A variant of Siegel's theorem for Drinfeld modules

TL;DR

This paper extends Siegel's theorem to finitely generated submodules under Drinfeld module action, providing a significant advancement in understanding the arithmetic of Drinfeld modules.

Contribution

It completes the proof of a Siegel type statement for finitely generated submodules of the additive group under Drinfeld module action.

Findings

Established a Siegel type finiteness result for Drinfeld modules

Extended classical Diophantine finiteness theorems to function field setting

Provided new tools for studying arithmetic properties of Drinfeld modules

Abstract

We complete the proof of a Siegel type statement for finitely generated $\Phi$-submodules of $\mathbb{G}_a$ under the action of a Drinfeld module $\Phi$.

Equations

$${\dim }_{\mathrm{Frac}(A)}\Gamma {\otimes }_A\mathrm{Frac}(A).$$

$$h(x)=\sum _{v\in M_K}\log \max (|x{|}_v,1).$$

$$\hat{h}(\beta )=\underset{n\to \infty }{\lim }\frac{h({\Phi }_{t^n}(\beta ))}{q^{nd}}.$$

$${\hat{h}}_v(\beta )=\underset{n\to \infty }{\lim }\frac{\log \max (|{\Phi }_{t^n}(\beta ){|}_v,1)}{q^{nd}};$$

$${\hat{h}}_v(x)=\frac{{\hat{h}}_v\left({\Phi }_{t^{\ell }}(x)\right)}{q^{d\ell }}=\frac{\log |{\Phi }_{t^{\ell }}(x){|}_v}{q^{d\ell }}.$$

$${\left|{\Phi }_Q(x)\right|}_v={\left|{\Phi }_{t^{\ell }}(x)\right|}_v^{q^{d(\mathrm{deg}(Q)-\ell )}}.$$

$${\hat{h}}_v({\gamma }_1)>\frac{{\max }_{i>1}{\hat{h}}_v({\gamma }_i)}{q^d}.$$

$$|{\Phi }_{Q_i}({\gamma }_i){|}_v<L_v.$$

$$\log {\left|{\Phi }_{Q_i}({\gamma }_i)\right|}_v=q^{d\mathrm{deg}(Q_i)}\cdot {\hat{h}}_v({\gamma }_i).$$

$$|{\Phi }_{Q_i}({\gamma }_i){|}_v\ge L_v>M_v.$$

$${\left|\sum _{i=1}^r{\Phi }_{P_i}({\gamma }_i)\right|}_v={\left|{\Phi }_{P_1}({\gamma }_1)\right|}_v>M_v.$$

$$\log {\left|\sum _{i=1}^r{\Phi }_{P_i}({\gamma }_i)\right|}_v=\log {\left|{\Phi }_{P_1}({\gamma }_1)\right|}_v=q^{d\cdot (\mathrm{deg}(P_1)-n_0)}\cdot \log {\left|{\Phi }_{t^{n_0}}({\gamma }_1)\right|}_v.$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{\log |{\sum }_{i=1}^r{\Phi }_{P_{n,i}}({\gamma }_i)-\alpha {|}_v}{{\sum }_{i=1}^r{q}^{d\mathrm{deg}{P}_{n,i}}}>0.$$

$$\underset{n\to \infty }{\lim }\sum _{i=1}^r{q}^{d\mathrm{deg}{P}_{n,i}}=+\infty .$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{\log |{\sum }_{i=1}^r{\Phi }_{P_{n,i}}({\gamma }_i){|}_v}{{\sum }_{i=1}^r{q}^{d\mathrm{deg}{P}_{n,i}}}>0.$$

$$\underset{n\to \infty }{\lim }\frac{q^{d\mathrm{deg}{P}_{n,j}}}{{\sum }_{i=1}^r{q}^{d\mathrm{deg}{P}_{n,i}}}=0$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{\log |{\sum }_{i\ne j}{\Phi }_{P_{n,i}}({\gamma }_i){|}_v}{{\sum }_{i\ne j}{q}^{d\mathrm{deg}{P}_{n,i}}}>0.$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{\log |{\sum }_{i\ne j}{\Phi }_{P_{n,i}}({\gamma }_i){|}_v}{{\sum }_{i=1}^r{q}^{d\mathrm{deg}{P}_{n,i}}}>0.$$

$${\left|\sum _{i\ne j}{\Phi }_{P_{n,i}}({\gamma }_i)\right|}_v={\left|\sum _{i=1}^r{\Phi }_{P_{n,i}}({\gamma }_i)\right|}_v$$

$$\underset{n\to \infty }{\lim }\frac{\log |{\Phi }_{P_{n,j}}({\gamma }_j){|}_v}{{\sum }_{i=1}^r{q}^{d\mathrm{deg}{P}_{n,i}}}={\hat{h}}_v({\gamma }_j)\cdot \underset{n\to \infty }{\lim }\frac{q^{d\mathrm{deg}{P}_{n,j}}}{{\sum }_{i=1}^r{q}^{d\mathrm{deg}{P}_{n,i}}}=0.$$

