The growth of the discriminant of the endomorphism ring of the reduction of a rank 2 generic Drinfeld module

Abstract

For q an odd prime power, $A={\mathbb{F}}_q[T]$, and $F={\mathbb{F}}_q(T)$, let $\psi :A\to F\{\tau \}$ be a Drinfeld A-module over F of rank 2 and without complex multiplication, and let $\mathfrak{p}=pA$ be a prime of A of good reduction for ψ, with residue field ${\mathbb{F}}_{\mathfrak{p}}$. We study the growth of the absolute value $|{\mathrm{\Delta }}_{\mathfrak{p}}|$ of the discriminant of the ${\mathbb{F}}_{\mathfrak{p}}$-endomorphism ring of the reduction of ψ modulo $\mathfrak{p}$ and prove that, for all $\mathfrak{p}$, $|{\mathrm{\Delta }}_{\mathfrak{p}}|$ grows with $|p|$. Moreover, we prove that, for a density 1 of primes $\mathfrak{p}$, $|{\mathrm{\Delta }}_{\mathfrak{p}}|$ is as close as possible to its upper bound $|a_{\mathfrak{p}}^2-4{\mu }_{\mathfrak{p}}p|$, where $X^2+a_{\mathfrak{p}}X+{\mu }_{\mathfrak{p}}p\in A[X]$ is the characteristic polynomial of ${\tau }^{\mathrm{deg}p}$.

Introduction

Let ${\mathbb{F}}_q$ be a finite field with q elements, $A:={\mathbb{F}}_q[T]$ be the ring of polynomials in T over ${\mathbb{F}}_q$, $F:={\mathbb{F}}_q(T)$ be the field of fractions of A, and $F^{\mathrm{alg}}$ be a fixed algebraic closure of F. We call a nonzero prime ideal of A simply a prime of A. The main results of this paper concern the reductions modulo primes of A of a fixed Drinfeld module over F. To state these results, we first recall some basic concepts from the theory of Drinfeld modules.

An A-field is a field L equipped with a homomorphism $\gamma :A\to L$. Two A-fields of particular prominence in this paper are F and ${\mathbb{F}}_{\mathfrak{p}}:=A/\mathfrak{p}$, where $\mathfrak{p}◁A$ is a prime. When L is an extension of F, we implicitly assume that $\gamma :A\hookrightarrow L$ is obtained from the natural embedding of A into its field of fractions $A\hookrightarrow F\hookrightarrow L$; when L is a finite extension of ${\mathbb{F}}_{\mathfrak{p}}$, we implicitly assume that $\gamma :A\to A/\mathfrak{p}\hookrightarrow L$ is obtained from the natural quotient map.

For an A-field L, denote by $L\{\tau \}$ the noncommutative ring of polynomials in τ with coefficients in L and subject to the commutation rule $\tau c=c^q\tau$, $c\in L$. A Drinfeld A-module of rank $r\ge 1$ defined over L is a ring homomorphism $\psi :A\to L\{\tau \}$, $a\mapsto {\psi }_a$, uniquely determined by the image of T:

$${\psi }_T=\gamma (T)+\sum _{1\le i\le r}{g}_i(T){\tau }^i,g_r(T)\ne 0.$$

The endomorphism ring of ψ is the centralizer of the image $\psi (A)$ of A in $L\{\tau \}$:

$${\mathrm{End}}_L(\psi ):=\{u\in L\{\tau \}:u{\psi }_a={\psi }_{a}u\text{ for all }a\in A\}=\{u\in L\{\tau \}:u{\psi }_T={\psi }_{T}u\}.$$

As ${\mathrm{End}}_L(\psi )$ contains $\psi (A)\cong A$ in its center, it is an A-algebra. It can be shown that ${\mathrm{End}}_L(\psi )$ is a free A-module of rank $\le r^2$; see [Dr74, Sec. 2].

