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Let $A^*=Pic^0(A)$ and apply $Pic^0$ to a surjective homomorphism $J\to A^*$. Added: as noted, the existence of a surjective homomorphism goes back to a theorem of Matsusaka. It is proved in Milne, Jacobian Varieties, 1986, 10.1.

Let $A^*=Pic^0(A)$ and apply $Pic^0$ to a surjective homomorphism $J\to A^*$. Added: as noted, the existence of a surjective homomorphism goes back to a theorem of Matsusaka. For the proof, see Milne, Jacobian Varieties, 1986, 10.1. The argument in Kleiman, Algebraic cycles and the Weil conjectures, 1968, 2A7, shows that the kernel of the dual homomorphism is finite and not divisible by any $l$ prime to the characteristic. I expect that the argument can be made to work also for $p$, but I haven't checked this.

Let $A^*=Pic^0(A)$ and apply $Pic^0$ to a surjective homomorphism $J\to A^*$. Added: as noted, the existence of a surjective homomorphism goes back to a theorem of Matsusaka. It is proved in Milne, Jacobian Varieties, 1986, 10.1. As far as the duality is concerned, in terms of the functor $Pic^0$ it is rather obvious that the dual of a surjective homomorphism is injective. (If it is an isogeny, then one even knows that the kernel of the dual isogeny is the Cartier dual of the kernel of the isogeny; see Mumford, Abelian Varieties, 1970, Theorem 1, p.143, where such things are explained in detail).

Let $A^*=Pic^0(A)$ and apply $Pic^0$ to a surjective homomorphism $J\to A^*$. Added: as noted, the existence of a surjective homomorphism goes back to a theorem of Matsusaka. It is proved in Milne, Jacobian Varieties, 1986, 10.1.

Let $A^*=Pic^0(A)$ and apply $Pic^0$ to a surjective homomorphism $J\to A^*$.

Let $A^*=Pic^0(A)$ and apply $Pic^0$ to a surjective homomorphism $J\to A^*$. Added: as noted, the existence of a surjective homomorphism goes back to a theorem of Matsusaka. It is proved in Milne, Jacobian Varieties, 1986, 10.1. As far as the duality is concerned, in terms of the functor $Pic^0$ it is rather obvious that the dual of a surjective homomorphism is injective. (If it is an isogeny, then one even knows that the kernel of the dual isogeny is the Cartier dual of the kernel of the isogeny; see Mumford, Abelian Varieties, 1970, Theorem 1, p.143, where such things are explained in detail).