$$\underset{n\to \infty }{\lim }\frac{\log |{\Phi }_{P_{n,j}}({\gamma }_j){|}_v}{{\sum }_{i=1}^r{q}^{d\mathrm{deg}{P}_{n,i}}}=0.$$

$${\left|\sum _{i\ne j}{\Phi }_{P_{n,i}}({\gamma }_i)\right|}_v={\left|\sum _{i=1}^r{\Phi }_{P_{n,i}}({\gamma }_i)\right|}_v$$

$$\frac{{\max }_{1\le i\le r}{q}^{d\mathrm{deg}{P}_{n,i}}}{{\min }_{1\le i\le r}{q}^{d\mathrm{deg}{P}_{n,i}}}\le B,$$

$$\underset{1\le i\le r}{\max }\mathrm{deg}{P}_{n,i}-\underset{1\le i\le r}{\min }\mathrm{deg}{P}_{n,i}\le \frac{{\log }_{q}B}{d}.$$

$$\sum _{i=1}^r{\Phi }_{P_{n_k,i}}({\gamma }_i)=\sum _{i=1}^r{\Phi }_{R_{k,i}}({\delta }_i)$$

$$0<\underset{k\to \infty }{\mathrm{liminf}}\frac{{\sum }_{i=1}^r{q}^{d\mathrm{deg}{P}_{n_k,i}}}{{\sum }_{i=1}^r{q}^{d\mathrm{deg}{R}_{k,i}}}\le \underset{k\to \infty }{\mathrm{limsup}}\frac{{\sum }_{i=1}^r{q}^{d\mathrm{deg}{P}_{n_k,i}}}{{\sum }_{i=1}^r{q}^{d\mathrm{deg}{R}_{k,i}}}<+\infty .$$

$$\underset{k\to \infty }{\mathrm{limsup}}\frac{\log |{\sum }_{i=1}^r{\Phi }_{R_{k,i}}({\delta }_i){|}_v}{{\sum }_{i=1}^r{q}^{d\mathrm{deg}{R}_{k,i}}}>0,$$

$$\underset{k\to \infty }{\mathrm{limsup}}\frac{\log |{\sum }_{i=1}^r{\Phi }_{P_{n_k,i}}({\gamma }_i){|}_v}{{\sum }_{i=1}^r{q}^{d\mathrm{deg}{P}_{n_k,i}}}>0.$$

$${\hat{h}}_v({\gamma }_j)>\frac{1}{q^d}\underset{i\ne j}{\max }{\hat{h}}_v({\gamma }_i)$$

$$\mathrm{deg}{P}_{n,j}>\underset{i\ne j}{\max }\mathrm{deg}{P}_{n,i}$$

Full text

1. Introduction

In 1929, Siegel ([Sie29]) proved that if $C$ is an irreducible affine curve defined over a number field $K$ and $C$ has at least three points at infinity, then there are at most finitely many $K$-rational points on $C$ that have integral coordinates. The two most important ingredients in the proof of Siegel’s theorem are Diophantine approximation, along with the fact that certain groups of rational points are finitely generated; when $C$ has genus greater than [math], the group in question is the Mordell-Weil group of the Jacobian of $C$, while when $C$ has genus [math], the group in question is the group of $S$-units in a finite extension of $K$. Motivated by the analogy between rank $2$ Drinfeld modules and elliptic curves (along with the understanding that higher rank Drinfeld modules of generic characteristic are the right vehicle in characteristic $p$ for similar conjectures to Diophantine questions one would pose for abelian varieties in characteristic [math]), Ghioca and Tucker conjectured in [GT08a, Conjecture 5.5] a Siegel type statement for finitely generated $\Phi$-submodules $\Gamma$ of ${\mathbb{G}}_{a}(K)$ (where $q$ is a power of $p$, $\Phi$ is a generic characteristic Drinfeld module of arbitrary rank and $K$ is a finite extension of ${\mathbb{F}}_{q}(t)$). In [GT07], Ghioca and Tucker proved the Siegel type statement for Drinfeld modules under the technical hypothesis that the ground field $K$ admits a single place which lies over the place at infinity of ${\mathbb{F}}_{q}(t)$. In the current paper, we are able to remove this technical hypothesis on the field $K$ and prove the following general result for Drinfeld modules in the spirit of the famous Siegel’s theorem.

Theorem 1.1**.**

Let $q$ be a power of the prime number $p$, let $K$ be a finite extension of the function field ${\mathbb{F}}_{q}(t)$ and let $\Phi$ be a Drinfeld module of generic characteristic defined over $K$. Let $\Gamma$ be a finitely generated $\Phi$-submodule of $\mathbb{G}_{a}(K)$, let $\alpha\in K$, and let $S$ be a finite set of places of the field $K$. Then there are finitely many $\gamma\in\Gamma$ such that $\gamma$ is $S$-integral with respect to $\alpha$.

