Let A be a non-isotrivial ordinary abelian surface over a global function field with good reduction everywhere. Suppose that A does not have real multiplication by any real quadratic field with discriminant a multiple of p. We prove that there are infinitely many places modulo which A is isogenous to the product of two elliptic curves. 3 4 I(m) by studying special endomorphisms. 1(3) We prove that the local contribution from a non-supersingular point is o(I(m)) as m → ∞. This allows us to conclude that, as m → ∞, more and more points of C contribute to the intersection C.Z(m). In order to prove Theorem 1(1), the sequence will consist only of squares, and in order to prove Theorem 2, the sequence will consist only of primes. Note that in A 2 , the Heegner divisor Z(m) for square m parametrizes abelian surfaces which are not geometrically simple, thereby allowing us to deduce Theorem 1(1). Similar arguments allow us to deduce part Theorem 1(2), and also Theorem 2.Compared to the number field situation, the main difficulty of the positive characteristic function field case is that the local contributions at supersingular points are of the same magnitude as the global contribution. More precisely, taking the Hilbert case as an example, Borcherds theory implies that the generating series of Z(m) is a non-cuspidal modular form of weight 2; on the other hand, the theta series attached to the special endomorphism lattice at a supersingular point is also a noncuspidal weight 2 modular form since the lattice is of rank 4. Therefore, even without considering 1 The ratio 3/4 is not a sharp upper bound.