Let \(A\) be an abelian variety over a field \(K\) of characteristic \(p>0\) and let \(K_s\) be the separable closure of \(K\). Let \(A^{(p^n)}\) be the image of \(A\) under the \(n\)-th power of the Frobenius and \(V_n: A^{(p^n)}\to A\) the dual isogeny. The period lattice of \(A\) is defined by \(\Lambda= \varprojlim \operatorname {Ker} V_n\). The author proves that there exists an exact sequence of \(G\)-modules \((G= \operatorname {Gal} (K_s/K))\): \[ \Lambda\to \widehat{K}_s^* \otimes \Lambda^{\otimes(-1)}\to \widehat{A(K_s)}\to 0. \] As an application it is proved that the natural homomorphism \(\operatorname {End}(A)\otimes \mathbb{Z}_p\to \operatorname {End}(\Lambda)\) is injective if \(A\) is sufficiently general.