On the independence of Heegner points in the function field case

Abstract

Let ∞ 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 ∞ and fix a modular parametrization ΦE:X0(N)→E. Let P1,…,Pr∈E(k¯) be Heegner points associated to the rings of integers of distinct quadratic “imaginary” fields K1,…,Kr over (k,∞). We prove that if the “prime-to-2p” part of the ideal class numbers of ring of integers of K1,…,Kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P1,…,Pr are independent in E(k¯)/Etors. 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.