Galois Theory and Torsion Points on Curves

We begin with a brief history of the problem of determining the set of points of a curve that map to torsion points of the curve’s Jacobian. Let K be a number field, and suppose that X/K is an algebraic curve of genus g ≥ 2. Assume, furthermore, that X is embedded in its Jacobian variety J via a K-rational Albanese map i; thus there is a K-rational divisor D of degree one on X such that i = iD : X ↪→ J is defined onK-valued points by the rule i(P ) = [(P )−D], where [ · ] denotes the linear equivalence class of a divisor on X. When D is a Krational point P0, we often refer to P0 as the base point of the embedding iQ. Let T := J(K) denote the torsion subgroup of J(K).