Does an Isogeny induce an Injective map on $Pic^0$

Question

Suppose that $\varphi \colon A_1 \to A_2$ is an isogeny between two abelian varieties (of the same dimension). Is the induced map $\varphi^* \colon Pic^\circ(A_2) \to Pic^\circ(A_1)$ always injective?

Answer 1

No this should essentially never be the case. If $f: E \to E'$ is a degree $d$ isogeny of elliptic curves (say, over $\mathbb{C}$) then the dual isogeny $\hat{f}: \operatorname{Pic}^0(E') \to \operatorname{Pic}^0(E)$ has degree $d$ as well. (cf. Hartshorne exercise IV.4.7) In particular $\hat{f}$ is not injective when $d > 1$.

This fact I used can be generalized to abelian varieties with some thought. See Proposition 5.3.2, on page 38 of these notes, for example.