44 problems

Bogomolov conjecture. If $X_n$ is Zariski dense in $X$ for every $n\geq 1$, then $X$ is a torsion subvariety of $\mathbb{G}_a^g$. This is presented as a consequence of the proposed…

Let $K$ be a finitely generated field, let $A$ be the coefficient ring of a Drinfeld module, and let $\phi:A\rightarrow K\{\tau\}$ be a Drinfeld module. For a point…

Let $g$ be a positive integer, and let $\phi_1:A\to K\{\tau\},\dots,\phi_g:A\to K\{\tau\}$ be Drinfeld modules. An algebraic $(\phi_1,\dots,\phi_g)$-submodule of $\mathbb{G}_a^g$ i…

Strong uniform boundedness conjecture. For fixed $A$, $r \ge 1$, and $d \ge 1$, there is a uniform bound on $\#({}^{\phi}L)_{\operatorname{tors}}$ as $L$ ranges over extensions of…

Let $K$ be a global function field, let $M_K$ be its set of places, let $S\subseteq M_K$ be finite, and let $\phi$ be a Drinfeld module over $K$. A point $\gamma$ is $S$-integral f…

Let $K$ be a global function field, let $M_K$ be the set of places of $K$, and let $\widehat h$ be the canonical height associated to the Drinfeld module $\phi$. For…

Let $K$ be a global function field, let $\boldsymbol{\phi}$ be a Drinfeld module of rank $d$ over $K$, let $M_K$ be the set of places of $K$, and let $\boldsymbol{\phi}_Q$ denote t…

Asymptotic cyclicity conjecture.

Let $A$ be the coefficient ring, let $\Phi$ be a rank-$2$ Drinfeld $A$-module with characteristic ideal $\chi_{\Phi}$, and let $P$ be the characteristic polynomial datum appearing…

Let $\phi:A\rightarrow K\{\tau\}$ be a Drinfeld module of generic characteristic, let $g\geq 1$, and let $\{x_k\}\subset\mathbb{G}_a^g(K^{\operatorname{sep}})$ be a strict sequence…

Let $\phi:A\rightarrow K\{\tau\}$ be a Drinfeld module of generic characteristic acting diagonally on $\mathbb{G}_a^g$, and let $\{x_k\}$ be a strict sequence of torsion points in…

Let $K$ be a function field of transcendence degree $1$ over $\mathbb{F}_p$, and let $\phi:A\rightarrow K\{\tau\}$ be a Drinfeld module. Let $n\ge 1$ and let $X$ be an irreducible…

Let $L=F_{q^{n}}$ be a finite field, let $P$ be the $A$-characteristic of $L$, let $m=[L,A/P]$, and let $d=\operatorname{deg}P$. Let $C(d,m,q)$ and $C_{0}(d,m,q)$ denote the two qu…

Let $A$ be the coefficient ring, let $[?]$ be the $A$-characteristic of a finite field $L$, and let $\chi _{\Phi }$ denote the characteristic ideal of a rank-two Drinfeld $A$-modul…

Let $\mathbf{A}=\mathbb{F}_q[t]$, $A=\mathbb{F}_q[\theta]$, and $K=\operatorname{Frac}(A)=\mathbb{F}_q(\theta)$. Let $E$ be a finite extension of $K$ and let…

Let $\phi$ and $\psi$ be two almost strictly pure $t$-modules over $K$, and let $\phi\otimes\psi$ denote their tensor product. Choose a $K\{\tau\}$-basis of the corresponding $t$-m…

Cartier--Nishi duality conjecture. The Cartier--Nishi theorem holds for $\Upsilon$.

Serre-type commensurability conjecture. The following conditions are equivalent:

Let $A$ be the coefficient ring and let … be a Drinfeld module. For each prime polynomial $P$, let $g_{P,\phi}$ denote the auxiliary polynomial associated with $P$ and $\phi$. Prim…

Let $A$ be the polynomial ring in the paper, let $phi$ be the given Drinfeld module, and let $a\in A$ be the fixed base. A polynomial has no square factors when it is squarefree, a…

Let $K$ be a global $Q$-field, let $E$ be a Drinfeld $A$-module over $\mathcal{O}_K$, and let $\mathfrak{p}$ be a prime of $A$. Write $S_{\mathfrak{p}}$ for the set of places of…

Let $\mathfrak{p}$ be a prime of $A$, let $J_0(\mathfrak{p})$ be the Jacobian of the Drinfeld modular curve $X_0(\mathfrak{p})$, let $\mathcal{S}(\mathfrak{p})$ be the Shimura subg…

Fix $q$, $r\geq 1$, and $d\geq 1$. Let $L$ range over extensions of $F$ of degree at most $d$, and let $\phi$ range over rank-$r$ Drinfeld $A$-modules over $L$. Poonen's Drinfeld-m…

Drinfeld cyclic-subgroup conjecture. If $\mathfrak{p}$ is a prime of $A$ with $\deg\mathfrak{p}\geq C$, then there is no $F$-rational $A$-submodule of $\phi[\mathfrak{p}]$ isomorph…

Schweizer's Drinfeld-module torsion conjecture.