On the Zero-Free Region of Certain Families of Dedekind Zeta-Functions

ON THE ZERO-FREE REGION OF CERTAIN FAMILIES OF DEDEKIND ZETA-FUNCTIONS

Nicholas Jian Hao Lai

B.Math., University of Waterloo, 2016

AN ESSAY SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

Master of Science

The Faculty of Graduate and Postdoctoral Studies (Mathematics)

The University of British Columbia (Vancouver)

February 2019

c Nicholas Jian Hao Lai, 2019 Abstract

In this thesis, we will study the analytical properties of subfamilies of Dedekind zeta-functions satisfying certain counting conditions. This was motivated by a result of Pierce, Turnage-Butterbaugh and Wood, giving a Chebotarev density result for subfamilies of Dedekind zeta-functions satisfying a zero-free region assumption. Their result is then applied to remove the requirement to assume the Generalised Riemann Hypothesis in a result of Ellenberg and Venkatesh towards the `-Torsion Conjecture. The Chebotarev density result of Pierce–Turnage-Butterbaugh–Wood is still conditional upon the validity of Strong Artin Conjecture. In particular, the reliance to the Strong Artin Conjecture is due to the application of an analytical method of Kowalski and Michel in their argument, which requires that the irreducible factors of the Dedekind zeta-functions be automorphic cuspidal. The goal of this thesis is to work towards the removal of the reliance to the Strong Artin Conjecture, by condensing down the conditional requirement to a requirement that poles (of pieces) of Dedekind zeta-functions satisfy some mild analytical condition in a family. This will pave a path towards an unconditional eﬀective version of the result of Pierce–Turnage-Butterbaugh–Wood. We will establish directly that certain families of Artin L-functions are ameanable to the analytical method of Kowalski–Michel under the aforementioned mild analytical condition. Applying Kowalski– Michel, we will prove the analog of an eﬀective Chebotarev density result for certain subfamilies of Dedekind zeta-functions. With regards to the result of Pierce–Turnage-Butterbaugh–Wood, this will show that the Strong Artin Conjecture can be replaced with a mild analytical condition in the assumption of Pierce–Turnage-Butterbaugh–Wood.

ii Table of Contents

Abstract ...... ii

Table of Contents ...... iii

1 Introduction ...... 1 1.1 Historical Background ...... 1 1.2 Summary of Results ...... 4

2 A Brief Overview of Artin L-functions ...... 7 2.1 L-Functions Before Artin ...... 7 2.1.1 Riemann Zeta-Function ...... 7 2.1.2 Dedekind Zeta-Functions ...... 8 2.1.3 Dirichlet Series and Dirichlet L-Functions ...... 9 2.1.4 Hecke L-Functions ...... 9 2.2 Artin L-Functions ...... 10 2.2.1 Decomposition Groups, Inertia Groups and Ramification Groups ...... 10 2.2.2 Unramified Factors of Artin L-Functions ...... 14 2.2.3 Ramified and Archimedean Factors of Artin L-Functions ...... 16 2.2.4 Convolution of Artin L-Functions ...... 19 2.3 Analyticity of Artin L-Functions ...... 23 2.3.1 Induction Theorems of Artin and Brauer ...... 23 2.3.2 Artin and Dedekind Conjecture ...... 25 2.3.3 Aramata–Brauer Theorem ...... 26 2.4 Artin L-Functions and Automorphic L-Functions ...... 28 2.4.1 Automorphic L-Functions ...... 29 2.4.2 Strong Artin Conjecture ...... 30 2.5 Convexity Bound on Artin L-Functions ...... 30

3 Kowalski–Michel Ameanable Families ...... 32 3.1 Families of Artin L-Functions ...... 32 3.1.1 Deﬁnition of Families of Artin L-Functions ...... 32 3.1.2 Kowalski–Michel Ameanable Families ...... 34 3.2 Components of Kowalski–Michel Estimate ...... 36 3.2.1 Dirichlet Series Expansion ...... 37 3.2.2 Pseudocharacters ...... 38 3.2.3 A Zero-Detector ...... 47 3.3 Zero-Region of Kowalski–Michel Q-Ameanable Families ...... 52 IV A New Family ...... 54 4.1 ν-Regular Representations ...... 54 4.1.1 Socle of a Group ...... 54 4.1.2 Faithful and Relatively Faithful Representations ...... 56

iii TABLE OF CONTENTS

4.1.3 ν-Regular and Relative ν-Regular Representations ...... 59 4.1.4 Variations on a Theorem of Aramata–Brauer ...... 61 4.2 Family of Artin L-Functions Associated to ν-Regular Representation ...... 63 4.2.1 A New Family and Its Kowalski–Michel Ameanability ...... 63 4.2.2 Zero-Free Region For Subfamilies of Dedekind Zeta-Functions ...... 65 4.2.3 Sterile Subfamilies of F 1 ...... 69 Q,G 5 An Application: Effective Chebotarev Density Theorem ...... 72 5.1 Previously Known Chebotarev Density Results ...... 72 5.2 Effective Chebotarev Density Theorem of Pierce–Turnage-Butterbaugh–Wood ...... 73 5.2.1 Statement of Effective Chebotarev Density Theorem of Pierce–Turnage-Butterbaugh– Wood ...... 74 5.2.2 Sketch of the Proof ...... 75 5.3 Removal of Strong Artin Conjecture ...... 80

6 Conclusion ...... 83

Bibliography ...... 84

iv Chapter 1

Introduction

1.1 Historical Background

To understand where the branch of mathematics that is analytic number theory comes from, it is perhaps most informative to start from the beginning of the subject. Let us consider the positive integers N and its multiplicative structure. One of the ﬁrst things one learns is the existence of prime numbers, and how they are the building block for N via the unique prime factorisation of integers. As such, to study N, the most basic and visible of mathematical objects, a study of prime numbers would yield many properties of N. Given their importance, the natural question one might ask is the following.

Question. How many prime numbers are there in N? Of course, with a basic course in number theory, one would be able to reproduce the arguments originally utilised by Euclid to show that there are inﬁnitely many prime numbers. Instead, we will like to direct the readers’ attention to the proof of Euler regarding the inﬁnitude of prime numbers. Central in Euler’s argument is the divergence of the harmonic series

8 1 fpkq “ nk n“1 ÿ at k “ 1. In particular, Euler’s proof exploits the analytical properties of the function f to deduce an arithmetic result. This is the essence of analytical number theory: using analytical properties to deduce arithmetic results. This line of investigation bore a number of arithmetical results under the study of Dirichlet, who was interested in primes in arithmetic progressions. Utilising character theory, Dirichlet rephrase the question of infinitude of primes in arithmetic progression to the question of divergence of Dirichlet series associated to Dirichlet characters, which will then yield results of infinitude of primes in arithmetic progression. This study of Dirichlet would then firmly establish analytical methods in multiplicative number theory. With the infinitude of prime numbers, one might then ask the following question.

Question. What is the distribution of prime numbers in N? Let us rephrase the question: Let πpxq be the prime-counting function, i.e.

πpxq “ |t primes p ď x u|

How does πpxq grows as x grows? In a landmark paper [Rie59], motivated by his study for formulas for the prime-counting function, Riemann did groundbreaking work on the study of ζpsq, the Riemann zeta-functions. In his paper, Riemann expanded fpkq as a function on the complex plane C, which he denotes as ζpsq, and proceed

1 CHAPTER 1. INTRODUCTION to deduce many analytical properties of ζpsq. Of particular note is Riemann’s sketch, later proved by von Mangoldt, of his explicit formula for the (normalised) prime-counting function involving non-trivial zeros of ζpsq. Riemann thus divised a general program for studying πpxq, by studying the non-trivial zeros of ζpsq. This is greatly expanded upon by Hadamard and de la ValéePoussin, culminating in the celebrated Prime Number Theorem. Prime Number Theorem. Let πpxq be the number of primes less than or equal to x. Then x πpxq „ log x Returning to Riemann’s paper [Rie59], Riemann proceeded to study the non-trivial zeros of ζpsq. Riemann calculated the first few zeroes of Zpsq, and discovered that they all lie on the critical line 1 Repsq “ 2 . From this, Riemann conjectured the most famous open conjecture in mathematics. 1 Riemann Hypothesis. All the non-trivial zeroes of ζpsq has real part 2 . The fame of Riemann Hypothesis is not unwarranted. Besides its notoriety as an extremely inadmissible open problem, its veracity also greatly improve the Prime Number Theorem, by explicitly ? quantifying the error term in the Prime Number Theorem to be of the magnitude of x log x. This cements the role of the study of non-trivial zeroes of ζpsq in analytical number theory as the premier direction towards the study of the distribution of prime numbers. Let us shift our focus. In light of the development of algebraic number theory, we can do an analogous study that we have been doing for N to a more general setting. Let K be a number field, and let OK be the integral closure of Z in K. Even though there is no notion of prime element in OK , we still have the notion of prime ideal in OK and that the unique prime factorisation of ideals holds in OK . In this general case, analogous to the Q case, we have now Dedekind zeta-functions ζK psq playing the role of ζpsq. Dedekind zeta-functions are very natural generalisation of ζpsq to general number fields, and they share similar properties. In fact, a version of Riemann Hypothesis for Dedekind zeta-functions is conjectured to hold.

1 Generalised Riemann Hypothesis. All the non-trivial zeroes of ζK psq has real part 2 .

The analogous question of prime distribution in OK will be the following.

Question. What is the distribution of prime ideals in OK ? We will rephrase the question similar to the case we have done for Z, but more generally, as follows: Let F be a number ﬁeld, and consider a normal extension K{F with Galois group GalpK{F q. Let

PpF q “ t p C OF u For C Ď GalpK{F q a conjugacy class, consider the set

PpC,K{F q “ t p C OF ; p is unramiﬁed in K, σp “ C u

We deﬁne the following prime counting function in OF , given by p F πpx, C,K{F q “ P PpC,K{F q; NQ ppq ď x How does the density of P C,K F in P Fˇ varies as C varies? How does(ˇ the growth of π x, C,K F p { q p ˇq ˇ p { q changes as C varies? The solution to the questions above is ﬁrst given by Chebotarev, in [Tsc26]. Chebotarev proved the following celebrated result which describes the density of the set PpC,K{F q.

