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 $|({\mathfrak{g}}_l{)}_x⊊{\mathfrak{g}}_l\}$ is independent of $l$ and confirm Conjecture 5.5 in [2].

arXiv comment4 pages