The electron mass from $g$-factor measurements on hydrogen-like carbon $^{12}$C$^{5+}$

The electron mass in atomic mass units has been determined with a relative uncertainty of $2.8\cdot 10^{-11}$, which represents a 13-fold improvement of the 2010 CODATA value. The underlying measurement principle combines a high-precision measurement of the Larmor-to-cyclotron frequency ratio on a single hydrogen-like carbon ion in a Penning trap with a corresponding very accurate $g$-factor calculation. Here, we present the measurement results in detail, including a comprehensive discussion of the systematic shifts and their uncertainties. A special focus is set on the various sources of phase jitters, which are essential for the understanding of the applied line-shape model for the $g$-factor resonance.