The Jacobian of a (smooth, projective, geometrically integral) curve of genus $g$ is an abelian variety of dimension $g$ that is canonically associated to the curve.
Points on the Jacobian correspond to certain formal sums of points on the curve, modulo an equivalence relation. By choosing a rational point on the curve, one can embed the curve in its Jacobian (this embedding is known as the Abel-Jacobi map); for elliptic curves this embedding is an isomorphism, but otherwise not.
Note that it is possible for an abelian variety to be isogenous to the Jacobian of a curve without being isomorphic to one.
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- Last edited by Bjorn Poonen on 2022-03-26 16:16:15