The MESA facility at Mainz will test the Standard Model (SM) and beyond with low-energy polarized electron scattering. One of the cornerstones of the experimental program at MESA, the P2 experiment, is a ~1.8% measurement of the proton’s weak charge via elastic parity-violating electron scattering (PVES), resulting in a ~0.15% determination of the weak mixing angle. PVES on 12C is under discussion as an additional avenue for several reasons. First of all, such a measurement will serve as a calibration measurement to study the performance of the P2 experimental apparatus. Moreover, a measurement on 12C is expected to enhance the physics output of PVES@MESA as it will probe a different isospin combination of effective 4-fermion operators. The authors of this proposal as part of a German-Mexican DFG-CONACyT cooperation have started to work out theoretical predictions needed for a conclusive interpretation of a 0.3% measurement of the weak charge of 12C at MESA. Some results are expected by mid 2020, but some questions will remain open. The Scientific Program is dedicated to presenting results of this cooperation and to identify those open problems which will have to be solved to ensure a successful 12C physics program at MESA. The physics case of PVES on 12C will be discussed in a more general context: the coherent scattering of charged leptons and neutrinos with nuclei, that is scattering without exciting the nuclei or producing new particles. Such coherent interactions involving conserved currents provide a framework for setting up precision tests of the Standard Model at low energies, e.g. by measuring weak charges of hadrons. Strong interaction effects cannot renormalize charges associated with conserved currents. However, precision requirements of modern experiments are such that generally small effects due to non-conserved currents appearing at higher order in the fine structure constant 𝛼, and symmetry breaking effects have to be taken into account when extracting those conserved charges from actual experimental measurements. The challenges arising in precision calculations of the observables, experimental techniques, exposure to possible effects due to non-standard particles and interactions allow us to cluster the main focus of the proposed workshop, the physics case of PVES on 12C.