2 CHAPTER 1. INTRODUCTION

Chebotarev Density Theorem. Let F be a number ﬁeld, and consider a normal extension K{F with Galois group GalpK{F q. For C Ď GalpK{F q a conjugacy class, the set PpC,K{F q has density

p P PpC,K{F q; N F ppq ď x |C| lim Q “ xÑ8 p P PpF q; N F ppq ď x |GalpK{F q| ˇ Q (ˇ ˇ ˇ ˇ (ˇ Chebotarev Density Theoremˇ is a celebrated result inˇ analytic number theory, being the principal tool in deducing many analytic property of subset of primes. It is also a generalisation of Dirichlet’s theorem on primes in arithmetic progressions. One can rephrase the Chebotarev Density Theorem as

|C| |C| x πpx, C,K{F q „ Lipxq „ |GalpK{F q| |GalpK{F q| log x where we denote x 1 Lipxq “ dt log t ż2 Observe that when K “ F “ Q, we get the Prime Number Theorem. Here, in the general case, the Generalised Riemann Hypothesis plays the same role as Riemann Hypothesis in the Prime Number Theorem, providing a description of the error term in Chebotarev Density Theorem conditional on the veracity of the Generalised Riemann Hypothesis. This was proven by Lagarias and Odlyzko in [LO77]. We will quote here the reﬁned version by Serre in [Ser81].

Conditional Eﬀective Chebotarev Density Theorem. Let F be a number ﬁeld, and consider a

normal extension K{F with Galois group GalpK{F q. Let the absolute discriminant of K be |∆K{Q|. If the Generalised Riemann Hypothesis holds for ζK psq, then there exists a constant C such that for C Ď GalpK{F q a conjugacy class, and for x ě 2, we have that

|C| |C| 1 rK: s πpx, C,K{F q ´ Lipxq ď C x 2 logp|∆ |x Q q |GalpK{F q| |GalpK{F q| K{Q ˇ ˇ ˇ ˇ ˇ ˇ Let us focusˇ on the reliance of Generalised Riemannˇ Hypothesis of these result. Given that the Generalised Riemann Hypothesis can be thought of as a result about the zero-free region of Dedekind zeta-functions, we can formulate the following question.

Question. Is there an unconditional result zero-free region on Dedekind zeta-functions which can replace Generalised Riemann Hypothesis in the Eﬀective Chebotarev Density Theorem?

Towards this question, Kowalski and Michel developed in [KM02] results about the zeroes of families of cuspidal automorphic L-functions. These results can be used in certain number-theoretic applications to replace the Generalised Riemann Hypothesis. Pierce, Turnage-Butterbaugh and Wood showed in [PTBW17] that by assuming the Strong Artin Conjecture, one can port the results of Kowalski and Michel to the case of Artin L-functions and replace the reliance of Generalised Riemann Hypothesis, giving an effective Chebotarev density result which does not rely on the veracity of Generalised Riemann Hypothesis. This comes at the cost of have the resulting effective Chebotarev density result to be true for all but a quantifiable number of exceptions.

3 CHAPTER 1. INTRODUCTION

1.2 Summary of Results

In this thesis, we will mainly be interested in working towards the removal of the reliance of Strong Artin Conjecture in the result of Pierce, Turnage-Butterbaugh and Wood. We will replace this reliance with an analytical condition involving the poles of Artin L-functions. To achieve that, we will need to reframe the result of Kowalski and Michel in a manner which is applicable to a suitable family of Artin L-function, along with some notion of a convolution of Galois representations which mirrors the Rankin-Selberg convolution for automorphic L-functions. We will first define convolution of representations in §2.2.4 of Chapter 2, proving some auxilary results in the process. In Chapter 3, we will proceed to define a family of Artin L-functions which is susceptible to the methods of Kowalski–Michel. We will call such a family a Kowalski–Michel ameanable family, and it mimics all the required properties of a family of cuspidal automorphic L-functions. Crucially, to remove the reliance of Strong Artin Conjecture, we require Kowalski–Michel ameanable family to be well-behaved under the convolution we introduced. We will then proceed as in [KM02], obtaining the following proposition. 1 Proposition 3.2.8. Fix z ě 1 large enough and 0 ă δ ă 2 , and let F be a Kowalski–Michel Q-ameanable 2 family. Let N ě R . Then, there exists a function GF pX,N,Rq such that for ε ą 0 satisfying

1 ´ε d` ma 2`2δ N 2 ą X 2 R log R there exists a constant Mpεq such that for an P C, 2 1 1 2 anψρ,rpnqprnsfps, ρqq ď MpεqNGF pX,N,Rq |an| sϕpρ, Rq |ψρprq| ˇ ˇ Lps,ρ;F {QqPFpXq rďR ˇn„N ˇ n„N ÿ rPRÿδ,zpρq ˇ ÿ ˇ ÿ ˇ ˇ We conjecture the following, which willˇ remove the reliance toˇ Strong Artin Conjecture if it is indeed proven to be true. Conjecture 3.2.9. Fix z ě 1 large enough, and let F be a Kowalski–Michel Q-ameanable family. Let 2 N ě R . Then, the function GF pX,N,Rq in Proposition 3.2.8 satisfy

GF pX,N,Rq ! 1 Proceeding as in [KM02] under the assumption of the above conjecture, we deduce the theorem of Kowalski–Michel in our setting. Theorem 3.3.2. Let F be Kowalski–Michel Q-ameanable family of Artin L-functions, and assume 3 Conjecture 3.2.9. Let α ě 4 and T ě 2. Then, there exists a constant B ą 0, depending on the family parameters, such that for all c ą 10mrF : Qsa ` 4d, there exists a constant Mpcq such that B c 1´α NF,α,T pXq ď MpcqT X 2α´1 for all X P N. In Chapter 4, we will start by studying faithful representations and ν-regular representations of G, deﬁned as ν regG “ pdim ρqρ ρPG kerpàρq“teu We will then prove a variant of Aramata–Brauer Theoremp for Artin L-functions associated to ν-regular representations.

4 CHAPTER 1. INTRODUCTION

Theorem 4.1.25. Let K{F be a Galois extension. The Artin L-functions associated to ν-regular representations of GalpK{F q is entire. We will also prove a variant of Aramata–Brauer Theorem for Artin L-functions associated to convolution of ν-regular representations. Theorem 4.1.27. Let K,K1{F be Galois extensions such that GalpK{F q – G and GalpK1{F q – G. Then, the Artin L-functions associated to convolution of ν-regular representations of GalpK{F q and GalpK1{F q is entire if and only if K ‰ K1. If K “ K1, then it has a pole only at s “ 1. We then define a family of Artin L-functions associated to ν-regular representations, and show that it is Kowalski–Michel ameanable. Specialising to the case where the base field of these families are Galois over Q, we will have a Kowalski–Michel zero-region estimate for each of these families. In particular, since each regular representation can be written as a direct sum of ν-regular representations inflated from its quotient group, we get a Kowalski–Michel zero-region estimate for each piece of the Dedekind zeta-function. We then considered sterile subfamilies of a family of Dedekind zeta-functions, that is subfamilies which satisfy some condition on the propogation of “bad” Artin L-functions. This allows us to deduce the following zero-free region result on sterile subfamilies.

Theorem 4.2.10. Let F {Q be Galois, and assume Conjecture 3.2.9. Let H be a sterile subfamily of F 1 with contamination index τ. Then, for any 0 ă ∆ ă 1 ´ τ and η ă 1 , there exists F,G dF,GpHq 4 ν B (depending on the Kowalski–Michel family parameter of F G pHq for each N G), a constant F, {N C 1 MpG, aF,GpHq, dF,GpHq, ∆, τq and 0 ă δ ď 4 , given by ε δ “ 20mrF : QsaQ,GpHq ` 8dQ,GpHq ` 4ε ν with m “ maxNCG dim reg G and {N τ ε ∆ “ 1 ´ ´ dF,GpHq 2dF,GpHq

p1´p1´ηq∆qdF,GpHq such that for all X P N, at most MpG, aF,GpHq, dF,GpHq, ∆, τqX members of HpXq that can have a zero in the region

η∆dF,GpHq η∆dF,GpHq r1 ´ δ, 1s ˆ ´X B ,X B „  We then reframed the results of Pierce–Turnage-Butterbaugh–Wood in this language. Finally, in Chapter 5, after briefly sketching the effective Chebotarev density result of Pierce–Turnage- Butterbaugh–Wood conditional on a zero-free region, we prove the their results holds unconditionally for set of fields which describe a sterile subfamily.

Theorem 5.3.1. Let F {Q be Galois, and assume Conjecture 3.2.9. Let

F Z Ď K{F ; rK : F s “ n, GalpK {F q “ G

! 1 ) Suppose HpZq is a sterile subfamily of FF,G with contamination index τ. Let A ě 2. Then, for every 0 ă ε ă 1 with 2 τ ε ∆ “ 1 ´ ´ ą 0 dF,GpHpZqq 2dF,GpHpZqq

5 CHAPTER 1. INTRODUCTION

such that ε 1 δ “ ď 20mrF : QsaF,GpHq ` 8dF,GpHq ` 4ε 2A there exists eﬀectively computable absolute constants C1,C2,C3,C4, a constant

pF q D “ DpZ, τ, ε, A, |G|, cF , β0 , rF : Qs,C1,C2,C3q

such that for X P N, all but DXτ`ε ﬁelds K{F in

K F Z; ∆ F X { P K {F ă ! ˇ ˇ ) ˇ ˇ has that there exists constants ˇ ˇ pF q F ki “ kipδ, A, |G|, cF , β0 , |∆F {Q|, rF : Qs, rK : Qs,C3,C4q

k 2 k2plog log |∆ F | 3 q for i “ 1, 2, 3 with for any conjugacy class C Ď G, we have for all x ě k1e K {Q ,

F |C| |C| x πpx, C, K {F q ´ Lipxq ď |G| |G| plog xqA ˇ ˇ ˇ ˇ ˇ ˇ Specialising to F “ Q and utilisingˇ their work on ﬁeld-counting,ˇ we thus remove the reliance of Strong Artin Conjecture in their main eﬀective Chebotarev density result under the assumption of Conjecture 3.2.9.

