Local–Global Principle of Affine Varieties Over a Subgroup of Units in a Function Field

Abstract

Over a large class of function fields, we show that the solutions of some linear equations in the topological closure of a certain subgroup of the group of units in the function field are exactly the solutions that are already in the subgroup. This result solves some cases of the function field analog of an old conjecture proposed by Skolem.

A Local-Global Criteria of Affine Varieties Admitting Points in Rank-One Subgroups of a Global Function Field

“…For any subgroup ∆ ⊂ K * , we denote by k(∆) the smallest subfield of K containing k and ∆, by ρ(∆) the subgroup m≥0 (K p m ) * ∆ of K * , and by K √ ∆ the subgroup {x ∈ K * : x n ∈ ∆ for some n ∈ N} of K * . The following result is a special case of Proposition 6 in [Sun14]. .…”

“…thus if Conjecture 1 holds for W , then we must have W (Γ) = i∈I W i (Γ). In fact, following the idea proposed by Stoll (Question 3.12 in [Sto07], so-called "Adelic Mordell-Lang Conjecture") and first realized by Poonen and Voloch [PV10], all previous results [Sun13,Sun14] dealing with Conjecture 1 for reducible W are established by reducing it to the assertion that Z(Γ) = i∈I Z i (Γ) for every finite union Z = i∈I Z i of irreducible zero-dimensional K-varieties in A M , and proving this assertion via an argument invented by Poonen and Voloch [PV10], who managed to bypass the difficulty encountered when one tries to develop the function-field analog of the proof by Stoll [Sto07] of the number-field counterpart of this assertion. In the present setting, the Mordell-Lang Conjecture is treated by Derksen and Masser [?]…”

“…in full generalities. The author [Sun14] establishes some "adelic analog" (restated as Proposition 3 below) of their result in certain case, and completes the aforementioned reduction under the artificial hypothesis induced from this analog; this hypothesis is then put in the main result in [Sun14] on Conjecture 1. In the remaining case, however, no adelic analog exists (see Remark 4); thus it is hopeless to completely solve Conjecture 1 via this "Mordell-Lang approach".…”

“…Letting (x ′′ n ) n∈N ⊂ A M (ΦΓ) be the sequence defined by x ′′ 2n−1 = x n and x ′′ 2n = x ′ n , we see that the sequence (x ′′ n ) n∈N ⊂ A M (ΦΓ) is Cauchy. As abelian groups, ΦΓ/Γ is isomorphic to Φ/(Γ∩Φ), which is finite by the construction of Φ; thus Corollary 2 of [Sun14] shows that Γ is open in ΦΓ. It follows that Φ ∩ Γ is open in ΦΓ.…”

“…For any closed K-variety W in A M , we consider the following conjecture. First formulated in this form by the author [Sun14], Conjecture 1 is an analog for split algebraic tori to Conjecture C in [PV10] for Abelian varieties. One of the deepest aspects of Conjecture 1 is explained as follows.…”