The $T$-adic Galois representation is surjective for a positive density of Drinfeld modules

Given an elliptic curve $E$ over $\mathbb{Q}$, and a prime natural number $n>1$, there is a Galois representation

on the $n$-torsion of $E(\bar{\mathbb{Q}})$. Moreover, given a prime $p$, the $p$-adic Tate-module $T_{p}(E)\mathrel{\mathop{\mathchar 58\relax}}=\varprojlim_{n}E[p^{n}]$ admits a natural Galois action, which is encoded by the representation

Passing to the inverse limit of $\rho_{E,n}$, one obtains the adelic Galois representation

which can also be identified with the product $\prod_{p}\hat{\rho}_{E,p}$. The celebrated open image theorem of Serre [Ser72] asserts that if $E$ does not have complex multiplication, then the image of $\hat{\rho}_{E}$ has finite index in $\operatorname{GL}_{2}(\widehat{\mathbb{Z}})$. In particular, this means that for all but finitely many primes $p$, the $\rho_{E,p}\mathrel{\mathop{\mathchar 58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{Z}/p\mathbb{Z})$ is surjective. A prime $p$ is said to be exceptional if $\rho_{E,p}$ is not surjective. Duke [Duk97] showed that almost all elliptic curves have no exceptional primes (when ordered according to height). An elliptic curve $E_{/\mathbb{Q}}$ is said to be a Serre curve if the index of $\hat{\rho}_{E}$ in $\operatorname{GL}_{2}(\widehat{\mathbb{Z}})$ is equal to $2$. This is the minimal possible index, and Jones [Jon10] showed that almost all elliptic curves are Serre curves. Generalized results have been obtained for principally polarized abelian varieties over a rational base, cf. [LSTX19].

In this article I explore natural analogues of such questions for Drinfeld modules of rank $2$. These arithmetic objects are natural function field analogues of elliptic curves, and they give rise to compatible families of Galois representations. Drinfeld modules (and their generalizations) have come to prominence for their central role in the Langlands program over function fields. Let $q$ be a power of a prime number and $F$ be the rational function field $\mathbb{F}_{q}(T)$. Let $F^{\operatorname{sep}}$ be a choice of separable closure of $F$. Take $A$ to be the polynomial ring $\mathbb{F}_{q}[T]$. Take $\widehat{A}\mathrel{\mathop{\mathchar 58\relax}}=\varprojlim_{\mathfrak{a}}A/\mathfrak{a}$, where $\mathfrak{a}$ runs over all non-zero ideals in $A$. Pink and Rütsche [PR09] proved an analogue of Serre’s open image theorem for Drinfeld modules $\phi$ over $F$ without complex multiplication. In greater detail, suppose that $\phi\mathrel{\mathop{\mathchar 58\relax}}A\rightarrow F\{\tau\}$ is a Drinfeld module of rank $r$ for which $\operatorname{End}_{\bar{F}}(\phi)=A$, then the image of the adelic Galois representation

has finite index in $\operatorname{GL}_{r}(\widehat{A})$. For odd prime powers $q\geq 5$, Zywina [Zyw11] was in fact able to construct an explicit Drinfeld module of rank $2$ for which the adelic Galois representation is surjective. These results have been generalized to certain higher ranks by Chen [Che22, Che21]. There has also been recent interest in the study of the mod-$T$ representation for certain infinite families of Drinfeld modules [GM23], as well as its application to the inverse Galois problem over the rational function field.

The strategy taken by Duke to prove that almost all elliptic curves have no expectional primes makes use of the large sieve. This involves local estimates that rely on bounds for certain sums of Hurwitz class numbers (in congruence classes). These sums are related to the fourier coefficients of certain modular forms, and the estimates rely on the Ramanujan bound (cf. [Duk97, Lemma 3]). The approach taken for Drinfeld modules in this article is indeed different, since Duke’s method does not readily generalize to this context. I consider only the Galois representation at the rational prime $\mathfrak{p}=(T)$. I then study the density of rank $2$ Drinfeld modules $\phi$ for which the $T$-adic Galois representation

is surjective.

One may refer to section 2.3, where a precise notion of density is given for Drinfeld modules over $A$.