6 Chapter 2

A Brief Overview of Artin L-functions

In this chapter, we will provide a basic overview on the study of Artin L-functions. In particular, we will place emphasis on reviewing properties of Artin L-functions that we will need later on in this thesis. We will not be exhaustive in our discussions, nor will we provide a novel approach to the subject except in §2.2.4. Experts can safely skip this chapter except §2.2.4, where we introduce convolution of Galois representations, and only referring back to this chapter as needed. Our exposition will mainly follow the exposition of Cogdell in [Cog07], with references to [IK04] and [MM12]. We also refer to other sources such as [CF67], [Neu86], [Neu13] and [Lan13a] for technical details and other number-theoretic background.

2.1 L-Functions Before Artin

Although we will not be thorough in listing the history behind the study of Artin L-functions, we will nevertheless sketch a brief timeline, to help motivate the deﬁnition that follows. Readers who are interested in the history of the development of L-functions should consult [Cog07] and its references.

2.1.1 Riemann Zeta-Function Let us start from the beginning: Consider the Riemann zeta-function, 1 ζpsq “ ns nP ÿN ζpsq is initially deﬁned as a function with integer domain. It is known since Euler’s time that one can write the Riemann zeta-function as an Euler product

1 ´1 ζpsq “ “ 1 ´ p´s ns nPN p prime ÿ ź ` ˘ for s ą 1. Using the divergence of harmonic series, Euler then deduced the inﬁnitude of rational primes. In a landmark paper [Rie59], motivated by his study for formulas for the prime-counting function, Riemann did groundbreaking work on the study of ζpsq, which now bore his name, and formulated the beginnings of analytic number theory. Riemann ﬁrst expanded ζpsq as a function on the complex plane C, and proceeded showed the existence of a meromorphic continuation of ζpsq to the half-plane Repsq ą 0 with a simple pole at s “ 1 of residue 1. By setting

´ s s Zpsq “ π 2 Γ ζpsq 2 ´ ¯ Riemann also showed that Zpsq has a meromorphic continuation to the entire complex plane with simple poles s “ 0 and s “ 1, and that it satisﬁes the functional equation Zpsq “ Zp1 ´ sq

7 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

Riemann also proved that ζpsq has infinitely many zeroes, which all non-trivial ones lie in the critical strip 0 ď Repsq ď 1. By calculating the first few zeroes of Zpsq, and discovering that they all lie on 1 the critical line Repsq “ 2 , Riemann proceeded to make his most famous conjecture, after supposedly “vergeblichen Versuchen (futile attempts)”, which still remains very much open. 1 Riemann Hypothesis. All the non-trivial zeroes of ζpsq has real part 2 . Riemann also sketched a proof, later proved rigorously by von Mangoldt, for his explicit formula for the (normalised) prime-counting function, of which involves non-trivial zeroes of ζpsq. This is greatly expanded upon by Hadamard and de la ValéePoussin, culminating in the celebrated Prime Number Theorem. Prime Number Theorem. Let πpxq be the number of primes less than or equal to x. Then x πpxq „ log x With the veracity of the Riemann Hypothesis, the error term is conjectured to be of the magnitude of ? x log x.

2.1.2 Dedekind Zeta-Functions Dedekind extended the notion of zeta-function to the context of an arbitrary number field. Definition 2.1.1. Let K be a number field. A Dedekind zeta-function of K is defined by 1 ζ psq “ K N K paqs aďO Q ÿK where a ď OK means that a is an ideal of OK . Hecke proved that Dedekind zeta-functions has properties very similar to ζpsq, which we will

henceforth rename as ζQpsq. Speciﬁcally, 1 1. Dedekind zeta-functions ζK psq has an analytic continuation to Repsq ą 1 ´ except a simple rK:Qs pole at s “ 1 with residue 2r1 p2πqr2 R h m |∆K{Q| where h is the class number of K, r1 is the numbera of real places of K, r2 is the number of complex places of K, R is the regulator, m is the number of roots of unity in K, and |∆K{Q| is the absolute discriminant of K.

2. Dedekind zeta-functions ζK psq has the Euler identity for Repsq ą 1 of K p ´s ´1 ζK psq “ 1 ´ NQ p q pCOK ź ` ˘ where p C OK means that p is a prime ideal of OK .

3. Dedekind zeta-functions ζK psq has a meromorphic continuation to the entire C, and satisfy a functional equation. Indeed, we even have a version of Riemann Hypothesis for Dedekind zeta-functions, which we will state here as Generalised Riemann Hypothesis. 1 Generalised Riemann Hypothesis. All the non-trivial zeroes of ζK psq has real part 2 .

8 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

2.1.3 Dirichlet Series and Dirichlet L-Functions

Let us go back to the original definition of ζQpsq. Dirichlet, in his study of infinitude of primes in arithmetic progression, expanded on the series form of ζQpsq and considered a general form of the series we define below.

Deﬁnition 2.1.2. A Dirichlet series is a series of the form

8 a Lpsq “ n ns n“1 ÿ for s P C.

ˆ ˆ To be precise, Dirichlet considered Dirichlet series with an “ χpnq, where χ : Z mZ Ñ C is a multiplicative character, called Dirichlet characters. In modern terminology, these are Dirichlet ˆ ˆ ` L ˘ L-functions associated to Dirichlet characters χ : Z mZ Ñ C , deﬁned as

χpnq ` L ˘ ´1 Lps, χq “ “ 1 ´ χppqp´s ns nPN p prime ÿ ź ` ˘ Dirichlet then showed the convergence and non-vanishing of Lps, χq at Repsq “ 1 associated to non- principal Dirichlet characters, and deduced the inﬁnitude of primes in arithmetic progression.

2.1.4 Hecke L-Functions The work of Weber and Hecke pushed the definitions further, resulting in a definition of a L-function of a number field F {Q associated to a ray-class character χ of F . These are hugely important in the early developments of abelian class field theory, and are a direct generalisation of Dirichlet L-functions. Due to elements of OF not having unique prime factorisation, the naive generalisation of Z mZ to OF breaks down quite significantly. However, basic algebraic number theory has provided us with a L another option, which is to instead consider characters of ideal class groups. L Let m be a modulus of F . We let Im be the group of nonzero fractional ideals of F which do not divide m, and let Pm be the subgroup generated by principal fractional ideals such that it is generated by an element α with for all p C OF ,

• ordppα ´ 1q ě ordpm0

• τpαq ą 0 for all τ P m8.

We call C Im to be the ray class group of modulus m, and this will now take the role of m “ Pm Z mZ in the case of F “ Q. We call the characters of Cm a Hecke character, which will now take the role of L L Dirichlet characters in the case of F “ Q. There is then a ﬁnite abelian extension F pmq{F for each modulus m, which is called the ray class ﬁeld of F of modulus m. By Artin–Takagi, we get that

Cm – GalpF pmq{F q and that every ﬁnite abelian extension of F is contained in some ray class ﬁeld of F . This means that we have successfully given a L-function of an abelian extension associated to its Galois representations.

9 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

Weber and Hecke then deﬁned the following: For a modulus m of F and a Hecke character χ, the associated L-function is deﬁned as

χpaq ´1 Lps, χ; F q “ “ 1 ´ χppqN F ppq´s N F paqs Q aďOF Q pCOF ÿ ź ` ˘ Hecke then proved that these L-functions has an analytic continuation to the entire C for non- principal χ, and satisfy an analogous functional equation. With this, we can state the following theorem, which relates Dedekind zeta-functions to Hecke L-functions.

Theorem 2.1.3. Let K{F be an abelian extension, and let m be a modulus of F such that K Ď F pmq. Let H be the corresponding normal subgroup of Cm. Then,

ζK psq “ Lps, χ; F q

χP Cźm H {N In particular, ζK psq is entire. ζF psq

2.2 Artin L-Functions

Class field theory gives a way of defining L-functions of an abelian extension K{F of number fields associated to (characters of) its Galois representations, by relating it to the Hecke L-functions associated to a ray class character of F . We will now extend this, and define L-functions associated to Galois representations of Galois extensions, first introduced by Artin in a series of papers written in 1923 and in the 1930s. For this section, let us set the following: Let K{F be a Galois extension of number fields, and let OK and OF be ring of integers of K and F respectively. Let G :“ GalpK{F q, and denote by its set of irreducible representations by Gˆ. Let ρ : G Ñ GLpV q be a representation of G, with character

χρpgq :“ Trpρpgqq

In this thesis, we will only consider complex representations.

2.2.1 Decomposition Groups, Inertia Groups and Ramification Groups We begin by giving a brief exposition on decomposition groups, inertia groups and ramification groups, which are ingredients central in the definition of Artin L-functions. Readers should consult, for example, [CF67] or [Lan13a] for more details and the number-theoretical backgrounds. Let p C OF be a prime ideal. Recall that the Galois group G acts transitively on the set of primes in OK lying over p. We define the following:

Deﬁnition 2.2.1. Let PCOK be a prime lying over p. The decomposition group of P is the subgroup

DP “ t σ P G; σP “ P u “ StabGpPq

The following lemma shows how the decomposition groups of all primes lying over p are related.

Lemma 2.2.2. The decomposition group of primes in OK lying over p are conjugates.

10 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

Proof. Let P C OK be a prime lying over p. Since G acts transitively on primes lying over p, we can 1 1 write any other prime P C OK lying over p as P “ τP, where τ P G. Now,

´1 ´1 σ P DτP ðñ στP “ τP ðñ τ στP “ P ðñ τ στ P DP and hence we get that ´1 DτP “ τDPτ and thus we get that the decomposition group of primes in OK lying over p are all conjugates.

