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Zbl 0840.11023
van Geemen, Bert; Top, Jaap
(van Geemen, Bert)
Selfdual and non-selfdual 3-dimensional Galois representations. (English)
[J] Compos. Math. 97, No.1-2, 51-70 (1995). ISSN 0010-437X; ISSN 1570-5846/e

The authors describe an explicit construction for compatible systems of three-dimensional $\lambda$-adic representations of $\text{Gal}(\overline\bbfQ/\bbfQ)$ of weight 2. In some cases, these representations appear to be essentially symmetric squares of two-dimensional representations, but in other cases, they are not even projectively self-dual. To obtain their representations, they pull back an elliptic surface over $P^1$ with prescribed singular fibres to $P^1$ via the quotient map $P^1\to P^1$ obtained from a cyclic 4-group in $\text{PGL}(2)$. The resulting surface $S$ has an automorphism $\sigma$ of order 4. Dividing $H^2(S)$ by the subspace generated by components of singular fibres and restricting to the $i$-eigenspace of $\sigma$, they obtain a system of representations which, with suitable choices, can be made three-dimensional.
[M.Larsen (Philadelphia)]

MSC 2000:: *11F70 Representation-theoretic methods in automorphic theory
14F20 Grothendieck cohomology and topology

Keywords: Galois representations; elliptic surface

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