On the independence of Heegner points in the function field case

Let infinity be a fixed place of a global function field k. Let E be an elliptic curve defined over k which has split multiplicative reduction at infinity and fix a modular parametrization phi(E):X(0)(n) -> E. Let P(1,) ..... P(r) is an element of E((k) over bar) be Heegner points associated to the rings of integers of distinct quadratic "imaginary" fields K(i), ... , K(r) over (k, 00). We prove that if the "prime-to-2p" part of the ideal class numbers of ring of integers of K(1), ... , K(r) are larger than a constant C = C(E, Phi(E)) depending only on E and Phi(E), then the points P(1), ... , P(r) are independent in E((k) over bar)/E(tors). Moreover, when k is rational, we show that there are infinitely many imaginary quadratic fields for which the prime-to-2p part of the class numbers are larger than C. (C) 2010 Elsevier Inc. All rights reserved.