Artin Conductor Explicit Computation

Introduction: Delving into Artin Conductors

In the realm of number theory, particularly within the fascinating subfields of representation theory, algebraic number theory, and Galois representations, the concept of Artin conductors emerges as a pivotal tool. These conductors, intricate invariants associated with Galois representations, provide deep insights into the ramification behavior of extensions of number fields. For anyone venturing into advanced number theory, understanding Artin conductors is essential. This article will delve into the explicit computation of these conductors, focusing on a specific example to illustrate the process clearly and comprehensively. The focus of this exploration is to elucidate the underlying principles and techniques involved in calculating Artin conductors, making it accessible to both students and researchers alike. We embark on this journey by first establishing the foundational concepts before diving into the computational aspects, ensuring a solid understanding of the theoretical underpinnings that govern these fascinating objects.

Foundational Concepts: Setting the Stage

Before embarking on an explicit computation, it's crucial to lay the groundwork by defining the fundamental concepts involved. Let's start with Galois extensions. A Galois extension $L/K$ is a field extension that is both normal and separable. The Galois group, denoted as $\text{Gal}(L/K)$, comprises all automorphisms of $L$ that fix $K$. This group, equipped with the operation of composition, encodes vital information about the structure of the extension. Understanding the structure of the Galois group is crucial, as it dictates many properties of the field extension. The elements of the Galois group, automorphisms, act as permutations on the roots of polynomials, and the fixed fields of subgroups of the Galois group provide a rich tapestry of intermediate fields within the extension. The study of these intermediate fields and their relationship to the subgroups of the Galois group is the cornerstone of Galois theory, providing a powerful framework for understanding field extensions.

Next, we introduce the concept of ramification. In simple terms, ramification describes how prime ideals in the base field $K$ behave when extended to the field $L$. If a prime ideal $\mathfrak{p}$ in $K$ factors as a product of distinct prime ideals in $L$, we say that $\mathfrak{p}$ is unramified. However, if any of the prime ideals in the factorization appear with an exponent greater than one, we say that $\mathfrak{p}$ is ramified. Ramification is a key phenomenon in algebraic number theory, providing crucial information about the structure of field extensions and the arithmetic properties of the fields involved. Understanding which primes ramify and how they ramify is essential for many number-theoretic investigations, including the study of class field theory and the arithmetic of elliptic curves.

The ramification groups form a filtration of subgroups of the Galois group, providing a refined understanding of the ramification structure. Specifically, for each integer $i \geq -1$, the $i$-th ramification group $G_i$ consists of elements in the Galois group that act trivially modulo a certain power of a prime ideal above $\mathfrak{p}$. These groups, denoted by $G_i$, capture the inertia and wild ramification behavior of the extension. The higher ramification groups, corresponding to larger values of $i$, provide finer information about the ramification structure. The study of ramification groups is a central theme in local class field theory, where they play a fundamental role in understanding the structure of local fields and their extensions.

Finally, we arrive at the definition of the Artin conductor. Given a Galois extension $L/K$ and a representation $\rho$ of the Galois group $\text{Gal}(L/K)$, the Artin conductor $f(\rho)$ is an integer that measures the ramification of the representation $\rho$. It is a crucial invariant that encapsulates the complexity of the ramification behavior associated with the representation. The Artin conductor appears in the Artin conductor-discriminant formula, a central result in algebraic number theory that relates the conductor to the discriminant of the extension. Understanding the Artin conductor is therefore essential for various applications, including the study of L-functions and the arithmetic of automorphic forms. The Artin conductor is formally defined using the ramification groups, providing a precise measure of the ramification complexity.

Explicit Computation: A Detailed Example

Now, let's delve into an explicit computation to solidify our understanding. We consider the field extension $L = \mathbb{Q}_2(\sqrt{2}, i)$ over $K = \mathbb{Q}_2$, where $\mathbb{Q}_2$ denotes the field of 2-adic numbers and $i$ is the imaginary unit (i.e., $i^2 = -1$). Our goal is to compute the Artin conductor for a particular representation of the Galois group $\text{Gal}(L/K)$. This specific example will serve as a blueprint for similar computations in other settings. The choice of this field extension is deliberate, as it exhibits interesting ramification behavior while remaining tractable for explicit calculations. The 2-adic field $\mathbb{Q}_2$ is a local field, and its extensions are crucial in number theory. Understanding the arithmetic of such extensions provides insights into global phenomena. The inclusion of both $\sqrt{2}$ and $i$ in the extension creates a rich interplay of ramification effects, making it an ideal example for illustrating the computation of Artin conductors.