DP In the following, let P C OK be a prime lying over p. Consider the subﬁeld F Ď K Ď K ﬁxed by DP. We show that

Lemma 2.2.3. P is the unique prime lying over P X OKDP . Furthermore, the injection of residue ﬁelds

OF ãÑ OKDP p P X OKDP M is an isomorphism. L

DP Proof. First, note that DP “ GalpK{K q acts transitively on primes in OL lying over PXOKDP , which includes P. However, since DP ﬁxes P, we get that therefore P is the only prime lying over P X OKDP . Further, ﬁrst note that for any σ P GzDP, we get that σP ‰ P, which implies that

σP X OKDP ‰ P X OKDP

Hence, for any x P OKDP , there exists a y P OKDP such that

y ” x pmod P X OKDP q and y ” 1 pmod σP X OKDP q ùñ y ” x pmod Pq and σy ” 1 pmod Pq G for all σ P GzDP. In particular, forσ ¯ P DP withσ ¯ ‰ 1, we get that M σ¯pyq ” 1 pmod Pq

KDP Thus, NF pyq P OF such that

KDP NF pyq “ σ¯pyq ” x pmod Pq

σ¯ G P źDP N and this shows the surjectivity of OF ãÑ OKDP . p P X OKDP M Note that by Orbit-Stabiliser TheoremL and since G acts transitively on primes in OL lying over p, DP we get that the number of primes in OL lying over p is rG : DPs “ rK : K s. We thus get the following proposition.

Proposition 2.2.4. p splits completely and does not ramify in KDP .

11 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

Here, we see the usefulness of the decomposition group, by allowing us to study the extension K{F as a tower of extensions where the splitting behaviour of p in each extension is well-understood. O O Consider now the residue fields kP “ K P and Fp “ F p. For any σ P DP, σ descends to a well-defined automorphism of kP which fixes Fp, and hence there exists a natural homomorphism L L

DP Ñ GalpkP{Fpq In fact, more is true, as demonstrated by the following proposition.

Proposition 2.2.5. The natural homomorphism

DP Ñ GalpkP{Fpq is surjective.

Let us deﬁne the following.

Deﬁnition 2.2.6. Let P C OK a prime lying over p. The inertia group of P is the group

IP “ kerpDP GalpkP{Fpqq Along with Proposition 2.2.5, we get the following exact sequence

1 IP DP GalpkP{Fpq 1

It is also therefore clear that |IP| “ epP{pq, the ramiﬁcation index of P over p. We also make the following observation.

Proposition 2.2.7. The inertia group of P is the subgroup of G deﬁned by

IP “ t σ P G; σpaq ” a pmod Pq, @a P OK u

´1 ´1 Proof. For σ R DP, we get that σ P ‰ P. Since P and σ P are maximal in OK , we get an element a P P such that a R σ´1P. Thus, σpaq ı a pmod Pq, and the result follows.

Indeed, the inertia groups of primes in OK lying over p are intimately related such as the case of its decomposition groups.

Lemma 2.2.8. The inertia group of primes in OK lying over p are conjugates.

1 1 Proof. Let P, P C OK be primes lying over p. By Lemma 2.2.2, let σ P G be such that σP “ P and ´1 DP1 “ σDPσ . Then, by Proposition 2.2.7, we get that

1 τ P IP1 ðñ τσpaq ” σpaq pmod P q, @a P OK ´1 ´1 ðñ σ τσpaq ” a pmod Pq, @a P OK ðñ σ τσ P IP

´1 and hence, we get that IP1 “ σIPσ .

N F ppq Now, recall that GalpkP{Fpq is cyclic with the generator x ÞÑ x Q . We deﬁne the following.

N F ppq Deﬁnition 2.2.9. The Frobenius element at P is an element σP P DP that is mapped to x ÞÑ x Q under the map DP GalpkP{Fpq.

12 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

Note that the Frobenius element is only defined mod IP. If however p is unramified in K, then IP “ teu, and in that case the Frobenius elements are defined, and further they are in the same conjugacy class by Lemma 2.2.2. In particular, this is true for all but finitely many primes of K. In this case, we can make the following well-defined definition.

Definition 2.2.10. Suppose p C OF is unramified in K. The Frobenius element σp of p is the conjugacy class of σP, where P C OK lies over p. For our purpose in Section 2.2.4, we shall also define the following.

Deﬁnition 2.2.11. Let PCOK a prime lying over p. Let i ě ´1 be an integer. The i-th ramiﬁcation group of P is the group

i`1 GipP{pq “ σ P G; σpaq ” a pmod P q, @a P OK ( We note that G´1pP{pq “ G and G0pP{pq “ IP, and that each GipP{pq are normal in G. We also note that ramiﬁcation groups are generalisations of the inertia group via Proposition 2.2.7. In later sections, we will consider the descending series of ramiﬁcation groups

G “ G´1pP{pq Ě G0pP{pq Ě G1pP{pq Ě ¨ ¨ ¨

For some background in ramification groups, c.f. [CF67] and [Lan13a] for an exposition. For our purpose, we wish to highlight the following result, regarding ramification groups of composite fields.

Proposition 2.2.12. Let K{F and K1{F be Galois extensions of number ﬁelds, with K Ď K1. Let p be a prime in F and let P be a prime in K1 lying over p. Then, for i ě 0,

|G pP{pq| |G ppP X O q{pq| 0 ě 0 K |GipP{pq| |GippP X OK q{pq| Proof. The quotient map q : GalpK1{F q Ñ GalpK{F q induces a surjection q : G0pP{pq Ñ G0ppP X OK q{pq Composing with the quotient map

G0ppP X OK q{pq πGippPXOK q{pq : G0ppP X OK q{pq Ñ GippP X OK q{pq M we get an isomorphism

G0pP{pq – G0ppP X OK q{pq kerpπGippPXOK q{pq ˝ qq GippP X OK q{pq M M Finally, note that since qpGipP{pqq Ď GippP X OK q{pq, we get that

GipP{pq Ď kerpπGippPXOK q{pq ˝ qq

and thus the result follows. We conclude this discussion with an example, produced using [LMF13].

13 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

Example 2.2.1. Consider the extension K{Q, where K is the splitting ﬁeld of the polynomial fpxq “ x4 ` x ` 1

Then, |∆K | “ 229, and so by basic number theory, the only prime of Q that ramifies in K is the prime 229. Let r1, r2, r3, r4 be the four distinct roots of fpxq. Now, we compute the Galois group GalpK{Qq ď S4 of K{Q. Clearly, the order of the Galois group is divisible by 4. Furthermore, the cubic resolvent of fpxq is x3 ´ 4x ` 1, which is irreducible mod 3, hence it is irreducible. Thus, K contains the cubic field Qpr1r2 ` r3r4q, and therefore the order of the GalpK{Qq is also divisible by 3. This means that GalpK{Qq is either A4 or S4. Finally, since the discriminant of fpxq is 229, which is crucially not a square, we therefore must have that GalpK{Qq “ S4. By brute force computation, we find that K “ Qpαq, where α is the root of fpxq “ 266962921 ` 2114616240x2 ` 67773664x4 ` 74190340x6 ` 50899280x8 ´ 462400x10 ´ 973378x12 ` 23120x14 ` 7520x16 ` 340x18 ´ 80x20 ` x24

Reducing fpxq modulo 229 gives us that

fpxq “ px2 `29q2px2 `32q2px2 `71x`174q2px2 `87x`32q2px2 `142x`32q2px2 `158x`174q2 pmod 229q

Hence, p229q in OK factorises as 6 2 p229q “ Pi i“1 ź where

2 P1 “ p229q ` pα ` 29q 2 P2 “ p229q ` pα ` 32q 2 P3 “ p229q ` pα ` 71α ` 174q 2 P4 “ p229q ` pα ` 87α ` 32q 2 P5 “ p229q ` pα ` 142α ` 32q 2 P6 “ p229q ` pα ` 158α ` 174q

Thus, we can now conclude that the ramification and residual index of p229q are both 2. Therefore, choosing P to be any of the six primes lying over p229q, we get that IP “ Z 2Z and DP is one of 2 Z 2Z or Z 4Z (it turns out to be the former). L ` L ˘ L 2.2.2 Unramified Factors of Artin L-Functions In this section, we suppose the prime p is unramified in K. Then, we get a well-defined Frobenius element σp. As σp is a conjugacy class, we get that any class function χ of G is constant on σp, and hence we denote this value by χpσpq. We can now define the unramified factor of an Artin L-function.

Deﬁnition 2.2.13. Let ρ : GalpK{F q Ñ GLpV q be a representation of GalpK{F q, and let χρ be its associated character. We deﬁne

F p ´s ´1 Lpps, ρ; K{F q “ det I ´ ρpσpqNQ p q ` ˘ 14 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS and

Lunramps, ρ; K{F q “ Lunramps, χρ; K{F q F p ´s ´1 “ Lpps, ρ; K{F q “ det I ´ ρpσpqNQ p q pCOF pCOK unramiﬁedź in K unramiﬁedź in L ` ˘

where ρpσpq “ ρpσPq for any prime P lying over p.

We note that there is no ambiguity in the deﬁnition: for any τ P G, we get that

´1 F p ´s F p ´s ´1 det I ´ ρpτσPτ qNQ p q “ detpρpτqq det I ´ ρpσPqNQ p q detpρpτqq det I ρ σ N F p ´s ` ˘ “ ´ p P`q Q p q ˘

Remark 2.2.14: Note that, since ρpσpq is a linear` transformation of˘ ﬁnite order on V , the eigenvalues λippq of ρpσpq are thus roots of unity. Hence, we get that

dim ρ F p ´s ´1 p F p ´s ´1 Lpps, ρ; K{F q “ det I ´ ρpσpqNQ p q “ 1 ´ λip qNQ p q i“1 ` ˘ ź ` ˘ where |λippq| “ 1. For each unramiﬁed prime p, note that

F ´s F ´s ´1 ´ log I´ρpσpqN ppq p p Q q det I ´ ρpσpqNQ p q “ det e

ρpσ ql ` ˘ ´ ´ p ¯ lě1 lNF ppqsl “ det e Q ˜ ř ¸ l l ρpσ q χρpσ q ´Tr p ´ p lě1 lNF pplqs lě1 lNF pplqs “ e Q “ e Q ř ř

l ´1 χρpσ q ùñ log det I ´ ρpσ qN F ppq´s “ ´ p p Q lN F pplqs lě1 Q ` ˘ ÿ and here we see the similarities of the above with the Hecke L-functions. In fact, this is probably where Artin started, leading to the deﬁnition we quoted above in his 1924 paper [Art24].

