The Mordel-Weil theorems for Drinfeld modules over finitely generated function fields

The Lehmer inequality and the Mordell–Weil theorem for Drinfeld modules

“…Mainly what will be needed for our result will be a better understanding of the heights associated to φ, both local and global heights. These heights were first introduced in [3] and then contributions towards their understanding were done in [7,14,18]. Lemma 4.14 appears also in our paper [8], in which we prove a local Lehmer inequality for Drinfeld modules.…”

“…We denote by φ tor the torsion submodule of . When K is a finitely generated extension of F q , it was proved in [14] (in the case that trdeg F p K = 1) and in [18] (for arbitrary finite, positive transcendence degree) that φ(K) is the direct sum of a finite torsion submodule with a free submodule of rank ℵ 0 . We will prove in Theorem 5.7 a similar structure result for certain infinitely generated extensions of F q .…”

The local Lehmer inequality for Drinfeld modules

“…Note that if α = 0, there might be no indices i j and i k as in (16). In that case, the construction of R v (0) from (17) …”

“…where the indices i j are the ones associated to α as in (16). Note that if α = 0, there might be no indices i j and i k as in (16).…”

“…Letĥ be the global height associated to the Drinfeld module φ as in [16] (see also Section 3). Before our work, the only known partial result towards Conjecture 1.1 was obtained in [3], which gave a lower bound for the canonical height of a nontorsion point x ∈ K sep restricted to the case in which φ is the Carlitz module.…”