Determining the Galois Group

The first step in our computation is to determine the Galois group $G = \text{Gal}(L/K)$. We observe that $L$ is obtained by adjoining $\sqrt{2}$ and $i$ to $\mathbb{Q}_2$. The extension $\mathbb{Q}_2(\sqrt{2})$ over $\mathbb{Q}_2$ has degree 2, and similarly, $\mathbb{Q}_2(i)$ over $\mathbb{Q}_2$ also has degree 2. Furthermore, these two extensions are linearly disjoint, meaning that their intersection is just $\mathbb{Q}_2$. Therefore, the extension $L/\mathbb{Q}_2$ has degree 4, and its Galois group $G$ has order 4. The Galois group $G$ can be identified as the Klein four-group $V_4 \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, which consists of four elements: the identity, the automorphism $\sigma$ that sends $\sqrt{2}$ to $-\sqrt{2}$ and fixes $i$, the automorphism $\tau$ that sends $i$ to $-i$ and fixes $\sqrt{2}$, and the automorphism $\sigma\tau$ that sends both $\sqrt{2}$ to $-\sqrt{2}$ and $i$ to $-i$. The explicit determination of the Galois group is crucial, as it forms the basis for understanding the symmetries of the field extension and the behavior of the ramification groups. Each element of the Galois group corresponds to a specific way of permuting the roots of the defining polynomials, and understanding these permutations is essential for analyzing the ramification structure.

Computing Ramification Groups

Next, we need to compute the ramification groups $G_i$ for $i \geq -1$. Recall that the ramification groups form a descending sequence of subgroups of the Galois group, providing finer and finer information about the ramification structure. The ramification groups are defined in terms of the action of the Galois group on the valuation ring of $L$. To compute these groups, we first need to determine the valuation ring and the maximal ideal. In this case, the valuation ring of $L$ is the ring of integers $\mathcal{O}_L$, and the maximal ideal is generated by a uniformizer. The uniformizer, in this context, is an element of valuation 1. Determining a uniformizer is a crucial step, as it allows us to measure the divisibility of elements in $L$ and understand the ramification structure. The ramification groups are defined in terms of how the Galois group acts modulo powers of the maximal ideal, so the choice of uniformizer plays a central role in the computations.

For $L = \mathbb{Q}_2(\sqrt{2}, i)$, we find that the ramification index $e$ is 4, meaning that the prime 2 factors as a fourth power of a prime ideal in $L$. The different $\mathfrak{D}_{L/K}$ of the extension $L/K$ is a measure of the ramification, and its valuation is related to the ramification groups. In this case, we can compute the ramification groups as follows:

• $G_{-1} = G_0 = G = \{1, \sigma, \tau, \sigma\tau\}$ (the entire Galois group)
• $G_1 = \{1, \sigma, \tau, \sigma\tau\}$ (the entire Galois group)
• $G_2 = \{1, \sigma\tau\}$
• $G_i = \{1\}$ for $i \geq 3$ (the trivial group)

The computation of these ramification groups involves careful analysis of the action of the Galois group elements on the valuation ring of $L$. The groups $G_0$ and $G_1$ coincide in this case, indicating a high degree of ramification. The group $G_2$ is a subgroup of order 2, reflecting a change in the ramification behavior at this level. The higher ramification groups $G_i$ for $i \geq 3$ are trivial, indicating that the ramification stabilizes at this point. The determination of these groups is a crucial step in computing the Artin conductor, as the lengths of these groups appear in the formula for the conductor.

Computing the Artin Conductor

Now, let's consider a representation $\rho$ of $G$. Since $G$ is isomorphic to the Klein four-group, it has four irreducible representations, all of which are one-dimensional. Let's focus on the representation $\rho$ that sends $\sigma$ to 1 and $\tau$ to -1. This representation is a homomorphism from $G$ to the multiplicative group of complex numbers, and it captures a specific aspect of the Galois group's action. The choice of this representation is illustrative, and the computation can be carried out similarly for other representations. The Artin conductor $f(\rho)$ is computed using the formula:

$$f(\rho) = \sum_{i=0}^{\infty} \frac{g_i}{g_0} \dim(V) - \dim(V^{G_i})$$

where $g_i$ is the order of the $i$-th ramification group $G_i$, $V$ is the representation space of $\rho$, and $V^{G_i}$ is the subspace of $V$ fixed by $G_i$. In our case, $V$ is a one-dimensional vector space, and we can simplify the formula accordingly. The dimensions of the fixed subspaces $V^{G_i}$ depend on the representation $\rho$ and the ramification groups $G_i$. Computing these dimensions is a crucial step in evaluating the Artin conductor. The formula for the Artin conductor involves an infinite sum, but in practice, the ramification groups become trivial for sufficiently large $i$, so the sum truncates to a finite number of terms.

