The ChemCam instrument package on the Mars rover, Curiosity, provides new capabilities to probe the abundances of certain trace elements in the rocks and soils on Mars using the laser-induced breakdown spectroscopy technique. We focus on detecting and quantifying Li, Ba, Rb, and Sr in targets analyzed during the first 100 sols, from Bradbury Landing Site to Rocknest. Univariate peak area models and multivariate partial least squares models are presented. Li, detected for the first time directly on Mars, is generally low (<15 ppm). The lack of soil enrichment in Li, which is highly fluid mobile, is consistent with limited influx of subsurface waters contributing to the upper soils. Localized enrichments of up to ~60 ppm Li have been observed in several rocks but the host mineral for Li is unclear. Bathurst_Inlet is a fine-grained bedrock unit in which several analysis locations show a decrease in Li and other alkalis with depth, which may imply that the unit has undergone low-level aqueous alteration that has preferentially drawn the alkalis to the surface. Ba (~1000 ppm) was detected in a buried pebble in the Akaitcho sand ripple and it appears to correlate with Si, Al, Na, and K, indicating a possible feldspathic composition. Rb and Sr are in the conglomerate Link at abundances >100 ppm and >1000 ppm, respectively. These analysis locations tend to have high Si and alkali abundances, consistent with a feldspar composition. Together, these trace element observations provide possible evidence of magma differentiation and aqueous alteration.