Elsevier

Journal of Number Theory

Volume 213, August 2020, Pages 285-318
Journal of Number Theory

General Section
On extending Artin's conjecture to composite moduli in function fields

https://doi.org/10.1016/j.jnt.2019.12.009Get rights and content

Abstract

In 1927, Artin hypothesized that for any given non-zero integer a other than 1, −1, or a perfect square, there exists infinitely many primes p for which a is a primitive root modulo p. In 1967, Hooley proved it under the assumption of the generalized Riemann hypothesis. Since then, there are many analogues and generalization of this conjecture. In this paper, we work on its generalization to composite moduli in the function fields setting. Let A=Fq[t] be the ring of polynomials over the finite field Fq and 0aA. Let C be the A-Carlitz module. Let a be a fixed element in A. For nA, C(A/nA) is a finite A-module. The set of all annihilators of C(A/nA) is an ideal and generated by a monic polynomial, denoted by λ(n). Similarly, The set of all annihilators of the submodule of C(A/nA) generated by a is an ideal and let la(n) be its monic generator. We say that a is a primitive root of n, if λ(n)=la(n). DefineNa(x):=|{nA|degn=x,nis monic,ais a primitiveroot of n}|We prove that for a given non-constant aA, aE, an exceptional set, there exists an unbounded set V of integers such thatliminfxVNa(x)/qx=0 This result is analogous to Li's theorem for Artin's conjecture on composite moduli. It is the first time that this kind of results holds in the setting of the function fields.

Introduction

Given a prime p, we can consider the set of invertible residue classes modulo p, denoted by (Z/pZ). It is well-known that under multiplication, (Z/pZ) is a cyclic group of order p1, with φ(p1) generators, where φ is the Euler totient function. A generator of (Z/pZ) is called a primitive root modulo p. For instance, if 10 is a primitive modulo a prime p if and only if 1/p has the decimal period length p1. Gauss raised the question of how often 10 is a primitive root modulo p, as p varies over primes. A precise conjecture was formulated by E. Artin in 1927. He hypothesized that for any given non-zero integer b other than 1, −1, and a prefect square, there exist infinitely many primes p for which a is a primitive modulo p. Moreover, if Pb(x) denotes the number of such primes up to x, he conjectured an asymptotic formula of the formPb(x)A(b)xlogx as x goes to infinity, where A(b) is a certain positive constant depending on b. Although the original conjecture is still open, under the assumption of the generalized Riemann hypothesis, in [6], Hooley proved both Artin's conjecture and the asymptotic formula for Pb(x).

One can extend the definition of primitive roots to general integers m. The following definition is due to Carmichael [1]. Suppose that b and m are coprime integers. If b modulo m generates a maximal cyclic subgroup of (Z/mZ), we say that b is a primitive of m. Similarly, let Mb(x) denote the number of m up to x, such that b is a primitive root of m. We can ask if there is an asymptotic formula for Mb(x) as the classical case of Artin's conjecture. It is a kind of surprise that Artin's conjecture of composite moduli is not true. In [11], Li proved

Theorem

For every integer b,liminfx1xMb(x)=0.

The function fields analogue of the multiplicative group scheme Gm over number fields the rank one Drinfeld modules. Here we consider the Carlitz module, a special case of Drinfeld modules of rank one. More precisely, let A=Fq[t],k=Fq(t) be the function fields. Let τ be the Frobenius element defined by τ(X)=Xq. We denote by k{τ} the ‘twisted polynomial ring’ whose multiplication is defined byτb=bqτ,bk. The A-Carlitz module C is the Fq-algebra homomorphismC:Ak{τ},fCf, characterized byCt=t+τ. Let B be a commutative k-algebra (or more generally, a commutative A-algebra since CT has coefficients in A) and B+ the additive group of B. We can view an element of k{τ} as an endomorphism of B+ in the following way: let uB and biτik{τ} (bik),(biτi)(u)=biuqi. Using the A-Carlitz module C, we can define a new multiplication on B as follows: for fA and uB,fu:=Cf(u)B. This gives B a new A-module structure; we denote it by C(B).

Let n be a polynomial in A. The subset{gA|Cg[C(A/MA)]=0}, the annihilator of C(A/nA), is an ideal of A. Define λ(n)A to be its monic generator. Let a be a polynomial in A. We say that a is a primitive root of n, if the cyclic A-submodule in C(A/nA) generated by a modulo n is isomorphic to A/λ(n)A. Let Qa(x) be the number of monic irreducible polynomials n of degree x such that a is a primitive root of n. In [8], Hsu proved

Theorem

For a non-constant polynomial a in A, with q2, there is a positive constant c(a) such thatQa(x)c(a)qxx as x goes to infinity.

