Appears in collection : SAGA - Symposium on Arithmetic Geometry and its Applications

Let $A$ be an abelian variety over a number field. The connected monodromy field of $A$ is the minimal field over which the image of the $l$-adic torsion representations have connected Zariski closure. We show that for all even $g \geq 4$, there exist infinitely many geometrically nonisogenous abelian varieties $A$ over $\mathbb{Q}$ of dimension $g$ where the connected monodromy field is strictly larger than the field of definition of the endomorphisms of $A$. Our construction arises from explicit families of hyperelliptic Jacobians with definite quaternionic multiplication. This is joint work with Victoria Cantoral-Farfan and Davide Lombardo.