Including the introduction, this article consists of three sections. In section 2, I discuss the basic properties of Drinfeld modules and their associated Galois representations. In the final section, I prove that the surjectivity of the $T$-adic representation can indeed be detected from certain very explicit congruence conditions on the coefficients of a Drinfeld module with coefficients in $A$ (cf. Theorem 3.1). These conditions are very explicit, and they are leveraged to prove that the density of Drinfeld modules satisfying these conditions is positive. In fact, a lower bound for this density can be given, and I refer to the Remark 3.6 for a discussion on the issues in obtaining a satisfactory lower bound.

One of the key methods used is the algorithm of Gekeler for computing characteristic polynomials of rank $2$ Drinfeld modules. I would expect that they can be generalized to prove similar results for the Galois representations on $\mathfrak{p}$-adic Tate modules, when $\mathfrak{p}$ is a prime with small degree. However, this has not been pursued since one does not have a general statement that applied to all $\mathfrak{p}$, and one only considers $\mathfrak{p}=(T)$ in this article. I am hopeful that Theorem 1.1 shall pave the way for interesting developments in the future.

I discuss the theory of Drinfeld modules; my notation is consistent with that in [Pap23]. Let $p$ be an odd prime number and $q=p^{n}$. Set $\mathbb{F}_{q}$ to be the field with $q$ elements and assume that $q\geq 5$. Let $A$ be the polynomial ring $\mathbb{F}_{q}[T]$ and $F$ its fraction field $\mathbb{F}_{q}(T)$. Given a non-zero ideal $\mathfrak{a}$ with generator $a$, set $\operatorname{deg}\mathfrak{a}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{deg}_{T}(a)$. Let $K$ be a field extension of $\mathbb{F}_{q}$, $K$ is called an $A$-field if it is equipped with an $\mathbb{F}_{q}$-algebra homomorphism $\gamma\mathrel{\mathop{\mathchar 58\relax}}A\rightarrow K$. Given an $A$-field $K$, the $A$-characteristic of $K$ is defined as follows

Then $K$ is of generic characteristic if $\operatorname{char}_{A}(K)=0$. Take $K^{\operatorname{sep}}$ to be a choice of separable closure of $K$, and set $\operatorname{G}_{K}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(K^{\operatorname{sep}}/K)$. The rational function field $F$ is an $A$-field of generic characteristic where the map $\gamma\mathrel{\mathop{\mathchar 58\relax}}A\hookrightarrow F$ is the natural inclusion map. Given a non-zero prime $\lambda$ of $A$, take $k_{\lambda}$ to denote the residue field of $A$ at $\lambda$. Then, $k_{\lambda}$ is an $A$-field of characteristic $\lambda$, where $\gamma_{\lambda}\mathrel{\mathop{\mathchar 58\relax}}A\rightarrow k_{\lambda}$ is the mod-$\lambda$ reduction map.

Given an $\mathbb{F}_{q}$-algebra $K$, set $K\{\tau\}$ to denote the noncommutative ring of twisted polynomials over $K$, the elements of which are polynomials $f(\tau)=\sum_{i=0}^{d}a_{i}\tau^{i}$, with coefficients $a_{i}\in K$. Addition is defined term-wise as follows

Given $f=\sum_{i=0}^{d}a_{i}\tau^{i}$, set

The polynomial $f(x)$ is an $\mathbb{F}_{q}$-linear polynomial, i.e., given $c_{1},c_{2}\in\mathbb{F}_{q}$, the following algebraic relation holds

in $K[x,y]$. I take $K\langle x\rangle$ to denote non-commutative ring of $\mathbb{F}_{q}$-linear polynomials defined by the relations

The map $f(\tau)\mapsto f(x)$ is an isomorphism of $\mathbb{F}_{q}$-algebras from $K\{\tau\}$ to $K\langle x\rangle$.

