Is every abelian variety a subvariety of a Jacobian? Let $k$ be an infinite field and $A$ be an abelian variety over $k$. Can $A$ be embedded into a Jacobian variety $J$ over $k$?
In these notes by William Stein this is stated without proof in remark 1.5.8; it is attributed to personal conversation with Brian Conrad. Unfortunately I was unable to locate or come up with any proof.
 A: You can find a detailed proof here (theorem 1.2) in the case of principally polarized abelian varieties. One reduces to this case using the Zarhin's trick.
The assumption of $k$ being infinite should not be necessary (see remark 1.3 in the paper)
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. 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. 
A: Embed the dual abelian variety into projective space. Take a smooth hyperplane section and interate until it's one-dimensional, obtaining a smooth curve $C$. By Lefschetz $C$ is irreducible, and the natural map $H_1(C, \mathbb Z) \to H_1(A^\vee, \mathbb Z)$ is surjective. Because $H_1(C, \mathbb Z)= H_1(J(C), \mathbb Z)$, the natural map $H_1(J(C), \mathbb Z) \to H_1(A^\vee, \mathbb Z)$ is surjective as well. Now this does imply that the map of dual abelian varieties is surjective because abelian varieties over $\mathbb C$ are Pontryagin dual to the integral homology of the dual abelian varieties.
In characteristic $p$, a similar argument should work using the injectivitity on etale cohomology with torsion coefficients and crystalline cohomology with torsion coefficients / algebraic de Rham cohomology, but I didn't check the details.
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 $\mathrm{def}(A)$.

