On the uniform boundedness conjecture for Drinfeld modules
According to similar results about elliptic curves, the uniform boundedness conjecture for Drinfeld modules states [see \textit{B. Poonen}, Math. Ann. 308, 571--586 (1997; Zbl 0891.11034)]: For a fixed Drinfeld ring \(A\) and natural numbers \(r\) and \(d\), there exists a uniform bound on \(\# (\phi_L)_{\text{tors}}\) as \(L\) ranges over extensions of \(K := \text{Fract} (A)\) of degree \(\leq d\), and \(\phi\) ranges over rank \(r\) Drinfeld \(A\)-modules over \(L\). There are obvious weakened versions of the conjecture, involving only fixed field extensions \(L\) of \(K\), or only the \({\mathfrak p}\)-torsion of \(\phi_L\) (where \({\mathfrak p}\) is a prime of \(A\)), etc. Poonen in his cited paper proved the conjecture for general Drinfeld rings and \(r=1\). The paper under review contributes to the case where \(A\) is a polynomial ring \({\mathbb F}_q[T]\) over a finite field \({\mathbb F}_q\) and \(r = 1\) or \(2\). For \(r = 2\), the following conjecture is stated: If \(\phi\) is a rank 2 Drinfeld \(A\)-module on \(K = \text{Fract}(A) = {\mathbb F}_q(T)\), and if \[ (^{\phi}K)_{\text{tors}} \cong A/{\mathfrak m} \oplus A/{\mathfrak n} \] with ideals \({\mathfrak m},{\mathfrak n}\) of \(A\) and \({\mathfrak m}| {\mathfrak n}\), then \(\deg {\mathfrak m} + \deg {\mathfrak n} \leq 2\). This conjecture would follow if the Drinfeld modular curve \(X_0(n)\) had no \(k\)-rational points other than cusps and CM-points (Lemma 4.5). The main result of the paper is Theorem 2.4, which states: Fix a prime \({\mathfrak p}\) of \(A = {\mathbb F}_q[T]\) and \(d \in {\mathbb N}\). As \(L\) ranges over the extensions of \({\mathbb F}_q(T)\) of degree \(d\) and \(\phi\) ranges over all rank 2 Drinfeld \(A\)-modules over \(L\), there is a uniform bound on the size of the \({\mathfrak p}\)-primary part of \((\phi_L)_{\text{tors}}\). Further, some effective estimates on the case \(r = 1\) are given which supplement Poonen's result.
scientific article; zbMATH DE number 1983488