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# Mathematical Research Letters

## Volume 18 (2011)

### Number 4

### On the Classification of Geometric Families of Four-Dimensional Galois Representations

Pages: 805 – 814

DOI: http://dx.doi.org/10.4310/MRL.2011.v18.n4.a17

#### Authors

#### Abstract

We give a classification theorem for certain four-dimensional families of geometric $\lambda$-adic Galois representations attached to a pure motive. More precisely, we consider families attached to the cohomology of a smooth projective variety defined over $\Q$ with coefficients in a quadratic imaginary field, non-selfdual and with four different Hodge–Tate weights. We prove that the image is as large as possible for almost every $\lambda$ provided that the family is irreducible and not induced from a family of smaller dimension. If we restrict to semistable families an even simpler classification is given. A version of the main result is given for the case where the family is attached to an automorphic form.