Diophantine equations of the form $Y^{n}=f(X)$ over function fields

Let $n\geq 2$ be an integer and $\kappa$ be a finite field of characteristic $\ell>0$. Let $F$ be a global function field with field of constants $\kappa$ and assume that there exists a prime $P_{\infty}$ of $F$ of degree $1$. In other words, we assume that there is a prime $P_{\infty}$, which is totally inert in the composite $\bar{\kappa}\cdot F$. The ring of integers $\mathcal{O}_{F}$ shall consist of all functions $f\in F$ with no poles away from $P_{\infty}$. Given a polynomial $f(X)$ with coefficients in $\kappa$, we study solutions to the diophantine equation $Y^{n}=f(X)$ for which both $X$ and $Y$ lie in $\mathcal{O}_{F}$. More precisely, we prove that if certain additional conditions are met, then there are no non-constant solutions, i.e., $X$ and $Y$ must both belong to the field of constants $\kappa$. Equations of the form $Y^{n}=f(X)$ are of significant interest. We shall apply our analysis to study a class of diophantine equations which involve perfect powers in arithmetic progressions. Let $m,n,d\geq 2$ be integers and let $r\geq 1$, then, there has been significant interest in the classification of integral solutions to the diophantine equation

cf. for instance, [BGP04, BZ13, Haj15, BPS17, BPSS18, KP18, Pat18, AGP20].

We also explore themes motivated by the Iwasawa theory of function fields. Mazur initiated the Iwasawa theory of elliptic curves over number fields (cf. [Maz72]), which had applications to the study of growth of Mordell–Weil ranks of elliptic curves in certain infinite towers of number fields. One hopes to extend such lines of investigation to curves of higher genus (cf. [Ray22]) and more generally, study the stability and growth of solutions to any diophantine equation in an infinite tower of global fields. In this paper, we shall study certain function field analogues of such questions, however, instead of elliptic curves, we consider the class of diophantine equations of the form $Y^{n}=f(X)$, where $f(X)$ has constant coefficients. Let us explain our results in greater detail. Given any integer $n$, there is a unique extension $\kappa_{n}/\kappa$ such that $\operatorname{Gal}(\kappa_{n}/\kappa)$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$. Given a prime $p$, set $\kappa_{n}^{(p)}$ to denote $\kappa_{p^{n}}$ and set $F_{n}^{(p)}$ to denote the composite $F\cdot\kappa_{n}^{(p)}$. This gives rise to a tower of function field extensions

Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, i.e., the valuation ring of $\mathbb{Q}_{p}$. The constant $\mathbb{Z}_{p}$-extension of $F$ is the infinite union

It is easy to see that the Galois group $\operatorname{Gal}\left(F_{\infty}^{(p)}/F\right)$ is isomorphic to $\mathbb{Z}_{p}$. Let $h_{F}$ denote the class number of $F$ (cf. [Ros02, Chapter 5]). Note that since $P_{\infty}$ is assumed to have degree $1$, it remains inert in $F_{n}^{(p)}$ for all $p$. Let $\mathcal{O}_{\infty}^{(p)}$ (resp. $\mathcal{O}_{n}^{(p)}$) be the ring of integers of $F_{\infty}^{(p)}$ (resp. $F_{n}^{(p)}$), i.e., the functions $f\in F_{\infty}$ with no poles away from $P_{\infty}$. We now state the main result.

As a consequence of the above result, we find that for any prime $p$, there are only finitely many numbers $n$, that are not powers of $\ell$, for which $Y^{n}=f(X)$ has solutions in $F_{\infty}^{(p)}$. A more specific criterion applies to any function field $F$, cf. Theorem 2.4. The methods used in proving the above result are applied to study another question of independent interest. Let $K$ be a field of positive characteristic $\ell$ and $A$ be the polynomial ring $K[T_{1},\dots,T_{n}]$.

It follows from the above result that if $n>1$ is not a power of $\ell$, then $X^{n}=f(X)$ does not have non-constant solutions in $A$.

Organization: Including the introduction, the manuscript consists of $5$ sections. In section 2, we prove criteria for the constancy of solutions to $Y^{n}=f(X)$ in global function fields $F$. The main result proven in section 2 is Theorem 2.4. In section 3, we extend the results in section 2 to prove the constancy of solutions to the above equation in $\mathbb{Z}_{p}$-extensions of $F$. It is in this section that we prove the main result of the paper, i.e., Theorem 3.2. In section 4, we prove similar results for the polynomial rings over a field. Finally, in section 5, we study the specific case when $f(X)=\sum_{i=1}^{k}(X+ir)^{m}$.

In this section, we introduce basic notions and prove results about the solutions to certain diophantine equations over global function fields. Throughout this section, $\ell$ be a prime number and $A$ be an integral domain of characteristic $\ell$ with field of constants $\kappa$. We introduce the notion of a discrete valuation on $A$.

Let $A_{0}$ be the subring of $A$ consisting of all elements $a\in A$ for which $d(a)\leq 0$. Given $f,g\in A$, we say that $f$ divides $g$ if $fh=g$ for some $h\in A$. It is clear that if $f$ divides $g$ then $d(f)\leq d(g)$.

We shall illustrate the above result in various cases of interest. In this section, we study diophantine equations over global function fields $F$. Let $\ell$ be a prime number and denote by $\mathbb{F}_{\ell}$ the finite field with $\ell$ elements (i.e. $\mathbb{Z}/\ell\mathbb{Z}$). Fix an algebraic closure $\bar{F}$ of $F$. Let $\kappa$ be the algebraic closure of $\mathbb{F}_{\ell}$ in $F$, and let $\bar{\kappa}$ be the algebraic closure of $\kappa$ in $\bar{F}$. Set $F^{\prime}$ to denote the composite of $F$ with $\bar{\kappa}$.