Now let $\psi :A\to F\{\tau \}$ be a Drinfeld A-module of rank r over F defined by

$${\psi }_T=T+g_1\tau +\dots +g_r{\tau }^r.$$

We say that a prime $\mathfrak{p}◁A$ is a prime of good reduction for ψ if ${\mathrm{ord}}_{\mathfrak{p}}(g_i)\ge 0$ for all $1\le i\le r-1$ and if ${\mathrm{ord}}_{\mathfrak{p}}(g_r)=0$. If that is the case, we view $g_1,\dots ,g_r$ as elements of the completion $A_{\mathfrak{p}}$ of A at $\mathfrak{p}$ and define the reduction of ψ at $\mathfrak{p}$ as the Drinfeld A-module $\psi \otimes {\mathbb{F}}_{\mathfrak{p}}:A\to {\mathbb{F}}_{\mathfrak{p}}\{\tau \}$ given by

$${(\psi \otimes {\mathbb{F}}_{\mathfrak{p}})}_T=\bar{T}+\overline{g_1}\tau +\cdots +\overline{g_r}{\tau }^r,$$

where $\bar{g}$ is the image of $g\in A_{\mathfrak{p}}$ under the canonical homomorphism $A_{\mathfrak{p}}\to A_{\mathfrak{p}}/\mathfrak{p}$. Note that $\psi \otimes {\mathbb{F}}_{\mathfrak{p}}$ has rank r since $\overline{g_r}\ne 0$. It is clear that all but finitely many primes of A are primes of good reduction for ψ; we denote the set of these primes by $\mathcal{P}(\psi )$.

Let $\mathfrak{p}=pA$ be a prime of good reduction of ψ, where p denotes the monic generator of $\mathfrak{p}$. Denote by ${\mathcal{E}}_{\psi ,\mathfrak{p}}={\mathrm{End}}_{{\mathbb{F}}_{\mathfrak{p}}}(\psi \otimes {\mathbb{F}}_{\mathfrak{p}})$ the endomorphism ring of $\psi \otimes {\mathbb{F}}_{\mathfrak{p}}$. It is easy to see that ${\pi }_{\mathfrak{p}}:={\tau }^{\mathrm{deg}p}$ is in the center of ${\mathbb{F}}_{\mathfrak{p}}\{\tau \}$, hence ${\pi }_{\mathfrak{p}}\in {\mathcal{E}}_{\psi ,\mathfrak{p}}$. Using the theory of Drinfeld modules over finite fields, it is easy to show that $A[{\pi }_{\mathfrak{p}}]$ and ${\mathcal{E}}_{\psi ,\mathfrak{p}}$ are A-orders in the imaginary field extension $F({\pi }_{\mathfrak{p}})$ of F of degree r (“imaginary” means that there is a unique place of $F({\pi }_{\mathfrak{p}})$ over the place $\infty :=1/T$ of F); see [GaPa19, Prop. 2.1]. Here we remind the reader that $A[{\pi }_{\mathfrak{p}}]=\psi (A)[{\pi }_{\mathfrak{p}}]$, i.e. that $A\cong \psi (A)$ also denotes the image of A under ψ. Then, denoting by ${\mathcal{O}}_{F({\pi }_{\mathfrak{p}})}$ the integral closure of A in $F({\pi }_{\mathfrak{p}})$, we obtain a natural inclusion of A-orders

$$A[{\pi }_{\mathfrak{p}}]\subseteq {\mathcal{E}}_{\psi ,\mathfrak{p}}\subseteq {\mathcal{O}}_{F({\pi }_{\mathfrak{p}})}.$$

It is an interesting problem, with important applications to the arithmetic of F, to compare the above orders as $\mathfrak{p}$ varies. For example, it is proved in [CoPa15, Thm. 1] (for $r=2$) and in [GaPa19, Thm. 1.2], [GaPa20, Thm. 1.1] (for $r\ge 2$) that the quotient ${\mathcal{E}}_{\psi ,\mathfrak{p}}/A[{\pi }_{\mathfrak{p}}]$ captures the splitting behavior of $\mathfrak{p}$ in the division fields of ψ; this result then leads to non-abelian reciprocity laws in function field arithmetic. Also, in [GaPa19, Thm. 1.1], it is shown that the quotients ${\mathcal{E}}_{\psi ,\mathfrak{p}}/A[{\pi }_{\mathfrak{p}}]$ and ${\mathcal{O}}_{F({\pi }_{\mathfrak{p}})}/{\mathcal{E}}_{\psi ,\mathfrak{p}}$ can be arbitrarily large as $\mathfrak{p}$ varies, whereas in [CoPa15, Thm. 6] an explicit formula is given for the density of primes for which $A[{\pi }_{\mathfrak{p}}]={\mathcal{E}}_{\psi ,\mathfrak{p}}$.

In this paper, we are interested in the growth of the discriminant of ${\mathcal{E}}_{\psi ,\mathfrak{p}}$ as $\mathfrak{p}$ varies, and in applications of this growth to the arithmetic of ψ. Our results assume that q is odd and $r=2$. These assumptions are to be kept from here on without further notice.