More generalizations of Artin's conjecture on function fields can be founded in [9] and [10].

In this paper, we would like to prove that the analogue of Li's theorem in this setting. More precisely, let Na(x) be the number of monic polynomials n of degree x such that a is a primitive root of n. We have the following theorem.

Theorem 1

Let E be the union of {t,t+1} if q=2, or the empty set otherwise, and the set of all a an A such that there exists an irreducible l in A, ClX=a has solutions in A. For every non-constant polynomial a in A, aE,liminfxNa(x)qx=0.

It is the first time that this kind of results holds in the setting of the function fields. Let us consider some technical difficulties involved. The main difficulty of this problem is the formulation. For the analogue of Li's theorem in function fields, a natural formulation is to consider the following question: give a non-zero polynomial aA, what is the density that a generates the largest cyclic groups of (A/fA), fA? However, surprisingly, this problem is unreachable by the current method.

The Carlitz module seems to be unnatural at first glance. Yet, it offers striking analogies with the multiplicative group Gm of Q. Developing an idea of Carlitz [1], D. Hayes [4] constructed explicitly the maximal abelian extension of k, which is closedly analogous to the classical theory of cyclotomic extensions of Q. Later developments, due independently to Drinfeld [2] and Hayes [5], showed that Carlitz's ideas can be generalized to give an explicit class field theory for any global function field. With Hsu's success of the proof of analogue of Artin's conjecture for the Carlitz module, it suggests the possibility for the analogue of Li's theorem in the setting of Carlitz modules. It turns out to be an approachable one.

Since the Carlitz module formulation is not a straightforward analogue in function fields, it is not trivial at all to formulate the corresponding statements in this new setting. Moreover, to go through the arguments in this new setting, there are two particular difficulties. One is on the algebraic number theory in function fields. Even though some results in [8] can be used, the algebraic extensions in our setting is more general than those in [8]. We need to obtain the corresponding results which are well-known in the classical cases but not yet considered in the function field cases. The other difficulty is sieve theory. The corresponding sieve theory statements are most unknown in the function field cases. We need to prove all of whatever we need. After our work, now we have many sieve theory results in functions fields in hands. Those results can be used to extend the current results to more general setting, such as Drinfeld modules.

The proof of Theorem 1 is divided by two parts. The first part, from Section 2 to 5, is the translation of the algebraic situation in the integer case to the function field case. In this part we apply the argument in [8] to establish the analogue results for function fields, which are critical for our cases. It involves the reduction of the Artin conjecture to algebraic number theory, the computation of the degree of the field extension involved in the reduction, and the computation of a bound on the discriminant. We also prove a version of the sieve of Eratosthenes which we will need later.

The second part, from Section 6 to 8, is the main argument. We follow the line of the proof in [11]. It is a combinational argument with various sieve techniques. After that, we obtain an upper bound and then show that there is a sequence such that the limit is the upper bound is zero.

Acknowledgment

All authors would like to thank the referee for the detailed comments and helpful suggestions that improved the paper.

Notations  For nA, define |n|=qdegn and Φ(n) to be the cardinality of multiplicative group (A/nA)×. Given a finite set S, |S| denotes the cardinality of S.

Let nA be a monic polynomial and nA the ideal of A generated by n. The C(A/nA) is the A-module whose A-module structure is from the Carlitz module. For gA, let g¯ be the reduction of g modulo nA. Consider the reduction of C modulo nA, i.e., the A-module C(A/nA) given by Ct(g¯)=tg¯+g¯q. We will drop the bar if there is no confusion. For the remainder of this paper let C() be the Carlitz module homomorphism. Let l be a polynomial in A. Then ClX means the polynomial in X which applies Cl on X. For example, CtX=tX+Xq. Let n and a be two polynomials of A. Define the la(n) be the monic generator of the ideal {lA|Cl(a¯)=0¯inC(A/nA)} and λ(n) be monic generator of the ideal {lA|aA,Cl(a¯)=0¯inC(A/nA)}; i.e., λ(n)=l1(n), where 1 is the multiplicative identity 1 in Fq. We say that a is a primitive root of n if la(n)=λ(n). A prime (irreducible) l always means monic irreducible. The exceptional set E is defined as the union of {t,t+1} if q=2, or the empty set otherwise, and the set of all a an A such that there exists an irreducible l in A, ClX=a has solutions in A.