As an aside, we note that for any finitely generated $\Phi$-module $\Gamma\subset\overline{K}$, we can always find a finite extension of $K$ containing $\Gamma$ and also containing the point $\alpha$; this explains why we state our Theorem 1.1 assuming $\Gamma$, $\alpha$ and also the coefficients of $\Phi_{t}$ are contained in $K$ (rather than contained in $\overline{K}$, as [GT08a, Conjecture 5.5] was previously stated). We refer the reader to Section 2 for more details on Drinfeld modules, including the notion of $S$-integral points with respect to a given point.

The strategy of our proof is identical with the strategy employed in [GT07]; actually, with one exception, the statements proven in [GT07] are valid in the generality of our Theorem 1.1. However, the main result from the aforementioned paper, i.e., [GT07, Proposition 3.12] is proven under the technical assumption that there exists a single place in the function field $K$ lying above the place at infinity from ${\mathbb{F}}_{q}(t)$. Moreover, the strategy of proof from [GT07, Proposition 3.12] does not extend to arbitrary function fields $K$, as explained in [GT07, Remark 3.14]. So, the main result of our current paper is to develop an alternative strategy for proving [GT07, Proposition 3.12] for arbitrary function fields $K$; the statement generalizing [GT07, Proposition 3.12] is proven in our Proposition 3.5. Finally, we note that in Theorem 1.1, if $\Phi$ were a Drinfeld module of special characteristic, then its conclusion could fail (as can be easily seen in the case the Drinfeld module is simply given by $\Phi_{t}(x):=x^{q}$; then we may take $K$ be a finite extension of the rational function field ${\mathbb{F}}_{q}(\theta)$ and so, there are infinitely many points in the cyclic $\Phi$-module generated by $\theta+1$, which are $S$-integral with respect to $1$, where $S$ consists of the places $v$ of $K$ for which $\theta$ is not a $v$-adic unit).

Our Theorem 1.1 completes the proof of Siegel’s theorem in the context of Drinfeld modules. Over the past 30 years, there was a significant increase in the study of the arithmetic of Drinfeld modules which established the validity of several classical theorems from the arithmetic geometry of abelian varieties in the context of Drinfeld modules. Indeed, Scanlon [Sca02] proved the Manin-Mumford type theorem for Drinfeld modules, conjectured by Denis [Den92a]. The second author proved in [Ghi06] an equidistribution statement for torsion points of Drinfeld modules, which may be interpreted as a weaker variant of the classical equidistribution theorem of Szpiro-Ullmo-Zhang [SUZ97] for torsion points of abelian varieties. Breuer [Bre05] proved an André-Oort type theorem for Drinfeld modules in the spirit of the classical results of Edixhoven-Yafaev [EY03] for Shimura varieties. The second author (both in a single author paper [Ghi05] and also in a joint paper with Tucker [GT08b]) proved various instances of a Mordell-Lang type statement for Drinfeld modules conjectured by Denis [Den92a]. Actually, the paper [GT08b] constituted the starting point of the Dynamical Mordell-Lang Conjecture (formulated in [GT09]), which by itself generated extensive research in the past 15 years (for a comprehensive discussion of the Dynamical Mordell-Lang Conjecture, see [BGT06], especially [BGT06, Chapter 12] which details the connection between the aforementioned conjecture and Denis’ conjecture for Drinfeld modules from [Den92a]).

The plan of our paper is as follows: in Section 2 we give the basic definitions and notation, and then, in Section 3 we prove the main result, which is Proposition 3.5 (note that Theorem 1.1 is an immediate consequence of our Proposition 3.5 using the exact same strategy of proof as for [GT07, Theorem 2.4], the only difference being the replacement of [GT07, Proposition 3.12] by our new Proposition 3.5).

Acknowledgments. We thank the referee for their many useful suggestions.

2. Notation

Our notational section has a significant overlap with [GT07, Section 2].

2.1. Drinfeld modules

We begin by defining a Drinfeld module. Let $p$ be a prime and let $q$ be a power of $p$. Let $A:=\mathbb{F}_{q}[t]$, let $K$ be a finite field extension of $\mathbb{F}_{q}(t)$, and let $\overline{K}$ be an algebraic closure of $K$. We let $\tau$ be the Frobenius on $\mathbb{F}_{q}$, and we extend its action on $\overline{K}$. Let $K\{\tau\}$ be the ring of polynomials in $\tau$ with coefficients from $K$ (the addition is the usual addition, while the multiplication is the composition of functions).

A Drinfeld module is a morphism $\Phi:A\rightarrow K\{\tau\}$ for which the coefficient of $\tau^{0}$ in $\Phi(a)=:\Phi_{a}$ is $a$ for every $a\in A$, and there exists $a\in A$ such that $\Phi_{a}\neq a\tau^{0}$. The definition given here represents what Goss [Gos96] calls a Drinfeld module of “generic characteristic”.

