I am in an ideal situation but I would appreciate a hint. First here is the scenario.


Let $\mathcal{J}$ be an the abelian variety obtained from the Jacobian of a genus $2$ curve $\mathcal{H}/k$ which has one $k$-rational point $\infty$ . Consider $\Theta\subset\mathcal{A}$ given by the the image of the Abel-Jacobi map $\iota:\mathcal{H}\to\mathcal{J}$ where $\iota(P)=[P-\infty]$. We have that $\Theta$ can be regarded as a divisor in $\text{Div}(\mathcal{J})$ which is ample since a basis of $\mathcal{L}(4\Theta)$ defines an embeding of $\mathcal{J}\hookrightarrow\mathbb{P}^{15}$.


The question:


Let $D\_1,D\_2\in\text{Div}(\mathcal{J})$. We denote by $D\_1\bullet D\_2$ the intersection number of the divisors $D\_1$ and $D\_2$ (In this case working with the abelian variety $\mathcal{J}$ of dimension $2$, the divisors $D\_1$ and $D\_2$ can be regarded as algebraic curves inside $\mathcal{J}$, therefore $\bullet$ denotes the cardinality of $D\_1\cap D\_2$ as curves inside $\mathcal{J}$ with counted multiplicities).


Let $\gamma\in\text{End}\_k(\mathcal{J})$ I want to know if under all these hypotheses the following two integers are equal: 


$ \gamma^\*\Theta\bullet\Theta=^?\gamma\_\*\Theta\bullet\Theta $.


And if not, how they are related.


Here $\gamma^\*$ and $\gamma\_\*$ are the pull back and push forward maps from $\text{Div}(\mathcal{J})\to\text{Div}(\mathcal{J})$.


I have checked in Fulton's but I cannot find some example or theorem what helps me deducing this but this should be something already known.



Intersection number of divisors with its pull back and its push forward
There is a very general projection formula that's valid for any proper morphism $f:X\to Y$ and any cycles $x$ and $y$ in the Chow groups of $X$ and $Y$ respectively:
$$ f\_\*(x\cdot f^\*y) = f\_\*(x)\cdot y. $$
In your case, the intersections are 0-cycles, so taking the degree of both sides gives the formula that you want. You can find the projection formula stated in Hartshorne *Algebraic Geometry*, Appendix A, Property A4, for example. 


