Let $X$ be a smooth, projective curve of genus $g \ge 2$. We know that the Jacobian $J(X)$ of the curve is a principally polarized abelian variety. The principal polarization is induced by the intersection form (or cup-product) on $H^1(X,\mathbb{Z})$. My question is: Is this the only principal polarization on $J(X)$? In other words, is there another unimodular, alternating, non-degenerate bilinear form on $H^1(X,\mathbb{Z})$ different from the cup-product (this will induce a different principal polarization on $J(X)$)?


I would think this is very basic, but I am not able find a good literature for this question. Any hint/reference will be most welcome.



Naive question on the Jacobian of a curve
It is possible for the Jacobian's of non-isomorphic curves to be isomorphic as abelian varieties, but obviously, not as principally polarized abelian varieties. This paper <https://arxiv.org/pdf/math/0304471.pdf> by Howe gives examples. Also from the same paper - "it has been known since the late 1800s that distinct curves
can have isomorphic unpolarized Jacobians. I"


