Conductor in Algebraic Number Theory

• Conductor is the largest common ideal between an order and its integral closure, pinpointing where non-integral behavior arises.
• In orders with prime conductor, the exceptional factorization phenomena are localized at one prime, allowing precise elasticity classification via the Davenport constant.
• The conductor concept bridges ideal theory and factorization, offering computational insights into non-maximal orders and class group structure.

Searching arXiv for the specified paper and closely related work on conductor ideals and arithmetic conductors. arxiv_search(query="id:(Kettinger et al., 24 Apr 2025) OR ti:\"Elasticity of Orders with Prime Conductor\" OR all:\"prime conductor order number field elasticity\" ", max_results=5) In algebraic number theory, the conductor of an order $R$ in a number field is the ideal

$$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$

where $\overline{R}$ denotes the integral closure of $R$, which for orders is the full ring of integers $\mathcal O_K$. It is the largest ideal of $\overline{R}$ contained in $R$, and therefore measures precisely where $R$ fails to be integrally closed. In the setting of orders with prime conductor, the non-Dedekind behavior is concentrated at a single prime ideal $P$, and this concentration permits a complete description of the order’s elasticity in terms of its class group (Kettinger et al., 24 Apr 2025).

1. Definition in the setting of orders

Let $K$ be a number field with ring of integers $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$0. An order in $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$1 is a subring $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$2 such that $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$3, $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$4, and $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$5 is a finitely generated $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$6-module. Its integral closure in $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$7 is

$$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$8

and for an order one has $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$9 (Kettinger et al., 24 Apr 2025).

The conductor is defined by

$\overline{R}$0

Equivalently, it is the largest ideal of $\overline{R}$1 contained in $\overline{R}$2, hence the largest common ideal of $\overline{R}$3 and $\overline{R}$4 (Kettinger et al., 24 Apr 2025). In this sense, the conductor is the ideal-theoretic locus of non-normality.

The paper "Elasticity of Orders with Prime Conductor" studies the case in which this conductor $\overline{R}$5 is prime in $\overline{R}$6. Formally, $\overline{R}$7 is prime when $\overline{R}$8 is an integral domain, equivalently when $\overline{R}$9 implies $R$0 or $R$1 for $R$2 (Kettinger et al., 24 Apr 2025). In that situation, one speaks of an order with prime conductor.

A broader algebraic use of the same term appears in commutative algebra: for a local ring $R$3 with normalization $R$4, the conductor ideal is likewise

$R$5

and it defines the non-normal locus up to radical (Asgharzadeh, 2024). This suggests a common conceptual role: the conductor encodes the discrepancy between a ring and its normalization.

2. Arithmetic role of the conductor

For orders in number fields, the conductor separates prime ideals into two qualitatively different classes. Ideals relatively prime to the conductor behave in a Dedekind-like manner: they are invertible, and they factor uniquely into prime ideals relatively prime to the conductor (Kettinger et al., 24 Apr 2025). More precisely, if $R$6 is an $R$7-ideal relatively prime to the conductor $R$8, then $R$9 is invertible, and every such ideal admits unique factorization into prime $\mathcal O_K$0-ideals relatively prime to $\mathcal O_K$1; moreover, all but finitely many prime ideals are of this kind (Kettinger et al., 24 Apr 2025).

By contrast, prime ideals dividing the conductor are exactly where non-Dedekind phenomena occur: non-invertibility, failure of unique ideal factorization, and subtle behavior in element factorization (Kettinger et al., 24 Apr 2025). The conductor therefore localizes the obstruction to Dedekind ideal theory.

The prime-conductor hypothesis is especially restrictive. When $\mathcal O_K$2 itself is prime, all exceptional behavior is concentrated at a single prime. This implies that principal ideals $\mathcal O_K$3 factor as powers of $\mathcal O_K$4 times products of primes coprime to $\mathcal O_K$5, and the contribution of the exceptional prime can be controlled uniformly (Kettinger et al., 24 Apr 2025). A central structural consequence is Theorem 2.6 of the paper: if $\mathcal O_K$6 has prime conductor $\mathcal O_K$7 and $\mathcal O_K$8, then the principal ideal $\mathcal O_K$9 factors into at most $\overline{R}$0 prime ideals of $\overline{R}$1, where $\overline{R}$2 is the Davenport constant of the class group (Kettinger et al., 24 Apr 2025).

This suggests that the conductor is not merely a local defect invariant. In the prime-conductor regime it acts as the mechanism that makes the transfer from ideal-theoretic control to factorization-theoretic control possible.

3. Conductor, factorization, and elasticity

The ambient ring $\overline{R}$3 is assumed atomic: every nonzero nonunit factors into irreducibles. For $\overline{R}$4, the set of lengths

$\overline{R}$5

records all possible irreducible factorization lengths (Kettinger et al., 24 Apr 2025). The elasticity of $\overline{R}$6 is

$\overline{R}$7

and the elasticity of the domain is

$\overline{R}$8

The domain is half-factorial if and only if $\overline{R}$9 (Kettinger et al., 24 Apr 2025).

The conductor intervenes in two ways. First, it determines which prime factors are “good,” namely those coprime to the conductor, and these can often be removed without decreasing elasticity. Lemma 2.5 states that if $R$0 has $R$1 and $R$2 with $R$3 a prime element relatively prime to the conductor, then $R$4 (Kettinger et al., 24 Apr 2025). Hence extremal elasticity can be studied after stripping away such prime factors.

