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In this paper, we consider the following magnetic nonlinear Choquard equation\[-(\nabla+iA(x))^2u+ V(x)u =\left(\frac{1}{|x|^{\alpha}}*|u|^{2_{\alpha}^*}\right) |u|^{2_{\alpha}^*-2} u +\lambda f(u)\ \textrm{ in }\ \R^N,\] where$2_{\alpha}^{*}=\frac{2N-\alpha}{N-2}$ is the critical exponent in the sense ofthe Hardy-Littlewood-Sobolev inequality, $\lambda>0$, $N\geq 3$, $0<\alpha< N$,$A: \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is an $C^1$,$\mathbb{Z}^N$-periodic vector potential and $V$ is a continuous scalarpotential given as a perturbation of a periodic potential. Under suitableassumptions on different types of nonlinearities $f$, namely,$f(x,u)=\left(\frac{1}{|x|^{\alpha}}*|u|^{p}\right)|u|^{p-2} u$ for$(2N-\alpha)/N<p<2^{*}_{\alpha}$, then $f(u)=|u|^{p-1} u$ for $1<p<2^*-1$ and$f(u)=|u|^{2^* - 2}u$ (where $2^*=2N/(N-2)$), we prove the existence of atleast one ground state solution for this equation by variational methods if $p$belongs to some intervals depending on $N$ and $\lambda$.

Nonlinear Perturbations of a periodic magnetic Choquard equation with Hardy-Littlewood-Sobolev critical exponent