Example 2.2.2. Consider the trivial representation 1 of G. Then, we get that

F p ´s ´1 Lunramps, 1,K{F q “ 1 ´ NQ p q pCOK unramiﬁedź in L ` ˘

which, up to finite number of Euler products, is ζK psq. Indeed, once we define the Euler products at the ramified primes, we will see that we do get ζK psq. The unramified factors of Artin L-functions satisfy desired properties, as demonstrated by the next few statements.

Proposition 2.2.15. Lunramps, ρ; K{F q is regular for Repsq ě 1 ` δ, for all δ ą 0.

15 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

Proposition 2.2.16. Lunram are additive: for representations ρ1, ρ2, we get that

Lunramps, ρ1 ‘ ρ2; K{F q “ Lunramps, ρ1; K{F qLunramps, ρ2; K{F q

Proof. We recall that χρ1‘ρ2 “ χρ1 `χρ2 . The result follows by considering log Lunramps, ρ1‘ρ2; K{F q.

Proposition 2.2.17. Lunram satisﬁes inﬂation: Let E{K{F be extensions such that E{K is also Galois. Let πGalpE{Kq : GalpE{F q G be the natural surjection. For ρ : G Ñ GLpV q a representation of G, let ρ˜ be the representation of GalpE{Kq given by

πGal E K 1 GalpE{Kq GalpE{F q p { q G 1

ρ˜ ρ

GLpV q

That is, ρ˜ “ ρ ˝ πGalpE{Kq. Then, we get that

Lunramps, ρ; K{F q “ Lunramps, ρ,˜ E{F q “ Lps, ρ ˝ πGalpE{Lq; E{F q

Proposition 2.2.18. Lunram is inductive: Let E be an intermediate ﬁeld, i.e. F Ď E Ď K. For ρ : GalpK{Eq Ñ GLpV q a representation of GalpK{Eq ď G, and consider the induced representation G IndGalpK{Eqρ. Then, we get that

G Lunramps, ρ; K{Eq “ Lunram s, IndGalpK{Eqρ, K{F We will not give a proof of Proposition 2.2.18 here,` refering the interested˘ reader to, for example, Lang’s [Lan13a] or Heilbronn’s essay in [CF67]. We conclude this section by remarking the propositions above are really the guiding principles behind Artin’s deﬁnition of his L-functions, along with the demand that it agrees with the Hecke L-functions in the abelian case.

2.2.3 Ramified and Archimedean Factors of Artin L-Functions In the previous section, we defined Artin L-functions at unramified primes of K. This accounts for all but finitely many Euler factors, but we still need to define the factors at the finitely many primes of F which ramifies in K to complete the definition of Artin L-functions. However, at a ramified prime p with P a prime lying over p, we note that the inertia group IP of P is non-trivial, as the size of the inertia group corresponds to the ramification index epP{pq. Thus, the Frobenius element σP is a coset, rather than an element of the decomposition group DP. Thus, in order to have a good definition of the Euler product at p, we need to instead consider a subspace of V such that IP acts trivially on this subspace, and that it is compatible with our definition of Euler products at unramified primes. For this, we simply consider the subspace

IP V :“ t v P V ; ρpgqv “ v, @g P IP u

16 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

and we note that this is indeed compatible with the deﬁnition of Euler product at unramiﬁed primes, for in that case IP IP “ teu ùñ V “ V

Thus, for any g1, g2 P σP, we get that

ρpg1q|V IP “ ρpg2q|V IP

1 so we deﬁne ρpσPq|V IP to be this value. Finally, for another P lying over p, since σP and σP1 are conjugates (mod IP), we get that equality of their characteristic polynomial

det xI ρ σ I det xI ρ σ I ´ p Pq|V P “ ´ p Pq|V P1 Hence, we arrive at the deﬁnition` given by Artin˘ in` [Art31b]. ˘

Deﬁnition 2.2.19. Let ρ : GalpK{F q Ñ GLpV q be a representation of GalpK{F q, and let χρ be its associated character. The Artin L-function associated to ρ is

F ´s ´1 I p Lps, ρ; K{F q “ Lps, χρ; K{F q “ det I ´ ρpσPq|V P NQ p q pCOK ź ` ˘ where P is any prime lying over p. We denote

F ´s ´1 I p Lpps, ρ; K{F q “ det I ´ ρpσPq|V P NQ p q ` ˘ Remark 2.2.20: As in Remark 2.2.14, note that, since ρpσPq|V IP is a linear transformation of ﬁnite order IP on V , the eigenvalues λippq of ρpσPq|V IP are thus roots of unity. Hence, we get that

dim ρ F ´s ´1 F ´s ´1 I p p p Lpps, ρ; K{F q “ det I ´ ρpσPq|V P NQ p q “ 1 ´ λip qNQ p q i“1 ` ˘ ź ` ˘ where |λippq| “ 1.

Artin then proceeded to prove the following proposition, mirroring those of Lunram. Proposition 2.2.21. The Artin L-functions are additive, satisfy inﬂation and are inductive.

This result can be deduced similarly to the case of Lunram. As a consequence of additivity and inductivity, we get the following pleasing result.

Corollary 2.2.22. Let regG be the regular representation of G. Then

χρpeq χρpeq ζK psq “ Lps, 1teu; K{Kq “ Lps, regG; K{F q “ Lps, ρ; K{F q “ ζF psq Lps, ρ; K{F q ˆ ˆ ρźPG ρPGźzt1Gu For completion, we will define the completed Artin L-functions, where Artin defined by trying to generalise the L-function for abelian extension. First, we define the Archimedean factors: Let v be an Archimedean prime of F , with w a prime lying over v in K. We set

n´ s s n` s`1 s 1 ´ 2 ´ 2 ` π Γ 2 π Γ 2 , if v is real γvps, ρ; K{F q “ ´s dimpρq #`pp2πq `Γp˘˘sqq ´ , ` ˘¯ if v is complex

17 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

where n` and n´ is the dimension of the eigenspace of V associated the the eigenvalues 1 and ´1 of ρpgq respectively, g the generator of Dw (of order 1 or 2). We denote

γps, ρ; K{F q “ γvps, ρ; K{F q v8 ź Next, let p be a prime of F , with P a prime lying over p in K. Let

G0pP{pq Ě G1pP{pq Ě ¨ ¨ ¨

be the decreasing series of ramiﬁcation groups of P. We deﬁne

8 |GipP{pq| npρ, p; K{F q “ codim V GipP{pq |G pP{pq| i“0 0 ÿ GipP{pq where V is the subspace of V fixed by ρpgq, for all g P GipP{pq. Artin showed in [Art31a] that npρ, p; K{F q is a well-defined integer, 0 for all but finitely many of them. We can now define the following. Definition 2.2.23. The Artin conductor of ρ is the ideal

fpρ; K{F q “ pnpρ,p;K{F q p ź The conductor of Lps, ρ; K{F q is deﬁned as

dim ρ F f Apρ; K{F q “ |∆F {Q| NQ p pρ; K{F qq

where ∆F {Q is the absolute discriminant of F . One can show that the Artin conductor satisfy the same relations as its associated Artin L-functions. Proposition 2.2.24. The Artin conductor is additive and satisfy inﬂation. It is also inductive, in the following sense: Let E be an intermediate ﬁeld, i.e. F Ď E Ď K. For ρ : GalpK{Eq Ñ GLpV q a G representation of GalpK{Eq ď G, and consider the induced representation IndGalpK{Eqρ. Then, we get that G dim ρ E f IndGalpK{Eqρ; K{F “ ∆E{F NF pf pρ, K{Eqq

where ∆E{F is the relative discriminant` of E{F . ˘ A proof of this can be found in [Neu13]. An important corollary, as a consequence of inductivity applied

to 1GalpK{Kq and additivity, is the following famous formula of Artin [Art31a]. Conductor-Discriminant Formula. For Galois extension K{F with Galois group G, we get that

dim ρ ∆K{F “ fpregG; K{F q “ fpρ; K{F q ˆ ρźPG We also get the following corollary regarding the conductor of Artin L-functions. Corollary 2.2.25. The conductor of Artin L-function is additive, satisfies inflation and is inductive. We can finally define the following.

18 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

Deﬁnition 2.2.26. The completed Artin L-function associated to ρ is

s Λps, ρ; K{F q “ Apρ; K{F q 2 γps, ρ; K{F qLps, ρ; K{F q Here, we summarise the result of Artin in [Art31a]. Theorem 2.2.27. The completed Artin L-function Λps, ρ; K{F q is additive, satisfies inflation, inductive, and satisfies the functional equation Λps, ρ; K{F q “ W pρqΛp1 ´ s, ρ¯; K{F q where W pρq is a constant of absolute value 1. Furthermore, Λps, ρ; K{F q has a meromorphic continuation to the whole complex plane. A proof of this fact can be found in [Mar77]. We conclude this section by remarking about the number W pρq. Definition 2.2.28. The Artin root number of ρ is W pρq. The Artin root number is a mysterious quantity. W pρq carries deep arithmetic information, such as information regarding its Galois module structre. Its factorisation was found by Deligne in 1974 and Langlands, much after they were described by Artin. We refer the interested reader to [Frö12]and [Mar77].

2.2.4 Convolution of Artin L-Functions We wish to now define a convolution on Galois representations, such that the associated Artin L-function “mirrors” the Rankin-Selberg convolution under the assumption of Strong Artin Conjecture. We thus define the following. Definition 2.2.29. Let K{F ,K1{F 1 be Galois extensions of number fields, and let ρ : GalpK{F q Ñ GLpV q and τ : GalpK1{F 1q Ñ GLpW q be representations. For number fields K Ď F XF 1, the convolution over K of ρ and τ is defined as

K K K K GalpK {KqˆGalpK1 {Kq GalpK {Kq GalpK1 {Kq ˚ ConvKpρ, τq “ Res K K Ind K ρ ˝ π K b Ind K τ ˝ π K GalpK K1 {Kq GalpK {F q GalpK {Kq GalpK1 {F 1q GalpK1 {K1q ˆ ˙ ˆ ˙ K K where K , K1 are the Galois closure of K and K1 respectively over K. The deﬁnition is a little tedious, but it comes from the following (hopefully) more intuitive picture.