Next, I define the notion of height $\operatorname{ht}_{\tau}(f)$ and degree $\operatorname{deg}_{\tau}(f)$ of a twisted polynomial $f(\tau)\in K\{\tau\}$. Write

where $a_{h},a_{d}\neq 0$, and set

Thus, one finds that the degree of $f(x)$ as a polynomial in $x$ is $q^{\operatorname{deg}_{\tau}(f)}$. When referring to the degree of the polynomial in $x$, I write

Let $\partial\mathrel{\mathop{\mathchar 58\relax}}K\{\tau\}\rightarrow K$ be the derivative map sending

Thus, $\operatorname{ht}(f)=0$ if and only if $\partial(f)\neq 0$. Note that $f(x)$ is separable if and only if its derivative in $x$ does not vanish. Since $\partial f=\frac{d}{dx}(f(x))$, one finds that $f(x)$ is separable if and only if $\operatorname{ht}(f)=0$. With these conventions at hand, I recall the definition of a Drinfeld module of rank $r\geq 1$. In this article, I shall only be concerned with the case in which $r=2$.

The second condition is equivalent to requiring that $\operatorname{deg}_{\tau}(\phi_{T})=r$. I shall write

and observe that since $\operatorname{deg}_{\tau}\phi_{T}=r$, $g_{r}\neq r$. The coefficients $(g_{1},\dots,g_{r})$ completely describe the module $\phi$.

Let $\phi$ and $\psi$ be Drinfeld modules, a morphism $u\mathrel{\mathop{\mathchar 58\relax}}\phi\rightarrow\psi$ is a function $u(\tau)\in K\{\tau\}$ such that $u\phi_{a}=\psi_{a}u$ for all $a\in A$. An isogeny is a non-zero morphism. It is easy to see that if there exists an isogeny, then, the rank of $\phi$ equals the rank of $\psi$. The group of all morphisms from $\phi$ to $\psi$ is denoted $\operatorname{Hom}_{K}(\phi,\psi)$. An isomorphism between Drinfeld module is an isogeny with an inverse. It is easy to see that the only invertible elements in $K\{\tau\}$ are the non-zero constant elements $c\in K$. Write $\phi_{T}=T+g_{1}\tau+\dots+g_{r}\tau^{r}$ and $\psi_{T}=T+g_{1}^{\prime}\tau+\dots+g_{r}^{\prime}\tau^{r}$, one finds that $\phi$ is isomorphic to $\psi$ if and only if there is a non-zero constant $c$ such that

for $i\in[1,r]$. When $r=2$, the $j$-invariant of $\phi$ is defined as

Two rank $2$ Drinfeld modules $\phi$ and $\phi^{\prime}$ are isomorphic if and only if $j(\phi)=j(\phi^{\prime})$ (cf. [Pap23, Ch. 3]).

Let $\Omega_{F}^{\ast}$ denote the set of discrete valuations of $F$, and $\Omega_{F}$ be the valuations corresponding to non-zero prime ideals of $A$. For $\lambda\in\Omega_{F}$, set $A_{\lambda}$ denote the completion of $A$ at $\lambda$, and $F_{\lambda}$ be its field of fractions. Taking $\mathfrak{m}_{\lambda}$ to be the maximal ideal of $A_{\lambda}$, set $k_{\lambda}\mathrel{\mathop{\mathchar 58\relax}}=A_{\lambda}/\mathfrak{m}_{\lambda}$. Let $\phi$ be a Drinfeld module over $F$ and $\lambda\in\Omega_{F}$. I denote by $\phi_{\lambda}$ the localized Drinfeld module over $F_{\lambda}$, defined to be the composite

where the second map is induced by the natural inclusion of $F$ into $F_{\lambda}$. Say that $\phi$ has stable reduction at $\lambda$ if there is a Drinfeld module $\psi$ over $F_{\lambda}$ that is isomorphic to $\phi_{\lambda}$ with coefficients in $A_{\lambda}$, for which the reduction

is a Drinfeld module. The rank of $\bar{\psi}$ is referred to as the reduction rank of $\phi$ at $\lambda$. If the reduction rank is $r$, then $\phi$ has good reduction at $\lambda$. Let $\lambda\in\Omega_{F}$ be a prime of good reduction, and consider the reduced Drinfeld module

Let $a$ be the monic generator of $\lambda$. The element $a$ is in the kernel of the reduction map $\gamma\mathrel{\mathop{\mathchar 58\relax}}A\rightarrow k_{\lambda}$. The constant term of $\phi_{a}$ is $\gamma(a)=0$, and thus, $\operatorname{ht}_{\tau}(\phi_{a})>0$. In fact, $\operatorname{deg}_{T}(a)$ divides $\operatorname{ht}_{\tau}(\phi_{a})$ and the height of $\bar{\psi}$ is defined to be the integer

which one refers to as the height of the reduction of $\phi$ at $v$. It is easy to see that $H(\bar{\psi})\in[1,r]$, I refer to [Pap23, section 3.2] for further details. The quantity $H(\bar{\psi})$ is called the height of the reduction of $\phi$ at $\lambda$.

