Arithmetic of Jacobi polynomials

We study the arithmetic of Jacobi polynomials. As applications, we prove the Mahlburg–Ono conjecture on the irreducibility and Galois groups of the four Kaneko–Zagier families, and the Cullinan–Hajir conjecture on the Galois groups of Legendre polynomials. Other applications concern common zeros of Jacobi polynomials, harmonic functions on cones, and hyperelliptic Jacobians.