Note: the below question doesn't make sense. I have mistakenly mixed two different $\pi_1$ and came up with a false claim. See my last comment.
Let $X$ be an abelian variety over an algebraically closed field $k$ of characteristic $p>0$. We fix a polarization $\lambda$ of $X$, of degree prime to $p$. This polarization induces the Weyl pairings $X[N]\times X[N]\rightarrow \mu_N$ on the $N$-torsion subgroups of $X$, for any integer $N$. These pairings are alternate (symplectic when $N$ is prime to the degree of $\lambda$).
Taking for $N$ the successive powers $l^m$ of a prime number $l\not = p$, these pairings are compatible with eachother and taking limit, tensoring with $\mathbb Q_l$, we obtain an alternate pairing $E_l:V_l(X)\times V_l(X)\rightarrow \mathbb Q_l(1)$ where $V_l(X)=T_l(X)\otimes_{\mathbb Z_l} \mathbb Q_l$ is the rational Tate-module.
Now, we have natural identifications of $T_l(X)$ (resp. $V_l(X)$) with the étale homology group $H_1(X,\mathbb Z_l)$ (resp. $H_1(X,\mathbb Q_l)$). Taking the restricted product of these groups for $l\not = p$, we obtain the group $H_1(X,\mathbb A^p_f)$ where $\mathbb A^p_f$ is the ring of finite adèles away from $p$, which is equipped with an alternate pairing to $\mathbb A_f^p(1)$.
Eventually, the étale fundamental group of $X$ is given by $\pi_1(X)=\prod_{l\text{ any prime}}T_l(X)$. This group acts on $H_1(X,\mathbb A^p_f)$ by addition componentwise, forgetting about the $p$-component.
In Kottwitz paper "Points on some Shimura varieties over finite field" in chapter $5$, it is claimed that $\pi_1(X)$ acts by similitudes on $H_1(X,\mathbb A^p_f)$. I fail to see why it is true, and I need help for this purpose.
If I interpret the statement componentwise, I believe it should mean that for any $l\not = p$, there exists some scalar $c\in \mathbb Q_l^{\times}$ such that for any $x,y\in V_l(X)$ and $a\in T_l(X)$, $E_l(x+a,y+a)=cE_l(x,y)$ in $\mathbb Q_l(1)$. I can't not see why this would hold...
Edit: I believe that my interpretation in the last paragraph must be totally wrong. If I consider $l$ prime to the degree of $\lambda$, then the pairing $E_l$ is symplectic. Assuming that my last statement were true, first I see that $c\not = 1$: otherwise I would have $E_l(x,a)+E_l(a,y)=E_l(x-y,a)=0$ for every $x,y$ and $a$ as above, which for a fixed $a$ implies $a=0$, that is absurd. But if $c\not = 1$, simply taking $a=0$ gives an absurdity.
This adds up to my confusion, I would gladly appreciate any explanation about how I ought to understand "by similitudes" here.