Choosing a basis ${\mathcal{E}}_{\psi ,\mathfrak{p}}=A{\alpha }_1+A{\alpha }_2$ of ${\mathcal{E}}_{\psi ,\mathfrak{p}}$ as a free A-module of rank 2, the discriminant ${\mathrm{\Delta }}_{\mathfrak{p}}:={\mathrm{\Delta }}_{\psi ,\mathfrak{p}}$ of ${\mathcal{E}}_{\psi ,\mathfrak{p}}$ is $\det {({\mathrm{Tr}}_{F({\pi }_{\mathfrak{p}})/F}({\alpha }_i{\alpha }_j))}_{1\le i,j\le 2}$, which is well-defined up to a multiple by a square in ${\mathbb{F}}_q^{\times }$. If we write ${\mathrm{\Delta }}_{\mathfrak{p}}=c_{\mathfrak{p}}^2\cdot {\mathrm{\Delta }}_{F({\pi }_{\mathfrak{p}})}$, where $c_{\mathfrak{p}},{\mathrm{\Delta }}_{F({\pi }_{\mathfrak{p}})}\in A$ with ${\mathrm{\Delta }}_{F({\pi }_{\mathfrak{p}})}$ square-free, then ${\mathcal{E}}_{\psi ,\mathfrak{p}}=A+c_{\mathfrak{p}}{\mathcal{O}}_{F({\pi }_{\mathfrak{p}})}$, ${\mathrm{\Delta }}_{F({\pi }_{\mathfrak{p}})}$ is the discriminant of ${\mathcal{O}}_{F({\pi }_{\mathfrak{p}})}$, and ${\mathcal{O}}_{F({\pi }_{\mathfrak{p}})}/{\mathcal{E}}_{\psi ,\mathfrak{p}}\cong A/c_{\mathfrak{p}}A$ as A-modules. Note that, similarly to ${\mathrm{\Delta }}_{\mathfrak{p}}$, the polynomial $c_{\mathfrak{p}}$ also depends on ψ, although this will not be explicitly indicated in our notation.

Denote by $|\cdot |=|\cdot {|}_{\infty }$ the absolute value on F corresponding to $1/T$, normalized so that $|a|=q^{\mathrm{deg}a}$ for $a\in A$, with deg a denoting the degree of a as a polynomial in T and subject to the convention that $\mathrm{deg}0=-\infty$. The first main result of this paper is the following:

Theorem 1

Assume ${\mathrm{End}}_{F^{\mathrm{alg}}}(\psi )=A$. Then

$$|{\mathit{\Delta }}_{\mathfrak{p}}|{\gg }_{\psi }\frac{\log |p|}{{(\log \log |p|)}^2},$$

where the implied ${\gg }_{\psi }$-constant depends on q and on the coefficients of the polynomial ${\psi }_T\in F\{\tau \}$.

By considering τ as the Frobenius automorphism of ${\mathbb{F}}_{\mathfrak{p}}$ relative to ${\mathbb{F}}_q$, that is, as the map $\alpha \mapsto {\alpha }^q$, we can also consider ${\mathbb{F}}_{\mathfrak{p}}$ as an A-module via $\psi \otimes {\mathbb{F}}_{\mathfrak{p}}$ (hence T acts on ${\mathbb{F}}_{\mathfrak{p}}$ as ${(\psi \otimes {\mathbb{F}}_{\mathfrak{p}})}_T$). This module will be denoted ${}^{\psi }\mathbb{F}_{\mathfrak{p}}$. The properties of the torsion elements of this module lead to an A-module isomorphism

$${}^{\psi }\mathbb{F}_{\mathfrak{p}}\cong A/d_{1,\mathfrak{p}}A\times A/d_{2,\mathfrak{p}}A$$

for uniquely determined nonzero monic polynomials $d_{1,\mathfrak{p}},d_{2,\mathfrak{p}}\in A$ such that $d_{1,\mathfrak{p}}|d_{2,\mathfrak{p}}$; as above, the polynomials $d_{1,\mathfrak{p}},d_{2,\mathfrak{p}}$ also depend on ψ, although this will not be explicitly indicated in our notation.

Note that $d_{2,\mathfrak{p}}$ may be regarded as the exponent of the A-module ${}^{\psi }\mathbb{F}_{\mathfrak{p}}$, and that $|d_{1,\mathfrak{p}}\cdot d_{2,\mathfrak{p}}|=|p|$; thus we have the trivial lower bound $|d_{2,\mathfrak{p}}|\ge |p{|}^{\frac{1}{2}}$. Theorem 1 allows us to deduce a stronger lower bound on $|d_{2,\mathfrak{p}}|$.