Second, once the conductor is prime, every irreducible has a principal ideal whose prime-ideal factorization is uniformly bounded by the Davenport constant (Kettinger et al., 24 Apr 2025). That bound translates into upper bounds on factorization lengths of arbitrary elements.

The lower bound is conductor-independent: for any order $R$5,

$R$6

as stated in Porism 2.4 (Kettinger et al., 24 Apr 2025). The proof uses minimal zero-sum sequences in the class group and prime ideals coprime to the conductor, producing an element with factorizations of lengths $R$7 and $R$8 (Kettinger et al., 24 Apr 2025).

4. Class group control and the exact elasticity formula

The decisive arithmetic invariant is the class group $R$9, a finite abelian group that records failure of principal generation at the ideal level (Kettinger et al., 24 Apr 2025). Every ideal class contains infinitely many prime $R$0-ideals (Kettinger et al., 24 Apr 2025), so minimal zero-sum relations in $R$1 can be realized by prime ideals. The relevant combinatorial invariant is the Davenport constant $R$2 of a finite abelian group $R$3, equivalently the maximal length of a zero-sum sequence with no proper zero-sum subsequence (Kettinger et al., 24 Apr 2025).

For rings of integers, Narkiewicz proved that if $R$4 is not a UFD, then

$R$5

(Kettinger et al., 24 Apr 2025). The prime-conductor paper extends this mechanism from maximal orders to non-maximal orders with one bad prime.

Its main theorem gives a complete classification. Let $R$6 be an order with prime conductor $R$7, and put $R$8. Then (Kettinger et al., 24 Apr 2025):

1. If $R$9 is principal in $P$0 and the equivalent conditions of Lemma 3.1 hold, then

$P$1

2. Otherwise,

$P$2

The exceptional case occurs only when the prime conductor is principal, say $P$3, and there exists a special element in the integral closure whose principal ideal factors into exactly $P$4 prime ideals and has no nonunit divisor from $P$5; equivalently, there exists an ideal class participating in a minimal zero-sum sequence of length $P$6 with a corresponding principality property after extension to $P$7 (Kettinger et al., 24 Apr 2025). When this occurs, the conductor contributes an extra $P$8 to the elasticity.

Two corollaries are explicitly singled out. If $P$9 is trivial, then the exceptional case occurs and $K$0 (Kettinger et al., 24 Apr 2025). If $K$1, then the non-exceptional case holds and $K$2 (Kettinger et al., 24 Apr 2025).

5. Comparison with other conductor notions

The conductor of an order belongs to a wider family of conductor constructions. In the classical commutative-algebraic setting, for a ring extension $K$3, the conductor is

$K$4

again the largest common ideal of $K$5 and $K$6 (Gasanova et al., 2024). The number-theoretic conductor $K$7 is a direct instance of this construction.

A distinct but related notion appears in the paper "On the Rees algebra and the conductor of an ideal" (Gasanova et al., 2024). For an ideal $K$8, with $K$9 the defining ideal of the Rees algebra and $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$00 the ideal of linear relations in a polynomial presentation, the conductor of $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$01 is

$$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$02

where $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$03 is the canonical map (Gasanova et al., 2024). There, the conductor measures the failure of linear type, rather than the failure of normality of a ring.

Another arithmetic use appears for curves. For Picard curves over $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$04, the conductor is the Artin conductor of the $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$05-Galois representation, equivalently the conductor of the Jacobian, and is expressed as

$$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$06

with $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$07 the local conductor exponent determined from stable reduction (Bouw et al., 2019). This is a different invariant from the conductor ideal of an order, though both encode bad arithmetic at finitely many primes.

These parallel usages indicate that “conductor” is consistently an interface invariant. In each case it marks where an object ceases to behave like its regularized or ambient counterpart: an order versus its normalization, a symmetric algebra versus a Rees algebra, or a curve versus good reduction.

6. Arithmetic consequences and computational significance

The prime-conductor formula has an immediate computational interpretation. Since

$$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$08

knowledge of elasticity constrains the Davenport constant, and hence the structure of the class group (Kettinger et al., 24 Apr 2025). Conversely, once $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$09 and the exceptional principal-conductor condition are known, the elasticity is completely determined (Kettinger et al., 24 Apr 2025).

The paper explicitly positions this as an application to class-group computation for certain orders (Kettinger et al., 24 Apr 2025). It also situates itself among work using factorization invariants to study class groups of orders, including work by Choi, Kettinger, Rago, Halter-Koch, and Moles, especially in quadratic settings (Kettinger et al., 24 Apr 2025).

A plausible implication is that the conductor, when prime, functions as an arithmetic simplifier. Because it compresses all non-integrally-closed behavior into one prime ideal, factorization invariants become sufficiently rigid to reflect the class group almost exactly. This is the mechanism behind the passage from abstract zero-sum combinatorics in $$P=(R:\overline{R})=\{x\in \overline{R}:x\,\overline{R}\subseteq R\},$$10 to exact formulas for elasticity.

The conductor therefore occupies a central place in the arithmetic of non-maximal orders. It is simultaneously the defect of integral closure, the boundary between Dedekind-like and exceptional ideal theory, and—in the prime case—the structural device that makes a complete factorization-theoretic classification possible (Kettinger et al., 24 Apr 2025).