Charm $CP$ violation measurements with $K^0_{\rm S}$ final states at LHCb

The LHCb collaboration achieved the world's most precise measurement of the $\mathcal{A}^{CP}(D^0 \rightarrow K^0_{\rm S} K^0_{\rm S})$ parameter using data collected during Run 2 (2015–2018), reaching a precision of $1.3\%$ and observing compatibility with the absence of  violation at a significance of $2.4\sigma$. Following a major Upgrade I, LHCb resumed operations in 2022 with a completely overhauled tracking system and a novel triggerless readout architecture, capable of reconstructing events at the full LHC bunch crossing rate of 30 MHz. This thesis presents the design and optimization of dedicated $K^0_{\rm S}$ triggers, aimed at enhancing the LHCb online selection efficiency of $D^0 \rightarrow K^0_{\rm S} K^0_{\rm S}$ decays. The new triggers have been commissioned with early Run 3 data; performed tests confirm the feasibility of this new trigger strategy, never implemented before at a hadron collider. With the availability of larger samples, the impact on signal efficiency has been evaluated, leading to a fourfold enhancement for the target decay $D^0 \rightarrow K^0_{\rm S} K^0_{\rm S}$. These performances motivated an early $\mathcal{A}^{CP}(K^0_{\rm S} K^0_{\rm S})$ measurement performed on $6$ fb$^{-1}$ of data collected by LHCb in 2024, that already surpassed the precision of the previous measurement. This measurement is a remarkable result, being the most precise determination of this parameter to date, and demonstrating the effectiveness of the new triggers in enhancing the physics reach of the LHCb experiment.