Given a Drinfeld module $\phi$ of rank $r$ over $F$, there is an isomorphic Drinfeld module $\psi$ for which

with coefficients $g_{i}\in A$. A tuple $(g_{1},\dots,g_{r})\in A^{r}$ is said to be minimal if there is no non-constant polynomial $f\in A$ such that $g_{i}$ is divisible by $c^{q^{i}-1}$ for all $i\in[1,r]$. It is easy to see that $\psi$ can be uniquely chosen so that $(g_{1},\dots,g_{r})$ is minimal in the above sense. Moreover, the primes of bad reduction are precisely those that divide the leading coefficient $g_{r}$. It is thus clear that there are only finitely many primes of bad reduction. The primes of unstable reduction are those that divide every single coefficient $g_{i}$ for $i\in[1,r]$.

I discuss Galois representations associated to Drinfeld modules over $F$ or rank $r$. Let $\phi$ be a Drinfeld module over $F$ and $a$ be a non-zero polynomial in $A$. Since the constant coefficient of $\phi_{a}$ is $\gamma(a)=a$, it follows that $\phi_{a}(x)$ is a separable polynomial. Let $\phi[a]\subset F^{\operatorname{sep}}$ denote the set of roots of $\phi_{a}(x)$. I note that if $b\in A$, then,

One defines a twisted $A$-module structure on $F^{\operatorname{sep}}$ by $b\ast\alpha\mathrel{\mathop{\mathchar 58\relax}}=\phi_{b}(\alpha)$. Then, the above computation shows that $\phi[a]$ is an $A$-submodule of $F^{\operatorname{sep}}$. Given a non-zero ideal $\mathfrak{a}$ in $A$, set $\phi[\mathfrak{a}]\mathrel{\mathop{\mathchar 58\relax}}=\phi[a]$, where $a$ is the monic generator of $\mathfrak{a}$. Then, one finds that $\phi[\mathfrak{a}]\simeq(A/\mathfrak{a})^{r}$. The Galois group $\operatorname{G}_{F}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Gal}(F^{\operatorname{sep}}/F)$ naturally acts on $\phi[\mathfrak{a}]$ by $A$-module automorphisms. I denote by

the associated Galois representation. Let $\mathfrak{p}$ be a non-zero prime ideal of $A$ and take $T_{\mathfrak{p}}(\phi)$ to denote the $\mathfrak{p}$-adic Tate-module, defined to be the inverse limit

Choosing a $A_{\mathfrak{p}}$-basis of $T_{\mathfrak{p}}(\phi)$ the associated Galois representation is denoted

The mod-$\mathfrak{p}$ reduction of $\hat{\rho}_{\phi,\mathfrak{p}}$ is identified with $\rho_{\phi,\mathfrak{p}}$. The Galois representation $\rho_{\phi,\mathfrak{a}}$ is unramified at all primes $\mathfrak{l}\nmid\mathfrak{a}$ of $A$ at which $\phi$ has good reduction. Moreover, given a non-zero prime $\mathfrak{l}\neq\mathfrak{p}$ of $A$, the representation $\hat{\rho}_{\phi,\mathfrak{p}}$ is unramified at $\mathfrak{l}$ if and only if $\mathfrak{l}$ is a prime of good reduction for $\phi$.

Assume that $\phi$ has good reduction at $\mathfrak{p}$. Let me briefly discuss the structure of $T_{\mathfrak{p}}(\phi)$ as a module over $\operatorname{G}_{\mathfrak{p}}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{G}_{F_{\mathfrak{p}}}$. Let $H$ denote the height of the reduction of $\phi$ at $\mathfrak{p}$. Let $n$ be a positive integer and $v=v_{\mathfrak{p}}$ be the valuation on $F_{\mathfrak{p}}$ normalized by $v(\pi_{v})=1$, where $\pi_{v}$ is a uniformizer of $F_{\mathfrak{p}}$.