K K K K1

K K 1 K K τ ˚˝π ρ˝π K 1K 1 GalpK {Kq GalpK {K q

K K1

ρ τ ˚ K K ρ τ ˚ F F 1

Figure 2.2.1: Hasse Diagram For Deﬁnition of ConvK

19 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

Here and henceforth, we denoted

K K 1 K GalpK {Kq ˚K GalpK {Kq ˚ ρ “ Ind K ρ ˝ π K and τ “ Ind K τ ˝ π K GalpK {F q GalpK {Kq GalpK1 {F 1q GalpK1 {K1q We will say more about this convolution in the later chapters. For now, we wish to prove an upper bound for the conductor of convolutions. The heart of the matter is the following proposition.

Proposition 2.2.30. Let K{F ,K1{F 1 be Galois extensions of number ﬁelds, and let ρ : GalpK{F q Ñ 1 1 GLpV q and τ : GalpK {F q Ñ GLpW q be representations. Let p be a prime in OK, and let P be a prime in O K K lying over p. Then, we have that K K1

K K K K 1 1 K ˚K 1 npConvKpρ, τq, p; K K {Kq ď pdim τqrF : Ksnpρ , p; K {Kq ` pdim ρqrF : Ksnpτ , p; K {Kq

K K Proof. Without loss of generality, we assume that K “ F , so that K “ K, K1 “ K1. First, we observe that for i ě 0, the natural quotient map

1 qK : GalpKK {F q Ñ GalpK{F q gives a corresponding surjection GipP{pq Ñ GippP X OK q{pq As such, by considering GalpKK1{F q ď GalpK{F q ˆ GalpK1{F q, we get that

GipP{pq Ď GippP X OK q{pq ˆ GippP X OK1 q{pq

ùñ pV b W ˚qGipP{pq Ě pV b W ˚qGippPXOK q{pqˆGippPXOK1 q{pq Ě V GippPXOK q{pq b pW ˚qGippPXOK1 q{pq

ùñ codimppV b W ˚qGipP{pqq ď pdim V qpdim W ˚q ´ dimpV GippPXOK q{pqq dimppW ˚qGippPXOK1 q{pqq ď codimpV GippPXOK q{pqq dim W ` codimppW ˚qGippPXOK1 q{pqq dim V

Thus, combining with Proposition 2.2.12, we get that

8 |GipP{pq| npρ b τ ˚, p; KK1{F q “ codimppV b W ˚qGipP{pqq |G pP{pq| i“0 0 ÿ 8 |GippP X OK q{pq| ď dim W codimpV GippPXOK q{pqq |G ppP X O q{pq| i“0 0 K ÿ 8 |GippP X OK1 q{pq| ` dim V codimppW ˚qGippPXOK1 q{pqq |G ppP X O 1 q{pq| i“0 0 K ÿ “ pdim τqnpρ, p; K{F q ` pdim ρqnpτ ˚, p; K1{F q which is what we wished to prove. We will point out that the above inequality is probably not sharp. See [BH97] for a possible candidate for a sharp upper bound. It will however be suﬃcient for our purposes. A direct consequence of the preceding proposition is the following result regarding upper bounds on the conductor of convolutions.

20 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

Corollary 2.2.31. Let K{F ,K1{F 1 be Galois extensions of number ﬁelds, and let ρ : GalpK{F q Ñ GLpV q and τ : GalpK1{F 1q Ñ GLpW q be representations. For number ﬁelds K Ď F X F 1, we have that

K K 1 1 pdim τqrF :Ks ˚ 1 1 pdim ρqrF :Ks ApConvKpρ, τq; K K {Kq ď Apρ; K{F q Apτ ; K {F q

Proof. The preceding proposition immediately gives

K K K 1 K 1 K pdim τqrF :Ks ˚K 1 pdim ρqrF :Ks ApConvKpρ, τq; K K {Kq ď Apρ ; K {Kq Apτ ; K {Kq

The result then follows via inductivity and inﬂation property of conductors.

1 Lastly, given K Ď K Ď F X F , we wish to prove a relation between ConvK and ConvK. Before proceeding, we will brieﬂy recall Mackey Decomposition Theorem, an important result in the representation theory of ﬁnite groups. For a proof of this, see for example [Ser12].

Mackey Decomposition Theorem. Let S,R ď G, and let pV, ρq be a representation of R. For g P G, deﬁne a representation pV, ρgq of Rg by ρgprq “ ρpg´1rgq for all r P Rg. Then

G G S Rg g ResS IndRρ “ IndSXRg ResSXRg ρ SgR à where the sum is taken over all double cosets SgR.

In general, it is currently unknown if there is a closed-form relation between ConvK and ConvK. The following diagram gives the picture of the Hasse diagram involved in general.

K K K K1

K K K K K K K1 K1

K K K K1

K K1

ρ τ ˚ K K ρ τ ˚ F F 1

K K ρ τ ˚ K

Figure 2.2.2: Full Hasse Diagram in Relating General ConvK and ConvK

We will, however, prove a relation between ConvKpρ, τq and ConvKpρ, τq when K{K is Galois, in which K K K K case K “ K and K1 “ K1 . In this case, the Hasse diagram involved is much simpler, as shown below.

21 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

K K K K K K1 “ K K1

K K K K K “ K K1 “ K1

K K1

ρ τ ˚ K K ρ τ ˚ F F 1

K ρK K τ ˚

Figure 2.2.3: Hasse Diagram in Relating ConvK and ConvK When K{K is Galois

Proposition 2.2.32. Let K{F ,K1{F 1 be Galois extensions of number ﬁelds, and let ρ : GalpK{F q Ñ GLpV q and τ : GalpK1{F 1q Ñ GLpW q be representations. For number ﬁelds K Ď K Ď F X F 1 with K{K a Galois extension, we have that

K K GalpK K1 {Kq ConvKpρ, τq “ rK : KsInd K K ConvKpρ, τq GalpK K1 {Kq

K K Proof. We note that since K “ K , we get that

K K K GalpK {Kq GalpK {Kq K ρ “ Ind K ρ ˝ π K “ Ind K ρ GalpK {F q GalpK {Kq GalpK {Kq We thus have that

K K K 1 K 1 K ˚K GalpK {Kq K GalpK {Kq ˚K GalpK {KqˆGalpK {Kq K ˚K ρ b τ “ Ind K ρ b Ind K τ “ Ind K K ρ b τ GalpK {Kq GalpK1 {Kq GalpK {KqˆGalpK1 {Kq ˆ ˙ ˆ ˙ K K K K Observe that since GalpK {Kq ˆ GalpK1 {Kq E GalpK {Kq ˆ GalpK1 {Kq, we have

K K pσ1,σ2q K K K K 1 1 1 GalpK {Kq ˆ GalpK {Kq “ GalpK {Kq ˆ GalpK {Kq, @pσ1, σ2q P GalpK {Kq ˆ GalpK {Kq ´ ¯ K K K K pσ1,σ2q K K ùñ GalpK K1 {Kq X GalpK {Kq ˆ GalpK1 {Kq “ GalpK K1 {Kq

´ K ¯ K 1 and the order of the double coset, for each pσ1, σ2q P GalpK {Kq ˆ GalpK {Kq, is

K K K K Gal K K1 K Gal K K Gal K1 K K K K K p { q p { q ˆ p { q GalpK K1 {Kqpσ , σ q GalpK {Kq ˆ GalpK1 {Kq “ 1 2 ˇ ˇ ˇ K K ˇ GalpK K1 {Kq ˇ ´ ¯ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇK Kˇ ˇ ˇ “ rK : Ks GalpKˇ {Kq ˆ GalpK1 ˇ{Kq ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 22 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

Thus, by Mackey Decomposition Theorem, we get that

K K K 1 K 1 GalpK {KqˆGalpK {Kq GalpK {KqˆGalpK {Kq K ˚K ConvKpρ, τq “ Res K K Ind K K ρ b τ GalpK K1 {Kq GalpK {KqˆGalpK1 {Kq K K K 1 K 1 GalpK K {Kq GalpK {KqˆGalpK {Kq K ˚K “ Ind K K Res K K ρ b τ GalpK K1 {Kq GalpK K1 {Kq Spσ1,σ2qR à K K 1 K K |GalpK {Kq ˆ GalpK {Kq| GalpK K1 {Kq “ Ind K K ConvKpρ, τq K K 1 rK : Ks GalpK {Kq ˆ GalpK1 {Kq GalpK K {Kq

K K ˇ GalpK K1 {Kq ˇ “ rK : KsIndˇ K K ConvKpρ, τq ˇ ˇ GalpK K1 {Kq ˇ

2.3 Analyticity of Artin L-Functions

In this section, we will give a brief exposition on the analyticity of Artin L-functions. We will primarily be interested in the holomorphicity of Artin L-functions. We refer interested readers to [MM12] for explicit discussions about zeroes and poles of Artin L-functions.

2.3.1 Induction Theorems of Artin and Brauer After defining the (unramified) Artin L-function in [Art24], Artin proceeded to establish the continuation and functional equation of his L-functions. Even though Artin will not have his Reciprocity Law to rely upon, Artin nevertheless realised that he needed to relate his L-functions to the abelian case, where he will have results of Hecke to draw upon, after he proved his Reciprocity Law later. This meant that Artin was looking for a way to push results from the abelian case to the nonabelian case. The way to make sense of this is via the inductivity of Artin L-functions. Specifically, with this in mind and assuming Artin Reciprocity Law, the question becomes Question. Can the representations of a finite group be expressed as a linear combination of representation induced from an abelian subgroup of the group? Since, over characteristic zero, reprsentations are uniquely determined by its character, we can instead consider the question in terms of characters of a group and thus expand the scope of the question. Artin then proceeded to prove in [Art24] a beautiful representation-theoretic result, from a paper in number theory. Artin Induction Theorem. Let G be a finite group. Then, every character of G is a rational linear combinations of characters induced from characters of cyclic subgroups of G. Even more is true: for every character χ of G, |G|χ is a integer linear combinations of characters induced from characters of cyclic subgroups of G. Along with Artin Reciprocity Law and inductivity, we get the following corollary. Corollary 2.3.1. Let K{F be a Galois extension, and let ρ : GalpK{F q Ñ V be a representation. Then, there exists representation ρi of cyclic subgroup Ci of GalpK{F q such that

k n |GalpK{F q| Ci i Lps, ρ; K{F q “ L s, ρi; K{K i“1 ź ` ˘ where ni P Z.