We note that usually, in the definition of a Drinfeld module, $A$ is the ring of functions defined on a projective nonsingular curve $C$, regular away from a closed point $\eta\in C$. For our definition of a Drinfeld module, $C=\mathbb{P}^{1}_{\mathbb{F}_{q}}$ and $\eta$ is the usual point at infinity on $\mathbb{P}^{1}$. On the other hand, every ring of regular functions $A$ as above contains $\mathbb{F}_{q}[t]$ as a subring, where $t$ is a nonconstant function in $A$. Furthermore, even for such a general ring of regular functions $A$, we have that $A$ is a finitely generated module over the ring ${\mathbb{F}}_{q}[t]$, which means that the statement of our Theorem 1.1 is left unchanged.

For every field extension $K\subset L$, the Drinfeld module $\Phi$ induces an action on $\mathbb{G}_{a}(L)$ by $a*x:=\Phi_{a}(x)$, for each $a\in A$. We call $\Phi$-submodules subgroups of $\mathbb{G}_{a}(\overline{K})$ which are invariant under the action of $\Phi$. We define the rank of a $\Phi$-submodule $\Gamma$ to be

[TABLE]

A point $\alpha\in\overline{K}$ is torsion for the Drinfeld module action if and only if there exists $Q\in A\setminus\{0\}$ such that $\Phi_{Q}(\alpha)=0$. It is immediate to see that the set of all torsion points is also a $\Phi$-submodule. Since each polynomial $\Phi_{Q}$ is separable, the torsion submodule $\Phi_{\operatorname{tor}}$ lies in the separable closure $K^{\operatorname{sep}}$ of $K$.

2.2. The coefficients of the Drinfeld module

Each Drinfeld module is isomorphic over $\overline{K}$ to a Drinfeld module for which the leading coefficient of $\Phi_{t}$ equals $1$; in particular, for each $a\in{\mathbb{F}}_{q}[t]\setminus{\mathbb{F}}_{q}$, we would then have that $\Phi_{a}$ is a polynomial whose leading coefficient lives in ${\mathbb{F}}_{q}$. This is a standard observation used previously in [Ghi07a, Ghi07b].

2.3. Valuations and Weil heights

We let $M_{K}$ be the set of valuations on $K$. Then $M_{K}$ is a set of valuations which satisfies a product formula (see [Ser97, Chapter 2]); thus (with an appropriate normalization for each absolute value $|\cdot|_{v}$) we have

• •

for each nonzero $x\in K$, there are finitely many $v\in M_{K}$ such that $|x|_{v}\neq 1$; and

• •

for each nonzero $x\in K$, we have $\prod_{v\in M_{K}}|x|_{v}=1$.

We may use these valuations to define a Weil height for each $x\in K$ as

[TABLE]

2.4. Canonical heights

Let $\Phi:A\rightarrow K\{\tau\}$ be a Drinfeld module of rank $d$ (i.e. the degree of $\Phi_{t}$ as a polynomial in $\tau$ equals $d$). The canonical height of $\beta\in K$ relative to $\Phi$ (see [Den92b]) is defined as

[TABLE]

Denis [Den92b] showed that a point is torsion if and only if its canonical height equals [math].

For every $v\in M_{K}$, we let the local canonical height of $\beta\in K$ at $v$ to be

[TABLE]

for more details on local canonical heights, see [Ghi07b]. Furthermore, for every $a\in{\mathbb{F}}_{q}[t]$, we have ${\widehat{h}}_{v}(\Phi_{a}(x))=\deg(\Phi_{a})\cdot{\widehat{h}}_{v}(x)$. It is clear that ${\widehat{h}}_{v}$ satisfies the triangle inequality, and also that $\sum_{v\in M_{K}}{\widehat{h}}_{v}(\beta)={\widehat{h}}(\beta)$ (therefore, also ${\widehat{h}}(\cdot)$ satisfies the triangle inequality).

2.5. Integrality and reduction

Since we can always replace $K$ by a finite extension, we will define the notion of $S$-integrality with respect to a given point $\alpha$ under the assumption that $\alpha\in K$ (note that we are interested in Theorem 1.1 for the $S$-integrality with respect to $\alpha$ within a given finitely generated $\Phi$-module $\Gamma$ and therefore, once again at the expense of replacing $K$ by a finite extension, we may assume $\Gamma\subset K$).

Definition 2.1**.**

For a finite set of places $S\subset M_{K}$ and $\alpha\in K$, we say that $\beta\in K$ is $S$-integral with respect to $\alpha$ if for every place $v\notin S$, the following are true:

• •

if $|\alpha|_{v}\leq 1$, then $|\alpha-\beta|_{v}\geq 1$.

• •

if $|\alpha|_{v}>1$, then $|\beta|_{v}\leq 1$.

We note that if $\beta$ is $S$-integral with respect to $\alpha$, then it is also $S^{\prime}$-integral with respect to $\alpha$, where $S^{\prime}$ is a finite set of places containing $S$. Furthermore (as noted in [GT07, Subsection 2.6]), the notion of $S$-integrality with respect to a point is invariant when replacing $K$ by a finite extension $L$ and also replacing $S$ by the set of places in $L$ lying above the places from $S$.

