The \(\mathfrak{p}\)-primary uniform boundedness conjecture for Drinfeld modules (Q6554582)

The \textit{Uniform Boundedness Conjecture} (UBC) for abelian varieties (resp., the \(p\)-primary UBC for abelian varieties) states that for a fixed integer \(d>0\) and a number field \(L\) (resp., \(d>0\), a number field \(L\), and a rational prime \(p\)), there exists \(C:=C(L,d)>0\) (resp., \(C:=C(L,d,p)>0\)) such that for any \(d\)-dimensional abelian variety \(X\) over \(L\), \(|X_{\mathrm{tors}} (L)|<C\) (resp., \(|X[p^{\infty}](L)|<C\)) holds.\N\NLet \(K\) be a global function field and \(\infty\) a fixed place of \(K\). Let \(A\) be the ring of elements of \(K\) regular away from \(\infty\). As for abelian varieties, the UBC for Drinfeld modules (resp., the \({\mathfrak p}\)-primary UBC for Drinfeld modules) states that for a fixed integer \(d>0\) and a finitely generated extension \(L\) of \(K\) (resp. a fixed integer \(d>0\), a finitely generated extension \(L\) of \(K\), and a maximal ideal \({\mathfrak p}\) of \(A\)), there exists \(C:= C(L,d)>0\) (resp., \(C:=C(L,d,{\mathfrak p})>0\)) such that for any Drinfeld module \(\phi\) of rank \(d\) over \(L\), \(| \phi_{\mathrm{tors}}(L)|<C\) (resp., \(|\phi[{\mathfrak p}^{\infty}] (L)|<C\)) holds.\N\N\textit{B. Poonen} [Math. Ann. 308, No. 4, 571--586 (1997; Zbl 0891.11034), Theorem 1] proved the UBC for Drinfeld modules of rank \(1\) using explicit class field theory and also the \({\mathfrak p}\)-primary UBC for Drinfeld modules of rank \(2\) for \(A={\mathbb F}_q[T]\) by means of Drinfeld modular curves.\N\NIn this paper, the author proves an analogue of the \({\mathfrak p}\)-primary UBC for one-dimensional families of Drinfeld modules over a finitely generated extension of \(K\). The main result is the following. Let \(S\) be a one-dimensional scheme that is of finite type over \(L\) and \(\phi\) a Drinfeld \(A\)-module over \(S\). Then there exists an integer \(N:= N(\phi, S,L,{\mathfrak p})\geq 0\) such that \(\phi_s[{\mathfrak p}^{\infty}](L)\subset \phi_s[{\mathfrak p}^N](L)\) holds for every \(s\in S(L)\).\N\NFor the proof, first an auxiliary result is proved by a formula of \textit{J. Oesterlé} [Invent. Math. 66, 325--341 (1982; Zbl 0473.12015)] regarding the asymptotic behavior of sizes of images of analytic sets. Next, the author uses a positive characteristic analogue of Mordell's conjecture proved by \textit{P. Samuel} [Exp. No. 287, 19 p. (1966; Zbl 0196.53404)] and uses this theorem to show the existence of a suitable model of the base field. Finally, by using a specialization argument, the result is reduced to the case when the base field is finite.\N\NAs a corollary, it is proved Poonen's result on the \({\mathfrak p}\)-primary UBC for Drinfeld modules of rank \(2\) over a finitely generated extension of \(K\).