For our chosen representation $\rho$, we have:

• $\dim(V) = 1$
• $\dim(V^{G_0}) = 0$ since $\rho(\tau) = -1$
• $\dim(V^{G_1}) = 0$ since $\rho(\tau) = -1$
• $\dim(V^{G_2}) = 1$ since $\rho(\sigma\tau) = -1 \cdot 1 = -1$
• $\dim(V^{G_i}) = 1$ for $i \geq 3$

Using these values, we can compute the Artin conductor:

$$f(\rho) = (\frac{4}{4}(1-0)) + (\frac{4}{4}(1-0)) + (\frac{2}{4}(1-1)) + \sum_{i=3}^{\infty} \frac{1}{4}(1-1) = 1 + 1 + 0 + 0 = 2$$

Therefore, the Artin conductor for the representation $\rho$ is 2. This computation provides a concrete example of how the ramification groups and the representation theory of the Galois group combine to determine the Artin conductor. The value of the conductor, 2 in this case, reflects the complexity of the ramification associated with the representation. Understanding the Artin conductor is crucial for various applications, including the study of L-functions and the arithmetic of automorphic forms.

Significance and Applications

The Artin conductor, as we've demonstrated through our explicit computation, is a powerful invariant that encapsulates the ramification behavior of Galois representations. Its significance extends far beyond theoretical curiosities, finding applications in diverse areas of number theory and related fields. One of the most prominent applications lies in the Artin conductor-discriminant formula, a cornerstone of algebraic number theory. This formula provides a deep connection between the Artin conductor and the discriminant of the field extension, linking ramification properties to the arithmetic structure of the extension. The discriminant, a fundamental invariant of a number field, measures the size and complexity of the ring of integers. The Artin conductor-discriminant formula provides a precise relationship between the conductor, which measures the ramification of the Galois representation, and the discriminant, which measures the ramification of the field extension itself.

Another crucial application of Artin conductors is in the study of Artin L-functions. These functions, generalizations of the Riemann zeta function, encode arithmetic information about Galois representations. The analytic properties of Artin L-functions, such as their meromorphic continuation and functional equation, are intimately related to the Artin conductors. The conductor appears in the functional equation as a key parameter, influencing the analytic behavior of the L-function. Understanding the Artin conductor is therefore essential for studying the analytic properties of L-functions, which play a central role in analytic number theory. The Artin L-functions are also deeply connected to the Langlands program, a vast network of conjectures relating number theory, representation theory, and automorphic forms.

Furthermore, Artin conductors play a vital role in class field theory, which aims to classify abelian extensions of number fields. The Artin map, a central object in class field theory, relates ideals in the base field to elements in the Galois group of an abelian extension. The Artin conductor appears in the description of the kernel of the Artin map, providing a precise characterization of the ideals that become trivial in the extension. Understanding the Artin conductor is therefore crucial for understanding the structure of abelian extensions and the reciprocity laws that govern them. Class field theory is a cornerstone of modern number theory, providing a deep understanding of the arithmetic of number fields and their extensions.

Conclusion

In conclusion, the explicit computation of Artin conductors, as illustrated through our detailed example, provides a tangible understanding of this abstract concept. The Artin conductor, deeply rooted in representation theory, algebraic number theory, and Galois representations, serves as a powerful tool for unraveling the intricate structure of field extensions. From the foundational concepts of Galois groups and ramification to the practical steps of computing ramification groups and applying the Artin conductor formula, we have navigated the landscape of this fascinating topic. The significance of Artin conductors resonates across diverse areas of number theory, from the Artin conductor-discriminant formula to the study of Artin L-functions and class field theory. By mastering the computation and understanding the applications of Artin conductors, we equip ourselves with a key to unlocking deeper insights into the arithmetic world.