Due to its biological importance in the fertilization process, the swimming of a solitary spermatozoon has been investigated intensively. However, the elasto-hydrodynamic interactions between spermatozoa remain unclear, and the collective swimming of cells has not been fully clarified. In this study, we numerically investigated pairwise interactions of sperm cells in terms of fluid and solid mechanics. To describe fluid-structure interactions between sperm cells, we developed a boundary element-finite element coupling method. When two sperm cells swim side-by-side, their swimming speed may increase compared to solitary swimming. On the other hand, when two sperm cells swim in line, the front sperm swims faster, while the rear sperm swims slower. To reproduce the experimentally observed flagellar synchronization, we employed a geometric clutch hypothesis and proposed a curvature-associated wave-propagation model. The elasto-hydrodynamic synchronization of flagella resulted in an increase in the swimming speeds of side-by-side sperm cells of up to 16%, indicating that elasto-hydrodynamic synchronization is beneficial for cells in terms of swimming speed. The results clarify the fluid-structure interactions of flagellar mechanics and are important in understanding the collective swimming of spermatozoa.