Rational Points on Entanglement Modular Curves

A degree $d$ point on a curve is isolated if it does not belong to an infinite family of degree $d$ points with a geometric parameterization. If an isolated point on the modular curve $X_1(N)$ comes from an elliptic curve $E$ with rational $j$-invariant, then $j(E)$ is expected to be one of 17 known ``isolated $j$-invariants” in $\mathbb{Q}$. This has been proven unconditionally for prime-power levels $N$ in joint work with Ejder. In this talk, we will discuss additional unconditional results in this direction, highlighting the role of finding rational points on modular curves which characterize unexpected entanglement of torsion point fields.