By a theorem of Matsusaka, every abelian variety $A$ over an algebraic closed field $k$ is a quotient of a Jacobian.

Now just apply Matsusaka's theorem to $A^{\vee}$, and dualize.

I think that the slightly weaker version where the field is infinite is proved somewhere in Milne's lecture notes on Abelian Varieties.

Reference.

T. Matsusaka: On a generating curve of an Abelian variety, Nat. Sci. Rep. Ochanomizu Univ. 3 1-4, (1952).

Quoting from P. Samuel review on MathSciNet:

An abelian variety $A$ is said to be generated by a variety $V$ (and a mapping $f$ of $V$ into $A$) if $A$ is the group generated by $f(V)$. It is proved that every abelian variety $A$ may be generated by a curve defined over the algebraic closure of $\mathrm{def}(A)$.

Let me give an answer for $k = \mathbb{C}$.

By a theorem of Matsusaka, every abelian variety $A$ over an algebraic closed field $k$ is a quotient of a Jacobian.

Now just apply Matsusaka's theorem to $A^{\vee}$, and dualize. Since we are over $\mathbb{C}$, dualization sends surjective morphisms of Abelian varieties into injective ones, so we are done.

I think that the slightly weaker version where the field is infinite is proved somewhere in Milne's lecture notes on Abelian Varieties.

Reference.

T. Matsusaka: On a generating curve of an Abelian variety, Nat. Sci. Rep. Ochanomizu Univ. 3 1-4, (1952).

Quoting from P. Samuel review on MathSciNet:

An abelian variety $A$ is said to be generated by a variety $V$ (and a mapping $f$ of $V$ into $A$) if $A$ is the group generated by $f(V)$. It is proved that every abelian variety $A$ may be generated by a curve defined over the algebraic closure of $\mathrm{def}(A)$.

By a theorem of Matsusaka, every abelian variety $A$ is a quotient of a Jacobian.

Now just apply Matsusaka's theorem to $A^{\vee}$, and dualize.

By a theorem of Matsusaka, every abelian variety $A$ over an algebraic closed field $k$ is a quotient of a Jacobian.

Now just apply Matsusaka's theorem to $A^{\vee}$, and dualize.

I think that the slightly weaker version where the field is infinite is proved somewhere in Milne's lecture notes on Abelian Varieties.

Reference.

T. Matsusaka: On a generating curve of an Abelian variety, Nat. Sci. Rep. Ochanomizu Univ. 3 1-4, (1952).

Quoting from P. Samuel review on MathSciNet:

An abelian variety $A$ is said to be generated by a variety $V$ (and a mapping $f$ of $V$ into $A$) if $A$ is the group generated by $f(V)$. It is proved that every abelian variety $A$ may be generated by a curve defined over the algebraic closure of $\mathrm{def}(A)$.