For ease of notation, denote by $\bar{\phi}$ the reduction of $\phi$ at $\mathfrak{p}$, and $\bar{\phi}[\mathfrak{p}^{n}]$ be its $\mathfrak{p}^{n}$-torsion module. Then, $\bar{\phi}[\mathfrak{p}^{n}]$ is an unramified $\operatorname{G}_{\mathfrak{p}}$-module and can be identified with the quotient $\phi[\mathfrak{p}^{n}]/\phi[\mathfrak{p}^{n}]^{0}$. One has a short exact sequence of $\operatorname{G}_{\mathfrak{p}}$-modules

cf. [Pap23, (6.3.1) on p. 371]. Taking

and obtain a short exact sequence

cf. [Pap23, Proposition 6.3.9]. As an $A_{\mathfrak{p}}$-module,

Thus, one finds that

Let $F_{\mathfrak{p}}(\phi[\mathfrak{p}^{n}]^{0})$ be the field extension of $F_{\mathfrak{p}}$ generated by $\phi[\mathfrak{p}^{n}]^{0}$. In other words, it is the fixed field of the kernel of the action map

In this section, I consider the data defining a Drinfeld module over $A$ of rank $2$. I say that $\phi$ is defined over $A$ if

where $(g_{1},g_{2})\in A^{2}$. It is easy to see that any Drinfeld module over $F$ is isomorphic to a Drinfeld module defined over $A$. Since I assume that $\phi$ has rank $2$, it follows that $g_{2}\neq 0$. The pair $w=(g_{1},g_{2})\in A^{2}$ with $g_{2}\neq 0$ shall be referred to as a Drinfeld datum. I shall take $\phi^{w}$ to be the Drinfeld module defined by

The pair $w$ does not uniquely determine the Drinfeld module $\phi$ up to isomorphism over $F$. Take $\mathcal{C}$ to be the set of Drinfelf data $w=(g_{1},g_{2})\in A^{2}$.

Fix a pair of positive integers $\vec{c}=(c_{1},c_{2})$. Given a polynomial $f$, set $|f|\mathrel{\mathop{\mathchar 58\relax}}=q^{\operatorname{deg}(f)}$. Then, the height

Let $X>0$ be an integer, take $\mathcal{C}(X)$ to be the set of Drinfeld modules with $|\phi|<q^{X}$. Then, this is the set of Drinfeld data $w=(g_{1},g_{2})$ such that

Thus, one finds that

Let $\mathcal{S}$ be a subset of $\mathcal{C}$ and given a positive integer $X>0$, take $\mathcal{S}(X)\mathrel{\mathop{\mathchar 58\relax}}=\{w\in\mathcal{S}\mid|w|<X\}$. Refer to the following limit

(provided it exists) as the density of $\mathcal{S}$. The upper and lower densities $\overline{\mathfrak{d}}(\mathcal{S})$ and $\underline{\mathfrak{d}}(\mathcal{S})$ are defined as follows

and unlike $\mathfrak{d}(\mathcal{S})$, these densities are guaranteed to exist. One says that $\mathcal{S}$ has positive density if $\underline{\mathfrak{d}}(\mathcal{S})>0$.

In this subsection, we prove the following result.

I prove some preparatory results. Let $\pi$ be the uniformizer of $A_{\mathfrak{p}}$ at $\mathfrak{p}$. The congruence filtration on $\operatorname{GL}_{2}(A_{\mathfrak{p}})$ is defined by

where $\operatorname{M}_{2}(A_{\mathfrak{p}})$ denotes the $2\times 2$ matrices with entries in $A_{\mathfrak{p}}$. Let $\mathcal{H}$ be a closed subgroup of $\operatorname{GL}_{2}(A_{\mathfrak{p}})$, take $\mathcal{H}^{i}\mathrel{\mathop{\mathchar 58\relax}}=\mathcal{H}\cap G_{\mathfrak{p}}^{i}$, and