General SectionOn extending Artin's conjecture to composite moduli in function fields
Introduction
Given a prime p, we can consider the set of invertible residue classes modulo p, denoted by . It is well-known that under multiplication, is a cyclic group of order , with generators, where φ is the Euler totient function. A generator of is called a primitive root modulo p. For instance, if 10 is a primitive modulo a prime p if and only if has the decimal period length . 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 denotes the number of such primes up to x, he conjectured an asymptotic formula of the form as x goes to infinity, where 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 .
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 , we say that b is a primitive of m. Similarly, let 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 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,
The function fields analogue of the multiplicative group scheme 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 be the function fields. Let τ be the Frobenius element defined by . We denote by the ‘twisted polynomial ring’ whose multiplication is defined by The A-Carlitz module C is the -algebra homomorphism characterized by Let B be a commutative k-algebra (or more generally, a commutative A-algebra since has coefficients in A) and the additive group of B. We can view an element of as an endomorphism of in the following way: let and (), Using the A-Carlitz module C, we can define a new multiplication on B as follows: for and , This gives B a new A-module structure; we denote it by .
Let n be a polynomial in A. The subset the annihilator of , is an ideal of A. Define 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 generated by a modulo n is isomorphic to . Let 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 , there is a positive constant such that 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 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 be the union of if , or the empty set otherwise, and the set of all a an A such that there exists an irreducible l in A, has solutions in A. For every non-constant polynomial a in A, ,
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 , what is the density that a generates the largest cyclic groups of , ? 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 of . 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 . 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 , define and to be the cardinality of multiplicative group . Given a finite set S, denotes the cardinality of S.
Let be a monic polynomial and nA the ideal of A generated by n. The is the A-module whose A-module structure is from the Carlitz module. For , let be the reduction of g modulo nA. Consider the reduction of C modulo nA, i.e., the A-module given by . We will drop the bar if there is no confusion. For the remainder of this paper let be the Carlitz module homomorphism. Let l be a polynomial in A. Then means the polynomial in X which applies on X. For example, . Let n and a be two polynomials of A. Define the be the monic generator of the ideal and be monic generator of the ideal ; i.e., , where 1 is the multiplicative identity 1 in . We say that a is a primitive root of n if . A prime (irreducible) l always means monic irreducible. The exceptional set is defined as the union of if , or the empty set otherwise, and the set of all a an A such that there exists an irreducible l in A, has solutions in A.
Section snippets
Computation of the degree of the field extension
For , let be the (additive) group of the roots of the polynomial in the algebraic closure of , which is the analogue of the -th roots of unity for the classical case. The Galois groups is isomorphic to and, hence, the degree of the extension is (see [12, Theorem 12.8]). Let be a root of , , and . We have the following diagram of fields.
The properties of the extension field have been studied intensively in [8]
Reduction to algebraic number theory
Let l be a monic non-constant irreducible polynomial and let be an integer. Define We would like deduce an algebraic condition for a prime to be in similar to the situation in the integers.
Theorem 7 A monic irreducible p is in if and only if p splits completely in . Proof By [12, Theorem 12.10], if and only if p splits completely in . 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 .
Theorem 8 Let ∞ be the place at infinity of k and a be a non-zero element in A. Let be the Galois group of and the total degree of the discriminant divisor . Then the discriminant divisor divides Moreover, we have as goes to infinity. Proof The theorem stated in [8] is for the case when .[8], Theorem 2.4
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 where and 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 be the set We choose to be a subset of monic irreducibles of . For each , let be a subset of distinct residue classes modulo P
A combinatorial argument
Recall that for , is a finite A-module. The set of all annihilators of is an ideal and generated by a monic polynomial, denoted by . Similarly, The set of all annihilators of the submodule of generated by an element is an ideal and let be its monic generator. By prime, we mean monic irreducible. For a prime and , define to be the exponent of Q in the prime decomposition of m; i.e., and . Let
A bound for
Lemma 32 Let be a set of primes Q such that . Then for any and sufficiently large integer x dependent on c alone Proof We begin by showing we can apply the previous lemma. for sufficiently large x. 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 has the limit 0. Firstly we need an auxiliary lemma.
Lemma 34 If and , and , then where and . Proof Let . There is an integer v such that It implies By our assumption, , therefore
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The research of the third author was supported by an NSERC discovery grant RGPIN-2015-03709.