For this question I will let the overly ambiguous word curve mean: smooth projective and connected curve over $\mathbb C$ (or equivalently a smooth compact Riemann-Surface). 


Let $C$ be a curve over $\mathbb C$. And suppose that $f: D \to C$ and $g:D \to E$ are dominant maps of curves. Then the pair $(f,g)$ forms a correspondence from $C$ to $E$, and one gets a map between Jacobians $f\_\* \circ g^\* : J(E) \to J(C)$. Then the image of $J(E)$ under this map is going to be a sub-abelian variety of $J(C)$. 


My questions is, is it possible to get all simple sub-abelian varieties of $J(C)$ from correspondences in this way?



Do all simple factors of jacobians of curves come from correspondences?
This argument is mostly contained in t3suji's comments, but with some of the proofs somewhat expanded.


As a convention (consistent with that of the theory of Chow motives), all actions of correspondences $\alpha \in \operatorname{CH}^\*(X \times Y)$ on cohomology (or Jacobians) are *covariant*: $\alpha\_\* \colon H^\*(X) \to H^\*(Y)$, etc.


**Lemma.** If $X$ and $Y$ are smooth projective varieties over $\mathbb C$, then there is an isomorphism
$$\operatorname{NS}(X \times Y) \cong \operatorname{NS}(X) \times \operatorname{NS}(Y) \times \operatorname{Hom}(\operatorname{Alb}\_X,\operatorname{Pic}^0\_Y).\label{Eq NS}\tag{1}$$


*Proof.* The Künneth formula gives an isomorphism of $\mathbb Z$-Hodge structures
$$H^2(X \times Y) = \big( H^2(X) \otimes H^0(Y)\big) \oplus \big( H^0(X) \otimes H^2(Y) \big) \oplus \big( H^1(X) \otimes H^1(Y) \big).\tag{2}\label{Eq Künneth}$$
By Poincaré duality, the last term equals $\operatorname{Hom}(H^{2\dim X-1}(X),H^1(Y))$. Taking $(1,1)$-parts in (\ref{Eq Künneth}) gives the result by Lefschetz's (1,1) theorem and the theory of abelian varieties over $\mathbb C$. $\square$


There is also a version for $\operatorname{Pic}$ for smooth projective varieties over an arbitrary field (or even over a base, under some mild hypotheses to ensure existence of $\operatorname{\textbf{Pic}}\_{X/S}$ and $\operatorname{\textbf{Alb}}\_{X/S}$). The above analytic proof has the advantage that it shows us how classes act on Jacobians: if $X$ and $Y$ are curves, then $\operatorname{Alb}\_X \cong \operatorname{Pic}\_X^0 \cong \operatorname{Jac}\_X$, and under these identifications, a class $\alpha \in \operatorname{CH}^1(X \times Y)$ induces the map $\alpha\_\* \colon \operatorname{Jac}\_X \to \operatorname{Jac}\_Y$ given by the corrresponding component of $\alpha$ under (\ref{Eq NS}).


**Construction.** Let $A \subseteq \operatorname{Jac}\_C$ be any abelian subvariety. Let $E$ be any curve admitting a surjection $\operatorname{Jac}\_E \to A$; for example, $E = C$ with the projection $\operatorname{Jac}\_C \to A$ given by $n \pi$ for $\pi \in \operatorname{End}^\circ(\operatorname{Jac}\_C)$ the projector corresponding to the isogeny factor $A$ of $\operatorname{Jac}\_C$, and $n \in \mathbb Z\_{> 0}$ such that $n\pi$ is integral. The composition gives a morphism $\phi \colon \operatorname{Jac}\_E \to \operatorname{Jac}\_C$, corresponding by (\ref{Eq NS}) to a class $\alpha \in \operatorname{NS}(X \times Y)$.


The class $(1,1,0)$ in (\ref{Eq NS}) is ample, hence for some $n \gg 0$ there exists a very ample line bundle $\mathscr L$ on $X \times Y$ mapping to $(n,n,\alpha) \in \operatorname{NS}(X \times Y)$. Let $D \in |\mathscr L|$ be a smooth member. By the discussion above, the cycle $[D] \in \operatorname{CH}^1(E \times C)$ induces the map $\phi$.


The graphs of the maps $g \colon D \to E$ and $f \colon D \to C$ give cycles $[\Gamma\_g]^\top \in \operatorname{CH}^1(E \times D)$ and $[\Gamma\_f] \in \operatorname{CH}^1(D \times C)$, whose actions on $\operatorname{Jac}$ are given by $g^\*$ and $f\_\*$ respectively. Moreover, we have
$$[\Gamma\_g]^\top \circ [\Gamma\_f] = [D] \in \operatorname{CH}^1(E \times C),$$
which shows that indeed $\phi = f\_\* \circ g^\*$. $\square$


