Abstract: In this paper, we are concerned with solutions to the fractional Schrödinger-Poisson system
$\begin{cases}(-\Delta)^su-\phi|u|^{2_s^*-3}u=\lambda u+\mu|u|^{q-2}u+|u|^{2_s^*-2}u,&x\in\mathbb R^3,\\(-\Delta)^s\phi=|u|^{2_s^*-1},&x\in\mathbb R^3,\end{cases} $
with prescribed mass $\int_{\mathbb{R}^3}|u|^2\mathrm{d}x = a^2,$ where $a>0$ is a prescribed number, $\mu>0$ is a paremeter, $s \in (0,1), 2<q<2^*_s$, and $ 2^*_s = \frac{6}{3-2s}$ is the fractional critical Sobolev exponent. In the $L^2$-subcritical case, we show the existence of multiple normalized solutions by using the genus theory and the truncation technique; in the $L^2$-supercritical case, we obtain a couple of normalized solutions by developing a fiber map. Under both cases, to recover the loss of compactness of the energy functional caused by the doubly critical growth, we need to adopt the concentration-compactness principle. Our results complement and improve upon some existing studies on the fractional Schrödinger-Poisson system with a nonlocal critical term.

Key words: fractional Schrödinger-Poisson system, normalized solutions, variational methods, $L^2$-subcritical, $L^2$-supercritical