How Do Generic Properties Spread?

Given a family of algebraic varieties, a natural question to ask is which properties of the generic fiber extend to the other fibers, and in what way they extend. Let us explore this topic, from an arithmetic point of view, in the following setting:

let $A \rightarrow X$ be a one-parameter family of principally polarized abelian varieties over a number field $K$, whose generic fiber has big monodromy.

How often does the specialization $A_x$ fail to be simple?

Equivalently, how often does the associated $\bmod \ell$ Galois representation drop into a proper subgroup of $\operatorname{GSp}_{2 g}\left(\mathbb{F}_{\ell}\right)$?

Can we give a quantitative estimate for the number of specializations of height at most $B$ where the monodromy drops?

I will give an answer to this question. Conceptually, this provides a quantitative illustration of the Mumford-Tate conjecture: failures of genericity are sparse and occur only on a height -density zero set of rational points.