  Topological crystalline superconductivity in the locally non-centrosymmetric
multilayer superconductors (SCs) is proposed. We study the odd-parity
pair-density wave (PDW) state induced by the spin-singlet pairing interaction
through the spin-orbit coupling. It is shown that the PDW state is a
topological crystalline SC protected by a mirror symmetry, although it is
topologically trivial according to the classification based on the standard
topological periodic table. The topological property of the mirror subsectors
is intuitively explained by adiabatically changing the BdG Hamiltonian. A
subsector of the bilayer PDW state reduces to the two-dimensional
non-centrosymmetric SC, while a subsector of trilayer PDW state is
topologically equivalent to the spinless p-wave SC. Chiral Majorana edge modes
in trilayers can be realized without Cooper pairs in the spin-triplet channel
and chemical potential tuning.
