Specialization of monodromy group and l-independence

TL;DR

This paper proves that the set of points where the specialized Lie algebra of an abelian scheme's Galois representation is strictly smaller than the generic one is independent of the prime l, confirming a conjecture in the field.

Contribution

It establishes l-independence of the specialization of Lie algebras of Galois representations for abelian schemes, confirming a specific conjecture.

Findings

The set of points with smaller specialized Lie algebra is l-independent.

Confirmed Conjecture 5.5 in the referenced work.

Provides new insights into the structure of Galois representations in algebraic geometry.

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].