Positively curved manifolds with maximal discrete symmetry rank

Let M be a closed simply connected n-manifold of positive sectional curvature. We determine its homeomorphism or homotopic type if M also admits an isometric elementary p-group action of large rank. Our main results are: There exists a constant p(n) > 0 such that (1) If M²ⁿ admits an effective isometric ℤ_p^k-action for a prime p ≥ p(n), then k ≤ n and "=" implies that M²ⁿ is homeomorphic to a sphere or a complex projective space. (2) If M²ⁿ⁺¹ admits an isometric S¹ ℤ_p^k-action for a prime p ≥ p(n), then k ≤ n and "=" implies that M is homeomorphic to a sphere. (3) For M in (1) or (2), if n ≥ 7 and k ≥ [3n/4] + 2, then M is homeomorphic to a sphere or homotopic to a complex projective space.