I am reading Fulton's but I cannot find a useful result for my problem that seems to be something well known.


Let $\mathcal{J}$ be the Jacobian of a hyperelliptic curve of genus $2$. Consider $D\_1,D\_2\in \text{Div}(\mathcal{J})$ (two curves inside $\mathcal{J}$) and let $D\_2'$ be a translation by a $2$-Torsion point of $\mathcal{J}$. 



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> **Q:** Is it true that $D\_1\cdot D\_2=D\_1\cdot D\_2'$ ? 
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I cannot see directly that $D\_2\sim D\_2'$ always. I will appreciate a source in case of being true, or a condition for $D\_2$ for this to happen.



Intersection number of divisors on abelian surfaces and its invariance under translation by 2-Torsion points
The answer is *yes* if $D\_2$ is an ample curve. More generally, the following result holds:



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> Two *ample* line bundles on an abelian variety
>  are algebraically equivalent if and only if they differ by a translation. 
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This is essentially Corollary 2.5.4 in Birkhenake-Lange book *Complex Abelian Varieties* (it is at p. 40 in the second edition), where it is stated for analytically equivalent line bundles on (non necessarily projective) complex tori.


