Let S be a hyperbolic oriented Riemann surface of finite type. The main purpose of this article is to show that non-trivial geometric intersection between closed curves on S is detected by some symplectic submodules they naturally determine in the homology groups of the compactifications of unramified p-coverings of S⁠, for p ≥ 2 a fixed prime. In particular, this gives a characterization of simple closed curves on S in terms of homology groups of p-coverings. In Section 4, we define a p-adic Reidemeister pairing on the fundamental group of S and show that the free homotopy classes of two loops have trivial geometric intersection if and only if they are orthogonal with respect to this pairing. As an application, we give a geometric argument to prove that oriented surface groups are conjugacy p-separable (a combinatorial proof of this fact was recentely given by Paris [14]). In the article [11, 20], Stallings and Jaco established the equivalence between the Poincaré conjecture, now a celebrated theorem by Perelman, and the following group-theoretical statement: * LetSgbe a closed oriented surface of genus g≥2⁠. Let Fg be a free group of rank g⁠. Let η: π1(Sg,s0)→Fg×Fg be an epimorphism. Then, there is a non-trivial element in the kernel of η which may be represented by a simple closed curve in Sg⁠. Of course, it is still an interesting problem to provide a group-theoretic proof of the above statement. The first step in this direction is to give an algebraic characterization of simple closed curves on the closed Riemann surface Sg⁠. A program in this sense was formulated by Turaev [21]. Progress in this direction have been recently accomplished by Chas, Gadgil, and Krongold (see [4–6]), who characterize simple closed curves on a Riemann surface in terms of the Goldman Lie algebra. In Section 2, we give an elementary criterion to characterize simple closed curves on any hyperbolic Riemann surface in terms of the intersection pairing on the closures of normal unramified p-coverings of the given Riemann surface, for a fixed prime p⁠. The proof is based on the hyperbolic geometry of the p-adic solenoid, developed by means of some elementary pro-p group theory.