Following [Ros02, Chapter 5], a prime $v$ of $F$ is by definition the maximal ideal of a discrete valuation ring $\mathcal{O}_{v}\subset F$ with fraction field equal to $F$. A divisor of $F$ is a finite linear combination $D=\sum_{v}n_{v}v$ of primes $v$. In the above sum, $n_{v}$ are all integers, and the set of primes $v$ for which $n_{v}\neq 0$ is referred to as the support of $D$. Given a function $g\in F$, denote by $\operatorname{div}(g)$ the associated principal divisor. Note that any principal divisor has degree $0$. Two divisors are considered equivalent if the differ by a principal divisor. The class group of $F$ is the group of divisor classes of degree $0$, and has finite cardinality (cf. [Ros02, Lemma 5.6]). Denote by $h_{F}$ the class number, i.e., the number of elements in the class group. Given a natural number $N$, denote by $h_{F}[N]$ the cardinality of the $N$-torsion in class group.

The field $F^{\prime}$ is identified with the field of fractions of a projective algebraic curve $\mathfrak{X}$ over $\bar{\kappa}$. A point $w\in\mathfrak{X}(\bar{\kappa})$ is also referred to a prime of $F^{\prime}$, since it corresponds to a valuation ring $\mathcal{O}_{w}\subset F^{\prime}$ with fraction field $F^{\prime}$. Given a prime $w$ of $F^{\prime}$ and a prime $v$ of $F$, we say that $w$ lies above (or divides) $v$ if the natural inclusion of fields $F\hookrightarrow F^{\prime}$ induces an inclusion of valuation rings $\mathcal{O}_{v}\hookrightarrow\mathcal{O}_{w}$. Given a function $g\in F$ (resp. $g\in F^{\prime}$), denote by $\operatorname{ord}_{v}(g)$ (resp. $\operatorname{ord}_{w}(g)$) the order of vanishing of $g$ at $v$ (resp. $w$). We refer to $d_{v}(g)\mathrel{\mathop{\mathchar 58\relax}}=-\operatorname{ord}_{v}(g)$ (resp. $d_{w}(g)\mathrel{\mathop{\mathchar 58\relax}}=-\operatorname{ord}_{w}(g)$) the order of the pole of $g$ at $v$ (resp. $w$). Given a finite and nonempty set of primes $S$ of $F$, the ring of $S$-integers $\mathcal{O}_{S}$ consists of all functions $g\in F$ such that $d_{v}(g)\leq 0$ for all primes $v\notin S$. Let $\bar{S}$ be the set of primes of $F^{\prime}$ that lie above $S$. Let $A$ denote the composite $\mathcal{O}_{S}\cdot\bar{\kappa}$. A function $g\in A$ satisfies the property that for all $w\notin\bar{S}$, $d_{w}(g)\leq 0$. According to our conventions, $\mathcal{O}_{F}$ is the ring of $S$ integers where $S\mathrel{\mathop{\mathchar 58\relax}}=\{P_{\infty}\}$. Since $P_{\infty}$ is a prime of degree $1$, is totally inert in $F^{\prime}$. By abuse of notation, the single prime in $\bar{S}$ is also denoted $P_{\infty}$.

We list some basic properties of the function $d_{w}$ on $A$. The following result applies for any ring of $\bar{S}$-integers.

Recall that $P_{\infty}$ is a prime of degree $1$ and $\mathcal{O}_{F}$ is the associated ring of integers in $F$.

In this section, we apply Theorem 2.4 proven in the previous section, to study questions motivated by Iwasawa theory. Given primes $p$ and $q$ (not necessarily distinct) let $h_{n}(p,q)$ denote $\#\operatorname{Cl}(F_{n}^{(p)})[q^{\infty}]$, the cardinality of the $q^{\infty}$-torsion in the class group of $F_{n}^{(p)}$.

Recall notation from the introduction. The prime $P_{\infty}$ is totally inert in $F_{\infty}^{(p)}$ for any prime $p$. We set $\mathcal{O}_{\infty}^{(p)}$ to denote the ring of integers of $F_{\infty}^{(p)}$, i.e., the functions with no poles outside $\{P_{\infty}\}$. The following is the main result of this manuscript.

In this section, we study solutions to equations of the form $Y^{n}=f(X)$ in polynomial rings over a field. Let $K$ be any field of characteristic $\ell>0$ and $A$ be the polynomial ring $K[T_{1},\dots,T_{r}]$. Given a polynomial $g$, let $d_{i}(g)$ be the degree of $g$ viewed as a polynomial in $T_{i}$ over the subring $K[T_{1},\dots,T_{i-1},T_{i+1},\dots,T_{r}]$. The pair $(A,d_{i})$ satisfies the conditions (1)–(5) of Definition 2.1. The class group $\operatorname{Cl}(A)$ denotes the group of equivalence classes of Weil divisors. Since $A$ is a unique factorization domain, we have that $\operatorname{Cl}(A)=0$.

In this section, we apply the results proven in previous sections to study the solutions of the diophantine equation involving perfect powers in arithmetic progressions

Here, $m,n,r,d$ are integers such that $m,n,d\geq 2$ and $r\geq 1$. Viewing $f(X)$ as a polynomial with integral coefficients let $\Delta$ denote its discriminant.