.

LA - eng

UR - http://eudml.org/doc/289601

ER -

## References

top- ATIYAH, M. F. - MACDONALD, I. G., Introduction to Commutative Algebra, University of Oxford, Addison-Wesley Publishing Company, 1969. Zbl0175.03601
- CARAYOL, H., Formes modulaires et representations galoisiennes valeurs dans un anneau local complet, Contemporary Mathematics, Volume 165, Amer. Math. Soc., Providence, RI, 1994, 213-237. Zbl0812.11036
- RIBET KENNETH, A. - PAPIER, E., Eisenstein ideals and $\lambda $-adic representations, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 281981, no. 3 (1982), 651-665. Zbl0508.12012
- MAZUR, B., An introduction to the deformation theory of Galois representation. In Modular Forms and Fermat's Last Theorem, G. Cornell, J. H. Silverman, G. Stevens, Eds. Springer, 43-311.
- WAGSTAFF, S., The irregular prime to 125.000, Math. Comp., 32 (1978), 583-591. Zbl0377.10002
- WILES, A., Modular elliptic curves and Fermat last Theorem, Ann. of Math., 141 (1995), 443-551. Zbl0823.11029

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