Specialization of monodromy group and l-independence

Abstract

Let $E$ be an abelian scheme over a geometrically connected variety $X$ defined over $k$, a finitely generated field over $\mathbb{Q}$. Let $\eta$ be the generic point of $X$ and $x\in X$ a closed point. If $\mathfrak{g}_l$ and $(\mathfrak{g}_l)_x$ are the Lie algebras of the $l$-adic Galois representations for abelian varieties $E_{\eta}$ and $E_x$, then $(\mathfrak{g}_l)_x$ is embedded in $\mathfrak{g}_l$ by specialization. We prove that the set $\{x\in X$ closed point $| (\mathfrak{g}_l)_x\subsetneq \mathfrak{g}_l\}$ is independent of $l$ and confirm Conjecture 5.5 in [2].