Definition 2.2**.**

The Drinfeld module $\Phi$ has good reduction at a place $v$ if for each nonzero $a\in A$, all coefficients of $\Phi_{a}$ are $v$-adic integers and the leading coefficient of $\Phi_{a}$ is a $v$-adic unit. If $\Phi$ does not have good reduction at $v$, then we say that $\Phi$ has bad reduction at $v$.

It is immediate to see that $\Phi$ has good reduction at $v$ if and only if all coefficients of $\Phi_{t}$ are $v$-adic integers, since we already assumed that the leading coefficient of $\Phi_{t}$ equals $1$. Finally, we note that replacing $\Phi$ by the isomorphic Drinfeld module $\Psi$ given by $\Psi_{t}(x)=c\Phi_{t}\left(c^{-1}x\right)$ (for some nonzero $c\in K$), if we were to show that there exist finitely many $S$-integral points in the finitely generated $\Phi$-module $\Gamma$ with respect to a given point $\alpha$, then it suffices to prove that there exist finitely many points in the $\Psi$-module $c\cdot\Gamma$, which are $S^{\prime}$-integral with respect to $c\alpha$, where $S^{\prime}$ consists of all places in $S$ along with the finitely many places $v$ of $K$ for which $|c|_{v}\neq 1$. Indeed, for each element $c\gamma\in c\Gamma$ which is $S^{\prime}$-integral with respect to $c\alpha$, we must have that $\gamma$ is $S^{\prime}$-integral with respect to $\alpha$. So, knowing there exist at most finitely many such elements $\gamma$ which are $S^{\prime}$-integral for $\alpha$ yields that there exist at most finitely many points in $\Gamma$, which are $S$-integral with respect to $\alpha$ (because $S\subseteq S^{\prime}$ and enlarging the set $S$ may only increase the number of $S$-integral points with respect to a given point).

3. Proofs of our main results

Before proceeding to the proof of Theorem 1.1, we prove several facts about local heights.

From now on, let $\Phi_{t}=\sum_{i=0}^{d}a_{i}\tau^{i}$; also, as explained in Section 2, we assume from now on that $a_{d}=1$. The following result follows using a similar proof as the one employed in [GT07, Fact 3.6] (in the case $\Phi_{t}$ is monic).

Lemma 3.1**.**

For every place $v$ of $K$, there exists a real number $M_{v}\geq 1$ such that for each $x\in K$, if $|x|_{v}>M_{v}$, then for every nonzero $Q\in{\mathbb{F}}_{q}[t]$, we have $|\Phi_{Q}(x)|_{v}=|x|_{v}^{q^{d\cdot\deg(Q)}}$. Moreover, if $|x|_{v}>M_{v}$, then ${\widehat{h}}_{v}(x)=\log|x|_{v}$.

Proof.

We let $M_{v}=\max\left\{1,\max_{i=0}^{d-1}|a_{i}|_{v}^{1/(q^{d}-q^{i})}\right\}$. Then for each $x\in K$ such that $|x|_{v}>M_{v}$, we have that $|\Phi_{t}(x)|_{v}=|x|_{v}^{q^{d}}$; so, in particular, $|\Phi_{t}(x)|_{v}>M_{v}$ and therefore, for each nonnegative integer $n$, we have that $\left|\Phi_{t^{n}}(x)\right|_{v}=|x|_{v}^{q^{dn}}$. In particular, for any nonzero polynomial $Q\in A$, we have $|\Phi_{Q}(x)|_{v}=|x|_{v}^{q^{d\deg(Q)}}$. Finally, this means that if $|x|_{v}>M_{v}$, then ${\widehat{h}}_{v}(x)=\log|x|_{v}$. ∎

Lemma 3.2**.**

For each place $v$ of $M_{K}$ and for each $x\in K$, if ${\widehat{h}}_{v}(x)>0$, then for all polynomials $Q\in{\mathbb{F}}_{q}[t]$ of sufficiently large degree, we have that ${\widehat{h}}_{v}(x)=\frac{\log\left|\Phi_{Q}(x)\right|_{v}}{q^{d\cdot\deg(Q)}}$.

Proof.

The proof is an immediate corollary of Lemma 3.1 once we note that if ${\widehat{h}}_{v}(x)>0$ then there exists a nonzero integer $\ell$ (depending on $x$ and $v$) such that $|\Phi_{t^{\ell}}(x)|_{v}>M_{v}$; moreover, we may assume $\ell$ is minimal with this property. Then Lemma 3.1 yields that

[TABLE]

Moreover, for each polynomial $Q\in{\mathbb{F}}_{q}[t]$ of degree at least equal to $\ell$, we have that

[TABLE]

Equations (3.2.1) and (3.2.2) finish the proof of Lemma 3.2. ∎

The following result is an immediate consequence of [Ghi07a, Theorem 4.5] (which provides a more general positive lower bound for the canonical height of non-torsion points $x\in\overline{K}$ depending only on the number of places of bad reduction for the given Drinfeld module in the field extension $K(x)$).

