The Sawada–Kotera hierarchy associated with a matrix spectral problem is derived with the aid of Lenard recursion equations. By using the characteristic polynomial of Lax matrix for the Sawada–Kotera hierarchy, we introduce a three-sheeted Riemann surface, from which we derive the associated Baker–Akhiezer function, the meromorphic function on it. Based on the theory of Riemann surfaces, we construct the explicit Riemann theta function representations of the Baker–Akhiezer function, the meromorphic function, and solutions for the Sawada–Kotera hierarchy.