We study the symmetries of twisted trilayer graphene's band structure under various extrinsic perturbations, and analyze the role of long-range electron-electron interactions near the first magic angle. The electronic structure is modified by these interactions in a similar way to twisted bilayer graphene. We analyze electron pairing due to long-wavelength charge fluctuations, which are coupled among themselves via the Coulomb interaction and additionally mediated by longitudinal acoustic phonons. We find superconducting phases with either spin-singlet/valley-triplet or spin-triplet/valley-singlet symmetry, with critical temperatures up to a few Kelvin for realistic choices of parameters.

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• Electronic structure
• Pairing mechanisms
• Phonons
• Plasmons
• Superconducting order parameter
• Superconductivity
• Graphene
• Band structure methods
• Bloch wave theory
• Mean field theory
• Symmetries in condensed matter