Lemma 3.3**.**

There exists a positive constant $c_{0}$ such that for all non-torsion points $x\in K$, we have ${\widehat{h}}(x)\geq c_{0}$.

The next result will be used in the proof of Proposition 3.5 (which is our main technical ingredient for the proof of Theorem 1.1).

Lemma 3.4**.**

Let $r\in{\mathbb{N}}$, let $\gamma_{1},\dots,\gamma_{r}\in K$ and let $v\in M_{K}$. Assume that

[TABLE]

Then there exists $n_{0}\in{\mathbb{N}}$ (depending only on $\Phi$, $v$ and on $\gamma_{1},\dots,\gamma_{r}$) and there exists a positive real number $c_{1}$ (depending only on $\Phi$ and $v$) such that for all polynomials $P_{1},\dots,P_{r}\in{\mathbb{F}}_{q}[t]$ satisfying

1. (i)

$\deg(P_{1})>\max\{\deg(P_{2}),\cdots,\deg(P_{r})\}$, and 2. (ii)

$\min\{\deg(P_{2}),\cdots,\deg(P_{r})\}\geq n_{0}$,

we have $\log\left|\Phi_{P_{1}}(\gamma_{1})+\cdots+\Phi_{P_{r}}(\gamma_{r})\right|_{v}>c_{1}\cdot q^{d\cdot(\deg(P_{1})-n_{0})}$.

Proof.

Let $M_{v}\geq 1$ be the real number as in the conclusion of Lemma 3.1 and let $L_{v}:=2M_{v}$. Then for each $i=2,\dots,r$ such that ${\widehat{h}}_{v}(\gamma_{i})=0$, we must have that for all polynomials $Q_{i}\in{\mathbb{F}}_{q}[t]$ then

[TABLE]

Similarly, according to Lemma 3.2, we have that for each $i=1,\dots,r$ for which ${\widehat{h}}_{v}(\gamma_{i})>0$ (note that our hypothesis yields that ${\widehat{h}}_{v}(\gamma_{1})>0$) then for all polynomials $Q_{i}\in{\mathbb{F}}_{q}[t]$ of degree larger than some positive integer $n_{0}$ (which depends only on $v$ and on the $\gamma_{i}$’s),

[TABLE]

Furthermore, for each $i$ as in equation (3.4.3), as long as $\deg(Q_{i})$ is sufficiently large, then we have that

[TABLE]

Combining equations (3.4.2), (3.4.3) and (3.4.4), coupled with our hypotheses (3.4.1) and (i) from Lemma 3.4, we obtain that $\left|\Phi_{P_{1}}(\gamma_{1})\right|_{v}>\max_{i>1}\left|\Phi_{P_{i}}(\gamma_{i})\right|_{v}$ and therefore,

[TABLE]

Finally, combining (3.4.5) with Lemma 3.1 (see also equation (3.4.3)), we obtain that

[TABLE]

Since $|\Phi_{t^{n_{0}}}(\gamma_{1})|_{v}\geq L_{v}>M_{v}$ and therefore, letting $c_{1}:=\log(2L_{v}/3)\geq\log(4/3)>0$, we obtain the desired conclusion in Lemma 3.4. ∎

The following proposition is the key technical result required to prove Theorem 1.1 and it is the generalization of [GT07, Proposition 3.12] in the case of arbitrary function fields $K$.

Proposition 3.5**.**

Let $\alpha\in K$, let $\Gamma$ be a torsion-free $\Phi$-submodule of ${\mathbb{G}}_{a}(K)$, and let $\gamma_{1},\dots,\gamma_{r}$ be a basis for the $\Phi$-submodule $\Gamma$. For each $i\in\{1,\dots,r\}$ let $(P_{n,i})_{n\in\mathbb{N}}\subset{\mathbb{F}}_{q}[t]$ be a sequence of polynomials such that for each $m\neq n$, the $r$-tuples $(P_{n,i})_{1\leq i\leq r}$ and $(P_{m,i})_{1\leq i\leq r}$ are distinct. Then there exists a place $v\in M_{K}$ such that

[TABLE]

Proof.

The hypothesis on the $r$-tuples $(P_{n,i})_{1\leq i\leq r}$ implies that

[TABLE]

Combining this with the triangle inequality for the $v$-adic norm, we have that the sought statement is equivalent to the existence of a place $v$ such that

[TABLE]

Observe that it is sufficient to prove it for a subsequence $(n_{k})_{k\geq 1}\subset{\mathbb{N}}$, since passing to a subsequence can only lower the $\limsup$. We will repeatedly use this fact during the proof and we will drop the extra indexes in order to lighten the notation.

The proof proceeds by induction on $r$. If $r=1$, since $\gamma_{1}$ is non-torsion, there exists a place $v$ such that ${\widehat{h}}_{v}(\gamma_{1})>0$ and the conclusion follows from Lemma 3.2. We can then assume that (3.5.2) holds true for all $\Phi$-submodules of rank less than $r$ and we are going to prove it for all $\Phi$-submodules of rank $r$.

