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](https://wstein.org/edu/Fall2003/252/lectures/10-17-03/10-17-03.pdf) 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.



Is every abelian variety a subvariety of a Jacobian?
You can find a detailed proof [here](https://webusers.imj-prg.fr/~anna.cadoret/TC.pdf "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)


