Detailed

Context. Until now, heavy interstellar extinction has meant that only a few studies of chemical abundances have been possible in the inner Galactic bulge. However, it is crucial to learn more about this structure in order to better understand the formation and evolution of the centre of the Galaxy and galaxies in general. Aims. In this paper, we aim to derive high-precision α-element abundances using CRIRES high-resolution IR spectra of 72 cool M giants of the inner Galactic bulge. Methods. Silicon, magnesium, and calcium abundances were determined by fitting a synthetic spectrum for each star. We also incorporated recent theoretical data into our spectroscopic analysis (i.e. updated K-band line list, better broadening parameter estimation, non-local thermodynamic equilibrium (NLTE) corrections). We compare these inner bulge α abundance trends with those of solar neighbourhood stars observed with IGRINS using the same line list and analysis technique; we also compare our sample to APOGEE DR17 abundances for inner bulge stars. We investigate bulge membership using spectro-photometric distances and orbital simulations. We construct a chemical-evolution model that fits our metallicity distribution function (MDF) and our α-element trends. Results. Among our 72 stars, we find four that are not bulge members. [Si/Fe] and [Mg/Fe] versus [Fe/H] trends show a typical thick disc α-element behaviour, except that we do not see any plateau at supersolar metallicities as seen in other works. The NLTE analysis lowers [Mg/Fe] typically by ∼0.1 dex, resulting in a noticeably lower trend of [Mg/Fe] versus [Fe/H]. The derived [Ca/Fe] versus [Fe/H] trend has a larger scatter than those for Si and Mg, but is in excellent agreement with local thin and thick disc trends. With our updated analysis, we constructed one of the most detailed studies of the α abundance trends of cool M giants in the inner Galactic bulge. We modelled these abundances by adopting a two-infall chemical-evolution model with two distinct gas-infall episodes with timescales of 0.4 Gyr and 2 Gyr, respectively. Conclusions. Based on a very meticulous spectral analysis, we have constructed detailed and precise chemical abundances of Mg, Si, and Ca for cool M giants. The present study can be used as a benchmark for future spectroscopic surveys