23 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

In 1947, Brauer improved on Artin’s result in [Bra47a]. For this, we need the following definition from group theory. Definition 2.3.2. Let p be a (rational) prime. A group is a p-elementary group is a direct product of a cyclic group of order prime to p and a p-group. A group is an elementary group if it is a q-elementary group for some (rational) prime q. We refer the reader to [Ser12] for properties of elementary groups especially with regards to Brauer Induction Theorem. We highlight the following property of elementary groups. Proposition 2.3.3. Every character of an elementary group is monomial, i.e. it is induced from a 1-dimensional character of a subgroup. Brauer is then able to prove the following in [Bra47a]. Brauer Induction Theorem. Let G be a finite group. Then, every character of G is an integer linear combinations of characters induced from one-dimensional characters of elementary subgroups of G. With this, we get the more pleasing corollary. Corollary 2.3.4. Let K{F be a Galois extension, and let ρ : GalpK{F q Ñ V be a representation. Then, there exists one-dimensional representation ρi of elementary subgroup Hi of GalpK{F q such that

k n Hi i Lps, ρ; K{F q “ L s, ρi; K{K i“1 ź ` ˘ where ni P Z. Thus, Lps, ρ; K{F q admits a meromorphic continuation to the complex plane. We conclude this section with an explicit example of Brauer Induction Theorem at work.

Example 2.3.1. The following example is quoted from [CF67]. Consider the group S3, with its three

irreducible representations: the trivial representation 1S3 , the sign representation sgn, and the standard representation ρ of S3 which is 2-dimensional. The following is the table of its characters. χ χ χ 1S3 sgn ρ 1 1 1 2 p123q, p132q 1 1 ´1 p12q, p13q, p23q 1 ´1 0

Table 2.3.1: Table of Characters of S3 We now wish to express sgn and ρ as direct sum of induced representations of an elementary subgroups of S3.

1. For ρ, consider the non-trivial 1-dimensional representation τ of A3 – Z 3Z, which is elementary. S3 One can check that the character of Ind τ matches χρ, and thus we have that A3 L χ IndS3 χ ρ IndS3 τ ρ “ A3 τ ùñ “ A3

2. For sgn, consider the non-trivial 1-dimensional representation ω of H “ t 1, p12q u, which is S3 elementary. One can check that the character of IndH ω matches χρ ` χsgn, and thus we have that χ IndS3 χ IndS3 χ sgn “ H ω ´ A3 τ

24 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

2.3.2 Artin and Dedekind Conjecture Via Artin and Brauer Induction Theorems, we get a stronger relation between Artin L-functions and Hecke L-functions. By observing that Hecke L-functions of non-principal characters are holomorphic, one expects the corresponding statment to be true of Artin L-function in general. This is, as of yet, an open conjecture, made by Artin.

Artin Holomorphicity Conjecture. Let K{F be a Galois extension, and let ρ be an irreducible representation of GalpK{F q. Then, Lps, ρ; K{F q is entire. Artin Holomorphicity Conjecture has only been veriﬁed when the Galois group of the extension is one of the following [CF67]:

• The symmetric groups S3 and S4. • Groups of square free order.

• Groups of prime power order.

• Groups with abelian commutator subgroup.

In particular, Artin Holomorphicity Conjecture has not been veriﬁed for the simple group A5. We show an example where Artin Holomorphicity Conjecture is known to hold.

Example 2.3.2. Consider S3, with its three irreducible representations: the trivial representation 1S3 , the sign representation sgn, and the standard representation ρ of S3 which is 2-dimensional. Let K{Q be a Galois extension with GalpK{Qq – S3. We have already worked out in Example 2.3.1 that

ρ IndS3 τ “ A3 so by inductivity, A Lps, ρ; K{Qq “ Lps, τ; K{K 3 q which is therefore entire. Further, we have that

sgn “ ω ˝ πH and by inﬂation, H Lps, sgn; K{Qq “ Lps, ω; K {Qq which is therefore entire. We thus have the veracity of Artin Holomorphicity Conjecture in this case.

Now, with Galois extension K{F with Galois group G, by Corollary 2.2.22 consider the factorisation

dim ρ ζK psq “ ζF psq Lps, ρ; K{F q ˆ ρPGźzt1Gu Then, the Artin Holomorphicity Conjecture would imply that ζK psq is entire. We will see in the next ζF psq section that in fact Artin Holomorphicity Conjecture is not needed to arrive at this conclusion. One can now ask if the above is still true for an arbitrary ﬁnite extension. Let K{F be a ﬁnite extension, and let E{F be its normal closure. Let G “ GalpE{F q and H “ GalpE{Kq. Then, we have by inductivity that G ζK psq “ Lps, 1H ; E{Kq “ Lps, IndH 1H ; E{F q

25 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

G Since IndH 1H is certainly a representation of G, and more precisely the permutation representation for the action of coset space G{H, we write

G IndH 1H “ 1G ‘ nρρ ρPGˆzt1 u à G where nρ P N. Thus, we get that

G nρ ζK psq “ Lps, IndH 1H ; E{F q “ ζF psq Lps, ρ; E{F q ˆ ρPGźzt1Gu Assuming the veracity of Artin Holomorphicity Conjecture, we get again that ζK psq is entire. This special ζF psq case of Artin Holomorphicity Conjecture has actually been noted by Dedekind prior, where he formed his conjecture.

Dedekind Conjecture. For any ﬁnite extension K{F of number ﬁelds, ζK psq is entire. ζF psq Dedekind Conjecture is known to hold when K{F is Galois, which we will explore in the next section, and when the normal closure E{F is solvable, by Uchida [Uch75] and van der Waall [vdW75]. For the latter, we refer the reader to [MM12].

2.3.3 Aramata–Brauer Theorem Even though we do not yet know the Artin Holomorphicity Conjecture in general, we can still deduce some analytical results of certain Artin L-functions. In this section, we will provide a proof of Aramata–Brauer Theorem, a central result that start oﬀ the major result of this thesis. In this section, let G be a ﬁnite group. Let ρ, ω, λ be representations of G such that ρ “ ω ‘ λ. We denote ρ a ω :“ λ. We also denote the inner product on characters of G by x, yG. We begin by proving the following lemma regarding the representations of G, following [MM12].

Lemma 2.3.5. Let C be the set of all cyclic subgroups of G. Then,

G |G| pregG a 1Gq “ nC,ψIndC ψ CPC ψPCˆzt1 u à à C where nC,ψ P N. Proof. We will prove this on the level of characters. For a cyclic group C, we deﬁne a character θC : C Ñ C by |C|, if C “ xgy θC pgq “ #0, otherwise and let

λC “ φp|C|qχregC ´ θC where φ is the Euler totient function. In particular,

φp|C|q “ 1 gPC xgÿy“C

26 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

Now, for any g P C, φp|C|q|C|, if g “ e λC pgq “ #´θC pgq, otherwise For any ψ P Cˆ, we get that

xλC , χψyC “ xφp|C|qχregC , χψyC ´ xθC , χψyC 1 “ φp|C|q ´ θ pgqχ pgq “ φp|C|q ´ χ pgq |C| C ψ ψ gPC gPC ÿ xgÿy“C Ğ Ğ

If ψ “ 1C , then the above is 0. Also, note that if ψ is non-trivial (so in particular |C| ą 1), then

t χψpgq; g P C, xgy “ C u is the complete set of |C|-th root of unity, and hence

xλC , χψyC “ φp|C|q ´ χψpgq “ φp|C|q ´ µp|C|q P N gPC xgÿy“C Ğ where µ is the M¨obiusfunction. Therefore, we get that

λC “ nψχψ ˆ ψPCÿzt1C u where nψ P N. Finally, we note that for any irreducible representation ρ of G, by Frobenius Reciprocity we get that

G G IndC λC , χρ “ xIndC λC , χρyG CCPC G CPC ÿ G ÿ “ xλC , χρ|C yC CPC ÿ

“ ¨φp|C|qχρpeq ´ χρpgq˛ “ χρpeq φp|C|q ´ χρpgq CPC gPC CPC gPG ÿ ˚ xgÿy“C ‹ ÿ ÿ ˚ Ę ‹ Ę ˝ ‚ and by noting that φp|C|q “ 1 “ 1 “ |G| CPC CPC gPC gPG ÿ ÿ xgÿy“C ÿ and thus

G IndC λC , χρ “ χρpeq φp|C|q ´ χρpgq “ χρpeq|G| ´ χρpgq “ |G|xχregG ´ χ1G , χρyG CCPC G CPC gPG gPG ÿ G ÿ ÿ ÿ Ę Ę G ùñ |G|pχregG ´ χ1G q “ IndC λC “ nC,ψχψ CPC CPC ˆ ÿ ÿ ψPCÿzt1C u

27 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

With this lemma, we can prove Aramata–Brauer Theorem, which was ﬁrst proven by Aramata [Ara33] and later by Brauer [Bra47b] independently.

Aramata–Brauer Theorem. Let K{F be a Galois extension. Then, ζK psq is entire. ζF psq Proof. By Lemma 2.3.5 and by inductivity, we get that

|G| ζK psq “ Lps, ψ; K{KC qnC,ψ ζ psq F CPC ˆ ˆ ˙ ź ψPCźzt1C u |G| C ζK psq where each nC,ψ P . We already know that each Lps, ψ; K{K q is entire, and thus is as well. N ζF psq Since ζK psq is meromorphic, we get that it must be entire. ´ ¯ ζF psq As a closing remark to this section, we will like to point out the usage of the theory of Heilbronn characters, ﬁrst introduced by Heilbronn in [Hei73], in the study of Artin Holomorphicity Conjecture. We deﬁne the following.

