# Galois representations with trivial determinant that do not factor through a number field

In arithmetic geometry one often encounters continuous representations $$\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \operatorname{GL}_n(\mathbb{Q}_l)$$ for some $$n\geq 1$$ and some prime number $$l$$ such that no Tate twist of $$\rho$$ factors through $$\operatorname{GL}_n(K)$$ for any number field $$K$$. An example is the first $$l$$-adic cohomology of an elliptic curve over $$\mathbb{Q}$$: if $$\rho \otimes \mathbb{Q}_l(i)$$ were to factor then $$\wedge^2 (\rho\otimes \mathbb{Q}_l(i))\approx \wedge^2 \rho \otimes \mathbb{Q}_l(2i)\approx \mathbb{Q}_l(2i-1)$$ would factor as well which is a contradiction since it is infinitely ramified.

This determinant trick is the only argument I know for proving that a representation does not factor. Are there explicit examples of representations unramified outside a finite set of primes and de Rham at $$l$$ with a trivial determinant and trivial Hodge-Tate weights that do not factor? Do such representations arise naturally in number theory or arithmetic geometry?

So e.g. if $$W$$ is $$H^1$$ of an elliptic curve, and $$V$$ is the quotient of $$W \otimes W^\vee$$ by its obvious 1-dimensional trivial subrep, then $$V$$ has trivial determinant, but its Hodge-Tate weights are $$\{-1, 0, 1\}$$.
• I now realize that I forgot to ask for irreducibility of the representation (otherwise one can take the direct sum of $H^1(E, \mathbb{Q}_l)$ and the cyclotomic character). Is your 3-dimensional representation irreducible?
• If the elliptic curve does not have CM then $V$ is irreducible. (This is easily seen from Serre's open image theorem, although that is using a sledgehammer to crack a nut.) Jan 22, 2020 at 15:53