Let $S_{0}$ be the set of places $v\in M_{K}$ such that ${\widehat{h}}_{v}(\gamma)>0$ for some $\gamma\in\Gamma$. This set is finite, as proved in [GT07, Fact 3.13]. Furthermore, for any place $w$ outside the set $S_{0}$, we have that $|\gamma|_{w}$ is uniformly bounded above for all $\gamma\in\Gamma$ (see Lemma 3.1).

If there exists $j$ such that

[TABLE]

then the conclusion follows from the inductive hypothesis. Indeed, by the inductive hypothesis, there exists a place $v$ such that

[TABLE]

So, in particular, there exists some $i\neq j$ such that $|\Phi_{P_{n,i}}(\gamma_{i})|_{v}\to\infty$ as $n\to\infty$ (also note that the denominator from (3.5.4) must go to infinity due to equation (3.5.3)). So, ${\widehat{h}}_{v}(\gamma_{i})>0$ (by Lemma 3.1), which means that actually $v\in S_{0}$. Combining (3.5.4) with (3.5.3) gives

[TABLE]

We distinguish two cases. If ${\widehat{h}}_{v}(\gamma_{j})=0$, then $\{|\Phi_{Q}(\gamma_{j})|_{v}\}_{Q\in{\mathbb{F}}_{q}[t]}$ is bounded, so that for large enough $n$ we have

[TABLE]

and the result follows from (3.5.5).

If ${\widehat{h}}_{v}(\gamma_{j})>0$ but $\deg(P_{n,j})\to\infty$ as $n\to\infty$, we apply Lemma 3.2 to get

[TABLE]

Now, if $\deg(P_{n,j})$ is bounded above, then also $\left|\Phi_{P_{n,j}}(\gamma_{j})\right|_{v}$ is bounded above and therefore, once again

[TABLE]

Thus, combining the fact that $\lim_{n\to\infty}\frac{\log|\Phi_{P_{n,j}}(\gamma_{j})|_{v}}{\sum_{i=1}^{r}q^{d\deg P_{n,i}}}=0$ with equation (3.5.5), gives

[TABLE]

for large enough $n$. We then derive (3.5.2) using (3.5.3) and (3.5.5).

We therefore assume that there is no $j$ for which (3.5.5) holds. Equivalently, there exists $B\geq 1$ such that for any $n$ we have

[TABLE]

which is the same as

[TABLE]

We will proceed doing analysis at the places $v\in S_{0}$; note that $S_{0}$ is non-empty since each $\gamma_{i}$ is non-torsion (and so, there are places $v$ such that ${\widehat{h}}_{v}(\gamma_{i})>0$).

The strategy of the proof goes as follows: if we cannot prove directly (3.5.2), then we find $\delta_{1},\dots,\delta_{r}\in\Gamma$, a sequence $(n_{k})_{k\geq 1}\subset{\mathbb{N}}$ and a sequence of $r$-tuples of polynomials $(R_{k,i})_{1\leq i\leq r}$ such that

[TABLE]

and

[TABLE]

Thanks to these two relations, we have that if there exists a place $v$ for which

[TABLE]

then also

[TABLE]

In this way we reduce to proving (3.5.2) for the $\delta_{i}$’s and the sequence of $r$-tuples of polynomials $(R_{k,i})_{1\leq i\leq r}$. We will choose them in such a way that the process cannot go on forever, thereby showing that (3.5.2) has to hold at a finite step for a suitable place $v$.

We claim that, if there exist $j$ and $v$ such that

[TABLE]

and

[TABLE]

for all $n$ big enough, then inequality (3.5.2) holds for the place $v$. Indeed, using (3.5.7), we have that for any $i$ the degrees $\deg P_{n,i}$ go to infinity as $n\to\infty$. This, together with the two inequalities above, allows us to apply Lemma 3.4 to obtain (for a suitable positive integer $n_{0}$) that

[TABLE]

where we used again (3.5.7).

In particular, if there exists $j$ such that

[TABLE]

for all $n$ big enough, either (3.5.2) follows, or we can assume that for all places $v$ we have

[TABLE]

We will now describe the construction of the $\delta_{i}$’s and $R_{n,i}$’s we referred to before. In the first step of our process, for all $n$ and each $i>1$, we divide (with quotient and remainder) $P_{n,i}$ by $P_{n,1}$, obtaining

[TABLE]

so that $\deg R_{n,i}<\deg P_{n,1}$; also, we note that $\deg R_{n,i}\leq\deg P_{n,i}$. We also let $R_{n,1}:=P_{n,1}$. From (3.5.7) the degrees $\deg C_{n,i}$ are uniformly bounded as $n\to\infty$. It follows that there are only finitely many possible polynomials $C_{n,i}$, so that, passing to a subsequence $(n_{k})_{k\geq 1}$, we may assume that there exist polynomials $C_{i}$ satisfying

[TABLE]

for all $n$ (where we dropped the index $k$ of the subsequence).