Theorem 2

Assume ${\mathrm{End}}_{F^{\mathrm{alg}}}(\psi )=A$. Then

$$|d_{2,\mathfrak{p}}|{\gg }_{\psi }|p{|}^{\frac{1}{2}}\frac{{(\log |p|)}^{\frac{1}{2}}}{\log \log |p|},$$

where the implied ${\gg }_{\psi }$-constant depends on q and on the coefficients of the polynomial ${\psi }_T\in F\{\tau \}$.

Theorem 1, Theorem 2 are the Drinfeld module analogues of results by Schoof for elliptic curves over $\mathbb{Q}$; see the statement and proof of the main result of [Sc91]. Our proof of Theorem 1 is inspired by Schoof's arguments. It relies on Drinfeld's now-classical function field analogue of the analytic theory of elliptic curves [Dr74], on the growth properties of the function field counterpart of the j-function, proved by Gekeler [Ge99], and on the more recent Drinfeld module analogue of Deuring's lifting lemma, proved in an earlier paper by the present authors [CoPa15].

Remark 3

According to Theorem 1, $|{\mathrm{\Delta }}_{\mathfrak{p}}|$ grows with deg p. In relation to the growth of $|{\mathrm{\Delta }}_{\mathfrak{p}}|$, Theorem 1.1 of [GaPa20] implies that, for any fixed number $\kappa >0$, we can find $\mathfrak{p}$ such that $|c_{\mathfrak{p}}|>\kappa$; therefore, for such $\mathfrak{p}$, $|{\mathrm{\Delta }}_{\mathfrak{p}}|=|c_{\mathfrak{p}}{|}^2\cdot |{\mathrm{\Delta }}_{F({\pi }_{\mathfrak{p}})}|>\kappa$. However, Theorem 1.1 of [GaPa20] does not imply that $|{\mathrm{\Delta }}_{\mathfrak{p}}|$ has to grow with deg p. In fact, computationally, Garai and the second author have found that it happens that $c_{\mathfrak{p}}=1$; in this case, their aforementioned theorem does not give any lower bound on $|{\mathrm{\Delta }}_{\mathfrak{p}}|$. On the other hand, Theorem 1 of the present paper does not imply that we can find some $\mathfrak{p}$ such that $|c_{\mathfrak{p}}|>\kappa$. Thus these two results are complementary to each other.

Remark 4

If ${\mathrm{End}}_{F^{\mathrm{alg}}}(\psi )\ne A$, then ${\mathrm{End}}_{F^{\mathrm{alg}}}(\psi )=\mathcal{O}$ is an order in an imaginary quadratic extension K of F, in which case the growth of $|{\mathrm{\Delta }}_{\mathfrak{p}}|$ is vastly different from that shown in Theorem 1. On one hand, if $\mathfrak{p}\in \mathcal{P}(\psi )$ splits in K, then

$$\mathcal{O}\subseteq {\mathrm{End}}_{{\mathbb{F}}_{\mathfrak{p}}}(\psi \otimes {\mathbb{F}}_{\mathfrak{p}})\subseteq {\mathcal{O}}_K$$

(see [Ge83, Lem. 3.3]), which implies that $|{\mathrm{\Delta }}_{\mathfrak{p}}|\le |{\mathrm{\Delta }}_{\mathcal{O}}|$, where ${\mathrm{\Delta }}_{\mathcal{O}}$ is the discriminant of $\mathcal{O}$. Hence $|{\mathrm{\Delta }}_{\mathfrak{p}}|$ remains bounded as $\mathfrak{p}$ varies over the primes that split in K. In particular, Theorem 1 is false without its assumption. On the other hand, if $\mathfrak{p}\in \mathcal{P}(\psi )$ is inert in K, then $\psi \otimes {\mathbb{F}}_{\mathfrak{p}}$ is supersingular, which implies that $A[{\pi }_{\mathfrak{p}}]=A[\sqrt{\alpha p}]$ for some $\alpha \in {\mathbb{F}}_q^{\times }$, and that $A[{\pi }_{\mathfrak{p}}]={\mathcal{E}}_{\mathfrak{p}}={\mathcal{O}}_{F({\pi }_{\mathfrak{p}})}$ (see Lemma 5.2 and Theorem 5.3 of [Ge83]). Hence $|{\mathrm{\Delta }}_{\mathfrak{p}}|=|p|$, a much larger growth than that shown in Theorem 1. One can also prove that, in the supersingular case, $d_{1,\mathfrak{p}}=1$ and $d_{2,\mathfrak{p}}=p-\beta$ for some $\beta \in {\mathbb{F}}_q^{\times }$ (see [CoPa15, Cor. 3]). Hence $|d_{2,\mathfrak{p}}|=|p|$, which is as large as possible.

