Constructing abelian varieties from rank 3 Galois representations with real trace field

Let U/K be a smooth affine curve over a number field and let L be an irreducible rank 3 Q_-local system on U with trivial determinant and infinite geometric monodromy around a cusp. Suppose further that L extends to an integral model such that the Frobenius traces are contained in a fixed totally real number field. Then, after potentially shrinking U, there exists an abelian scheme f BU U such that L is a summand of R²f_* Q_ (1). The key ingredients are: (1) the totally real assumption implies L admits a square root M; (2) the trace field of M is sufficiently bounded, allowing us to use recent work of Krishnamoorthy-Yang-Zuo to construct an abelian scheme over U ₊ geometrically realizing L; and (3) Deligne's weight-monodromy theorem and the Rapoport-Zink spectral sequence, which allow us to pin down the arithmetizations using the total degeneration.