Let

[TABLE]

Observe that for each $n$, we have

[TABLE]

as it follows from the definition of the $\delta_{i}$’s and of the $R_{n,i}$’s.

Also, since $\deg R_{n,i}\leq\deg P_{n,i}$ for all $n$ and $i$, we have

[TABLE]

which implies

[TABLE]

Using that $R_{n,1}=P_{n,1}$ we obtain

[TABLE]

where we used (3.5.7) in the last step. In particular we derive

[TABLE]

As explained before, thanks to (3.5.9) and (3.5.10), we can reduce the problem to the study of the $\delta_{i}$’s and the sequence of $r$-tuples of polynomials $(R_{n,i})_{1\leq i\leq r}$.

In order for our strategy to work, we will also need that (3.5.7) has to hold for the polynomials $R_{n,i}$, that is, we want

[TABLE]

for a suitable constant $B^{\prime}$. However, arguing as we did from (3.5.3) to (3.5.6), if the above is not satisfied, then there exists a place $w$ such that

[TABLE]

Using (3.5.9) and (3.5.10) we obtain

[TABLE]

thereby proving the statement. We may then assume that (3.5.7) holds also for the $R_{n,i}$’s (possibly with a different constant $B$).

Since

[TABLE]

for all $n$, as we previously observed, if there exists a place $v$ such that

[TABLE]

then (3.5.2) holds at the place $v$. Therefore, we may assume that for all places $v$ we have

[TABLE]

from which we get

[TABLE]

where we used that $\delta_{i}=\gamma_{i}$ for $i>1$.

We now repeat the above construction of the $\delta_{i}$’s and $R_{n,i}$’s using $R_{n,2}$ in place of $P_{n,1}$, that is, we will proceed dividing by $R_{n,2}$ each of the $R_{n,i}$’s for $i\neq 2$. We will still denote the new polynomials with the letter $R$ to not cluster the notation. We will then find a subsequence of $\mathbb{N}$, polynomials $R_{n,i}$ and $\delta_{2}$ (given by a suitable $\Phi$-linear combination of $\delta_{1}$ and of $\gamma_{i}$ for $i\geq 2$) for which both (3.5.9) and (3.5.10) hold (where $\delta_{i}=\gamma_{i}$ for $i\geq 3$).

As before, either the result follows or we can assume that (3.5.7) holds and for all places $v$ we have

[TABLE]

From this and (3.5.11) we get

[TABLE]

for all places $v$.

Continuing this process for all $i$ up to $r$, either we obtain (3.5.2), or we find a subsequence of $\mathbb{N}$, polynomials $R_{n,i}$ satisfying (3.5.7) and $\delta_{1},\dots,\delta_{r}$ (given by $\Phi$-linear combinations of the original $\gamma_{i}$’s) satisfying (3.5.9) and (3.5.10), and such that

[TABLE]

for all $j$ and all places $v$. By summing over $v\in S_{0}$, we find

[TABLE]

for any $j$.

Repeating the above procedure with the $\delta_{i}$’s in place of the $\gamma_{i}$’s, either we can find $v$ such that (3.5.2) holds, or we construct $\epsilon_{1},\dots,\epsilon_{r}$ such that for all $j$ and $v$ we have

[TABLE]

This, combined with (3.5.12), gives

[TABLE]

for all $j$ and $v$. Summing over $v\in S_{0}$ we find

[TABLE]

for all $j$. More generally, repeating this process $\ell$ times, one would find elements $\alpha_{1}^{(\ell)},\dots,\alpha_{r}^{(\ell)}\in\Gamma$ for which

[TABLE]

for each $j=1,\dots,r$. However, by Lemma 3.3 there exists a positive constant $c_{0}$ such that ${\widehat{h}}(x)\geq c_{0}$ for all non-torsion $x\in K$. Since $\Gamma$ is torsion-free, the process has to stop at some finite step. Indeed, otherwise we would get that for large enough $\ell$, the right hand side in the inequality (3.5.13) is smaller than $c_{0}$ because the $\gamma_{i}$’s are given and therefore $\alpha_{j}^{(\ell)}$ must be a torsion point in $\Gamma$ and so, $\alpha_{j}^{(\ell)}=0$ for each $j$. But then $\sum_{i=1}^{r}\Phi_{P_{n,i}}(\gamma_{i})=0$ (see (3.5.9), which holds at each step in our process) for all $n$ sufficiently large, which contradicts the fact that the $\gamma_{i}$’s are a basis for the $\Phi$-submodule $\Gamma$ (and the $r$-tuples $(P_{n,i})_{i=1}^{r}$ are distinct). Therefore, there exists a place $v$ for which (3.5.2) must hold. ∎

Theorem 1.1 follows identically as the proof of [GT07, Theorem 2.4] once we replace [GT07, Proposition 3.12] by Proposition 3.5.