Let $M$ be an affine complex manifold, let $A$ be an abelian scheme over $M$. Let $\mathcal{A}$ be the sheaf of local sections of $A$ over $M$. We can equip $M$ with etale topology $M_{et}$ or complex topology $M_{an}$. There is a natural comparison map $$\gamma\colon H^1(M_{et},\mathcal{A})\to H^1(M_{an},\mathcal{A})$$ between the corresponding sheaf cohomologies.

(a) Are there explicit examples such that $\mathrm{ker}(\gamma)\neq 0$?

(b) Are there explicit examples such that $\mathrm{coker}(\gamma)\neq 0$?

[In case (a) we need to find an algebraic $A$-torsor $T$, such that $T/M$ admits an analytic section but not an algebraic section. Suppose such $T$ exist, we can consider its relative albanese map $a:T\to \mathrm{Alb}(T)$, this is a family of algebraic maps over $M$, I think there should be plenty of non-algebraic family of algebraic maps, but I am not sure how to make such an example..]

[In case (b), I think we need to find a non-algebraic family of complex torus $T'/M$, whose albanese is algebraic, not sure how to find one...]