= Theta autoduality of a Jacobian
{title2=$\lambda_\Theta:J(C)\xrightarrow{\sim}\widehat{J(C)}$}
For $T_q(x)=x+q$, the <homomorphism associated to a line bundle on an abelian variety> gives $\lambda_\Theta(q)=T_q^*\mathcal O(\Theta)\otimes\mathcal O(\Theta)^{-1}$. Restrict to the negative <Abel map of an algebraic curve>. The <restriction of the theta divisor to an Abel curve> identifies this restriction with the degree-zero class $q$. Pullback of algebraically trivial <line bundles> does not change when the embedding is translated. Thus pullback $\rho:\widehat{J(C)}\to J(C)$ satisfies $\rho\lambda_\Theta=1$. Equal dimensions and properness make $\lambda_\Theta$ an isomorphism, including on schemes, rather than only on closed points.
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