$\newcommand{\Pic}{\operatorname{Pic}}$$\newcommand{\mcL}{\mathcal{L}}$ The context for this is Hartshorne's exercise IV.4.10. I am having trouble defining this map $\Pic(X \times X) \to R$ where $R$ is the endomorphism ring of an elliptic curve $X$ (over some algebraically closed field). That is, $R$ is the ring of finite morphisms $f: X \to X$ which send the basepoint $P_0$ to itself.
Since $X$ is its own Jacobian variety, I see how we can view $R$ as a subgroup of $\Pic^0(X/X)$. I am having trouble projecting $\Pic(X \times X)$ to $\Pic^0(X \times X) = \{[\mcL] \in \Pic(X \times X)\;|\;\deg(\mcL_y) = 0\:\forall y \in X\}$ where these are fibers of the second projection $p_2: X \times X \to X$. I tried to use semicontinuity and flatness to find that $\mcL$ has constant degree along the fibers, so that I could perhaps replace $\mcL$ by some $\mcL - p_1^*\mathcal{O}(dP_0) \in \Pic^0(X \times X)$. I can't seem to get this to work, though, and I somewhat doubt that it's true that the degree is constant over each fiber.
Is there some other projection I could use to find this surjective map?
Thanks!
Short answer: try the endomorphism $1-\pi_2^{\star}$ of $\mathrm{Pic}(X \times X)$, where $\pi_2=(O,\mathrm{id}): X \times X \rightarrow X \times X$.
Long answer: Let $X_1, X_2$ be smooth projective geometrically connected curves of some positive genus with $k$-rational base points $O_1,O_2$. Let $J_i$ be the Jacobian of $X_i$, endowed with an Abel-Jacobi immersion $O_i \rightarrow 0$. We want to construct a natural map $\mathrm{Pic}(X_1\times X_2) \rightarrow \mathrm{Hom}(J_1,J_2)$.
(I find this general setting easier to manipulate – all that remains for you to do is specialize to $X_1=X_2=J_1=J_2=X$ for some elliptic curve $X$).
By Albanese functoriality, $\mathrm{Hom}(J_1,J_2)=\mathrm{Hom}_{O_1 \mapsto 0}(X_1,J_2)$. It’s well-known that this set is the set of equivalence classes $L \in \mathrm{Pic}(X_1 \times X_2)$ such that for all geometric points $t$ of $X_1$, $L_{t}$ has degree zero, modulo the pullbacks of line bundles on $X_1$ and such that $L_{|\{O_1\} \times X_2}$ is trivial.
Since $X_1$ is connected, the last condition is sufficient: $\mathrm{Hom}_{O_1 \mapsto 0}(X_1,J_2)$ is the set of line bundles $L$ on $X_1 \times X_2$ such that $L_{|\{O_1\} \times X_2}$ is trivial, modulo the pullbacks of line bundles on $X_1$.
So the goal is to find a good map from $\mathrm{Pic}(X_1 \times X_2)$ to the subset of line bundles that become trivial when restricted to $\{O_1\} \times X_2$.
Let $\pi_2= (O_1, \mathrm{id}): X_1 \times X_2 \rightarrow X_1 \times X_2$: then $\pi_2^2=\pi_2$, and if $i_2$ is the natural injection $\{O_1\} \times X_2 \subset X_1 \times X_2$, $\pi_2 \circ i_2= \pi_2$.
The projection we seek is then $1-\pi_2^{\star}$.