Section snippets

Computation of the degree of the field extension

For lA, let Λls be the (additive) group of the roots of the polynomial ClsX in the algebraic closure of Fq(t), which is the analogue of the ls-th roots of unity for the classical case. The Galois groups Gal(k(Λls)/k) is isomorphic to (A/(ls))× and, hence, the degree of the extension is Φ(ls) (see [12, Theorem 12.8]). Let a1/l be a root of ClXa, Kls=k(a1/l,Λls), and kls=k(Λls). We have the following diagram of fields.

The properties of the extension field Kls have been studied intensively in [8]

Reduction to algebraic number theory

Let l be a monic non-constant irreducible polynomial and let s>0 be an integer. DefinePl,s={PA|Pmonic irreducible,ls|(P1)andC(P1)/la=0} We would like deduce an algebraic condition for a prime to be in Pl,s similar to the situation in the integers.

Theorem 7

A monic irreducible p is in Pl,s if and only if p splits completely in Kl,s.

Proof

By [12, Theorem 12.10], ls|p1 if and only if p splits completely in kls. Following this, we invoke the proof of [8, Proposition 1.1]. The relevant parts of the proof for

Applying Chebotarev

Before we apply the Chebotarev density theorem, we need to estimate the discriminant divisor Δ(Kls/k).

Theorem 8

[8], Theorem 2.4

Letbe the place at infinity of k and a be a non-zero element in A. Let Gls be the Galois group of Kls/k and dls the total degree of the discriminant divisor Δ(Kls/k). Then the discriminant divisor Δ(Kls/k) divides(ls)2|Gls|[1+2degls+dega+qdega(dega+2degls)]|Gls|. Moreover, we have dls|Gls|=O(deg(ls)) as degl goes to infinity.

Proof

The theorem stated in [8] is for the case when s=1.

The sieve of Eratosthenes

In this section, we establish a sieve of Eratosthenes in function fields. This sieve will allow us to compute sums of the form deg(n)=x(n,P(z))=11 where P(z)=PPdegP<zP and P is a subset of primes.

We begin with some basic definitions and conditions. Let x be a fixed sufficiently large positive number. As usual, let A(x) be the setA(x):={nFq[t]:n monic deg(n)=x}. We choose P to be a subset of monic irreducibles of Fq[t]. For each PP, let P˜ be a subset of distinct residue classes modulo P

A combinatorial argument

Recall that for nA, C(A/nA) is a finite A-module. The set of all annihilators of C(A/nA) is an ideal and generated by a monic polynomial, denoted by λ(n). Similarly, The set of all annihilators of the submodule of C(A/nA) generated by an element aA is an ideal and let la(n) be its monic generator. By prime, we mean monic irreducible. For QA a prime and mA, define νQ(m) to be the exponent of Q in the prime decomposition of m; i.e., QνQ(m)|m and QνQ(m)+1m. Let Δ˜Q(n)={P:P|n and νQ(P1)=νQ(λ(n

A bound for Na(x)

Lemma 32

Let R be a set of primes Q such that |Q|(lnlnx)1/2. Then for any c>1 and sufficiently large integer x dependent on c aloneNa(x)cqxQR(1e2|Q|2{lnlnxln|Q|}).

Proof

We begin by showing we can apply the previous lemma.|R||{nFq[t]:deg(n)<logq(ln(lnx)1/2)}|rlogq((lnlnx)1/2)qr=qlogq((lnlnx)1/2)+1qq1qq1(lnlnx)1/2ln(x)1/4 for sufficiently large x.Na(x)qx+BR(1)|B||{nA:n monic, deg(n)=x,νQ(la(n))=νQ(λ(n)),H1(Q),H2(Q),QB}| From the above we see that the size of the

An unbounded set

In the following we will employ Dirichlet's theorem to construct an unbounded set on which Na(x)/qx has the limit 0. Firstly we need an auxiliary lemma.

Lemma 34

If u>5 and T1, and ln(2T)lnu<ln22lnu, then122<1u1{lnTlnu}<12, where {x}=xx and x=min{xx,xx}.

Proof

Let ϵ=ln(2T)lnu. There is an integer v such thatln(2T)lnu=v±ϵ. It impliesv±ϵ=ln2lnu+lnTlnulnTlnu=v(ln2lnuϵ). By our assumption, 1>ln2lnuϵ>0, therefore{lnTlnu}=1ln2lnuϵ132ln2lnu<{lnTlnu}<112ln2lnu

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The research of the third author was supported by an NSERC discovery grant RGPIN-2015-03709.

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