Remark 5

From the theory of Drinfeld modules over finite fields, one can deduce that the discriminant of $A[{\pi }_{\mathfrak{p}}]$ has degree $\le \mathrm{deg}p$. More precisely, the characteristic polynomial of ${\pi }_{\mathfrak{p}}$ is of the form $X^2+a_{\mathfrak{p}}X+{\mu }_{\mathfrak{p}}p\in A[X]$, where ${\mu }_{\mathfrak{p}}\in {\mathbb{F}}_q^{\times }$ and $\mathrm{deg}{a}_{\mathfrak{p}}\le \frac{\mathrm{deg}p}{2}$. This implies that $|{\mathrm{\Delta }}_{\mathfrak{p}}|\le |a_{\mathfrak{p}}^2-4{\mu }_{\mathfrak{p}}p|\le |p|$. Note that the coefficients $a_{\mathfrak{p}}$ and ${\mu }_{\mathfrak{p}}$ also depend on ψ, although this will not be explicitly indicated in our notation.

The lower bound on $|{\mathrm{\Delta }}_{\mathfrak{p}}|$ in Theorem 1 holds for all primes $\mathfrak{p}=pA◁A$, with finitely many exceptions. The next theorem gives a stronger lower bound, almost as close as the upper bound of Remark 5, which holds for a set of primes $\mathfrak{p}=pA◁A$ of Dirichlet density 1:

Theorem 6

Assume ${\mathrm{End}}_{F^{\mathrm{alg}}}(\psi )=A$. For any positive valued function $f:\mathbb{N}\to (0,\infty )$ with $\underset{x\to \infty }{\lim }f(x)=\infty$, we have that, as $x\to \infty$,

$$\mathrm{\#}\{\mathfrak{p}\in \mathcal{P}(\psi ):\mathrm{deg}p=x,|{\mathit{\Delta }}_{\mathfrak{p}}|>\frac{|a_{\mathfrak{p}}^2-4{\mu }_{\mathfrak{p}}p|}{q^{f(\mathrm{deg}p)}}\}\sim {\pi }_F(x),$$

where ${\pi }_F(x):=\mathrm{\#}\{\mathfrak{p}◁A:\mathrm{deg}p=x\}$. Moreover, the Dirichlet density of the set

$$\{\mathfrak{p}\in \mathcal{P}(\psi ):|{\mathit{\Delta }}_{\mathfrak{p}}|>\frac{|a_{\mathfrak{p}}^2-4{\mu }_{\mathfrak{p}}p|}{q^{f(\mathrm{deg}p)}}\}$$

exists and equals 1.

Theorem 6 is a Drinfeld module unconditional analogue of a recent result of the first author and Fitzpatrick for elliptic curves over $\mathbb{Q}$; see [CoFi21]. It is inspired by the results of [CoSh15] and relies on some of the main results of [CoPa15].

In Sections 2, 3, and 4 we review and prove several results about orders, quadratic forms, and j-invariants of Drinfeld modules needed in the proofs of the main theorems. In Sections 5 and 6 we present the proofs of Theorem 1, Theorem 2, Theorem 6.

Notation. Throughout the paper, we use the standard ∼, o, O, ≪, ≫ notation, which we now recall: given suitably defined real functions $h_1,h_2$, we say that $h_1\sim h_2$ if $\underset{x\to \infty }{\lim }{h}_1(x)/h_2(x)=1$; we say that $h_1=\mathrm{o}(h_2)$ if $\underset{x\to \infty }{\lim }{h}_1(x)/h_2(x)=0$; we say that $h_1=\mathrm{O}(h_2)$ or $h_1\ll h_2$ or $h_2\gg h_1$ if $h_2$ is positive valued and there exists a positive constant C such that $|h_1(x)|\le C{h}_2(x)$ for all x in the domain of $h_1$; we say that $h_1={\mathrm{O}}_D(h_2)$ or $h_1{\ll }_D{h}_2$ or $h_2{\gg }_D{h}_1$ if $h_1\ll h_2$ and the implied O-constant C depends on priorly given data D. We make the convention that any implied O-constant may depend on q without any explicit specification.