Rotational modulation and long-term evolution of the small-scale magnetic fields of M dwarfs observed with SPIRou

M dwarfs are known to host magnetic fields, impacting exoplanet studies and playing a key role in stellar and planetary formation and evolution. Observational constraints are essential to guide theories of dynamo processes believed to be at the origin of those fields, in particular for fully convective stars whose internal structure differs from that of partially convective stars. Observations reveal long-term evolution of the large-scale magnetic field reconstructed with Zeeman-Doppler imaging, and a diversity of their topologies. These large-scale magnetic fields, however, only account for a small amount of the unsigned magnetic flux at the stellar surface that can be probed by directly modeling the Zeeman broadening of spectral lines in unpolarized spectra. We aim to investigate the long-term behavior of the average small-scale magnetic field of fully convective and partially convective M dwarfs with time, and assess our ability to detect rotational modulation and retrieve rotation periods from time series of field measurements derived from unpolarized spectra. We performed fits of synthetic spectra computed with ZeeTurbo to near-infrared high-resolution spectra recorded with SPIRou between 2019 and 2024 in the context of the SLS and SPICE large programs. The analysis was performed on the spectra of two partially convective (AD Leo and DS Leo) and three fully convective (PM J18482+0741, CN Leo, and Barnard's star) M dwarfs, along with EV Lac, whose mass is close to the fully convective limit. Our analysis provides measurements of the average small-scale magnetic field, which are compared to longitudinal magnetic field and temperature variation measurements (dTemp) obtained from the same data. We detected the rotation period in the small-scale magnetic field series for four of the stars in our sample. We find that the average magnetic field can vary by up to 0.3,kG throughout the year (e.g., CN Leo), or of up to 1,kG across rotation phases (e.g., EV Lac). The rotation periods retrieved from longitudinal and small-scale magnetic fields are found to agree within error bars. The dTemp measurements are found to anti-correlate with small-scale magnetic field measurements for three stars (EV Lac, DS Leo, and Barnard's star). The results demonstrate our ability to measure rotation periods from high-resolution data through small-scale magnetic field measurements, provided that the inclination of the observed targets is sufficiently large. We observe long-term fluctuations of the average magnetic field that could indicate magnetic cycles in the parent dynamo processes. These long-term variations appear mainly uncorrelated with large-scale magnetic field variations probed through longitudinal field measurements. Large variations in the amplitude of the rotationally modulated signals, in particular, hint towards a change in the distribution of the surface inhomogeneities accessible to Zeeman broadening measurements.