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Revealing the nature of the ULX and X-ray population of the spiral galaxy NGC 4088

We present the first \textit{Chandra} and \textit{Swift} X-ray study of the spiral galaxy NGC\,4088 and its ultraluminous X-ray source (ULX N4088--X1). We also report very long baseline interferometry (VLBI) observations at 1.6 and 5 GHz performed quasi-simultaneously with the \textit{Swift} and \textit{Chandra} observations, respectively. Fifteen X-ray sources are detected by \textit{Chandra} within the D25 ellipse of NGC\,4088, from which we derive the X-ray luminosity function (XLF) of this galaxy. We find the XLF is very similar to those of star-forming galaxies and estimate a star-formation rate of 4.5 $M_{\odot}$ yr$^{-1}$. The \textit{Chandra} detection of the ULX yields its most accurate X-ray position, which is spatially coincident with compact radio emission at 1.6 GHz. The ULX \textit{Chandra} X-ray luminosity, $L_\mathrm{0.2-10.0 keV} = 3.4\ \times$ 10$^{39}$ erg s$^{-1}$, indicates that N4088--X1 could be located at the high-luminosity end of the high-mass X-ray binary (HMXB) population of NGC\,4088. The estimates of the black hole mass and ratio of radio to X-ray luminosity of N4088--X1 rule out a supermassive black hole nature. The \textit{Swift} X-ray spectrum of N4088--X1 is best described by a thermal Comptonization model and presents a statistically significant high-energy cut-off. We conclude that N4088--X1 is most likely a stellar remnant black hole in a HMXB, probably fed by Roche lobe overflow, residing in a super-Eddington ultraluminous state. The 1.6 GHz VLBI source is consistent with radio emission from possible ballistic jet ejections in this state.