Deﬁnition 2.3.6. Let K{F be a Galois extension with Galois group G. Let s0 P Czt1u. The Heilbronn character of G at s0 is the virtual character

θG,s0 “ ord Lps, ρ; K{F q χρ s“s0 ˆ ρÿPG ˆ ˙ An immediate consequence is the following.

Proposition 2.3.7. Let K{F be a Galois extension with Galois group G. Every Artin L-function associated to representations of G is analytic at s0 if and only if θG,s0 is either identically zero or is a (proper) character of G.

One can also use the theory of Heilbronn characters to prove a generalisation of Aramata–Brauer Theorem. Speciﬁcally, one can prove the following:

Theorem 2.3.8. Let K{F be a Galois extension with Galois group G. For every irreducible representation ˆ ζK psq ρ P G of G, we get that Lps,ρ;K{F q are entire. For more details, we suggest [FGM15] as a good starting point.

2.4 Artin L-Functions and Automorphic L-Functions

In the work of Pierce, Turnage-Butterbaugh and Wood in [PTBW17], they assumed the veracity of the Strong Artin Conjecture to port the results of Kowalski and Michel in [KM02] for the automorphic cuspidal L-functions to Artin L-functions. Since the main result of this thesis is the relaxation of that assumption, we will spend this section to give a very barebones introduction to automorphic L-functions and Strong Artin Conjecture.

28 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

2.4.1 Automorphic L-Functions In order to define automorphic L-functions, one must have a good understanding of automorphic representations. The subject of automorphic representation is a vast one, one that witness a flurry of research work being done today. We will however not have the capacity to give a proper treatment of automorphic representation, lest we lengthen this chapter to an unreasonable length. We proceed by assuming some knowledge of automorphic representations. We will only need the theory of automorphic L-functions to state the Strong Artin Conjecture and to remark on the result of Kowalski and Michel in [KM02] in the next chapter. Thus, one can also treat these automorphic representation as a “black box”, as we will not use any properties of automorphic representations in this thesis beyond what is needed for the aforementioned purposes. We will mainly follow the exposition of [Mic07]. Let π “ bpπp be a unitary irreducible cuspidal automorphic representation of GLnpAF q. Godement and Jacquet in [GJ06], and later Jacquet and Shalika in [JS81], attached a L-function to π. (See also [Lan71].) Definition 2.4.1. The automorphic L-function associated to π is n p F p ´s ´1 Lps, πq “ Lpps, πpq “ 1 ´ απp , iqNQ p q p p i“1 ź ź ź ` ˘ where απpp, iq is the local parameter of π at p.

For us, we will note that απpp, iq for unramified p are eigenvalues which parametrise a semisimple conjugacy class associated to πp [PTBW17]. An important example is the following. Example 2.4.1. When n “ 1, then we get that Lps, πq is a Dirichlet L-function, where now π is a Dirichlet character of conductor the conductor of Lps, πq. This shows that automorphic L-functions are, in this sense, a non-abelian generalisation of Dirichlet L-functions. We can, as in the case of Artin L-functions, define the Euler factors at the infinite prime, to get an extended or completed automorphic L-function which satisfy a functional equation similar to those satisfied by the completed Artin L-functions. We will highlight one more property of automorphic L-functions, before concluding our very barebones overview of the subject. It is known that for 1 i n, by writting N F p pk for some k, ď ď Q p q “ 1 log |α pp, iq| ď p π 2 for all finite primes, and further for unramified primes 1 log |α pp, iq| “ p π 2 However, even more is conjectured to be true. Ramanujan-Petersson Conjecture. At all finite primes,

|απpp, iq| ď 1 with equality when p is unramiﬁed. Here, we only quoted the conjecture for ﬁnite primes, as those are the ones we are mainly interested in in this thesis.

29 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

2.4.2 Strong Artin Conjecture We noted previously in Example 2.4.1 that automorphic L-functions are non-abelian generalisations of Dirichlet L-functions. However, recall also that abelian Artin L-functions are also Dirichlet L-functions. If automorphic L-functions are to truly be a generalisation of Dirichlet L-function in a sensible way, we expect some relation between Artin L-functions and automorphic L-functions. A more precise statement is the following.

Strong Artin Conjecture. Let K{F be a ﬁnite Galois extension, with GalpK{F q “ G. Let ρ be an n-dimensional representation of G. Then, there exists an automorphic representation πpρq on GLnpAF q such that the L-functions Lps, ρ; K{F q “ Lps, πpρqq except at a ﬁnite number of primes. Moreover, if ρ is irreducible, then π is cuspidal.

In fact, we have a stronger conjecture, by combining with the following theorem.

Theorem 2.4.2. Let π be cuspidal automorphic on GLnpAF q, and ρ a n-dimensional representation of G. If for all but ﬁnitely many primes, we have that

Lpps, ρ; K{F q “ Lps, πpq then, we have that Lps, ρ; K{F q “ Lps, πq. The assumption of Strong Artin Conjecture is what allowed Pierce et al. to justify the usage of the results of Kowalski and Michel in [KM02], originally for the setting of automorphic L-functions, in the setting of Artin L-functions in [PTBW17]. The Strong Artin Conjecture was also used to get a convexity bound on Artin L-functions, by transfering known convexity bounds for automorphic L-functions via this conjecture. We conclude this section by quoting [PTBW17], and note that the Strong Artin Conjecture has been veriﬁed for the following cases.

• One-dimensional representations, by Artin [Art31b].

• Nilpotent Galois extensions, by Arthur and Clozel [AC16].

• A4, by Langlands [Lan80].

• S4, by Tunnell [Tun81]. • Dihedral groups, by Langlands [Lan80].

2.5 Convexity Bound on Artin L-Functions

We conclude this brief primer on Artin L-functions with a discussion regarding its convexity bound. We first need a standard fact in complex analysis. In short, we need a version of the maximus modulus principle for bounding holomorphic functions on vertical strips. This generalisation we needed is the various versions of the Phragmén–Lindelöfprinciple, which has been covered extensively in many texts on complex analysis such as [Lan13b]. We will quote the following version, regarding the bound of a holomorphic function on vertical strips.

30 CHAPTER 2. A BRIEF OVERVIEW OF ARTIN L-FUNCTIONS

Phragm´en–Lindel¨ofPrinciple. Let f be continuous on the strip

S “ t s P C; σ1 ď Repsq ď σ2 u with f being holomorphic on the interior and satisfying the ﬁnite order inequality on S, i.e. there exists some λ ą 0 such that λ fpsq “ Ope|s| q If there exists an M ą 0 such that M fpσi ` ıtq “ Op|t| q then we have that for all s “ σ ` ıt P S,

fpsq “ Op|t|M q

We wish to point out that the implied constant depends only on M. A consequence of Phragm´en–Lindel¨ofprinciple is the following convexity result.

Theorem 2.5.1. Let f be holomorphic on the strip

S “ t s P C; σ1 ď Repsq ď σ2 u and satisfying the ﬁnite order inequality on S, i.e. there exists some λ ą 0 such that

λ fpsq “ Ope|s| q

Suppose f grows at most like a power of |t| on Repsq “ σi, for both i “ 1, 2, and let ψpσq be the least number such that for s “ σ ` ıt, ψpσq`ε fpsq “ Oεp|t| q

Then, ψ is convex in rσ1, σ2s. Note that through the proof of the theorem, the implied constant is the maximum of the constant implied on the side of the strip. Further, we can ﬁnd a constant which allows one to bound the function along the vertical strip Repsq “ σ. We can now describe a convexity bound on Artin L-function.

Lemma 2.5.2. For any Artin L-function Lps, ρ; K{F q which is holomorphic, we get that for any ε ą 0, there exists a constant Cpdim ρ, εq such that

1´Repsq `ε |Lps, ρ; K{F q| ď Cpdim ρ, εq Apρ; K{F qp|t| ` 2qdim ρ 2

Proof. By considering the Dirichlet series of Lps, ρ`; K{F q for Repsq “ 1 `˘2ε, we get that Lps, ρ; K{F q is bounded along the strip Repsq “ 1 ` 2ε. Recalling the Stirling formula (see for example [IK04]) and the functional equation, we get that there exists a constant C1pdim ρ, εq such that on the strip Repsq “ ´2ε,

1 `2ε |Lps, ρ; K{F q| ď C1pdim ρ, εq Apρ; K{F qp|t| ` 2qdim ρ 2

Thus, by the convexity corollary of Phragm´en–Lindel¨ofprinciple,` we get the desired˘ convexity result.

31 Chapter 3

Kowalski–Michel Ameanable Families

In this chapter, we will deﬁne and study families of Artin L-functions which satisfy certain technical hypothesis. We will then follow the spirit of Kowalski and Michel in [KM02] in showing that such families admit analytical results about their zeroes akin to those obtained by [KM02] for families of cuspidal automorphic L-functions. In particular, we do not assume the Strong Artin Conjecture in the following, choosing instead to (re)prove all the analytical results by hand.

3.1 Families of Artin L-Functions

In this section, we will describe a special class of family of Artin L-functions. This special class will be the families such that the analytical method of Kowalski–Michel regarding the zero-region of the family is applicable, in the case of Artin L-functions instead of the case of cuspidal automorphic L-functions as was considered in [KM02].

3.1.1 Definition of Families of Artin L-Functions Following [KM02] and [PTBW17], we deﬁne the following.

Deﬁnition 3.1.1. A family of Artin L-functions of degree m is a set

F “ t FpXq; X P N u

where for each X P N, FpXq is a ﬁnite collection of Artin L-functions associated to m-dimensional representations satisfying:

• There exists constants a ą 0 and Ccond such that for every X P N,

a Apρ; K{F q ď CcondX

for all Lps, ρ; K{F q P FpXq.

• There exists constants d ą 0 and Corder such that for every X P N,

d |FpXq| ď CorderX

We say an Artin L-function Lps, ρ; K{F q belongs to F, denoted Lps, ρ; K{F q P F, if Lps, ρ; K{F q P FpXq for some X P N. On first glance, the above definition seems to differ slightly from the definitions of [KM02] and Condition 6.1 of [PTBW17]. Specifically, comparing with Condition 6.1 of [PTBW17], the property that the family satisfies Ramanujan-Petersson Conjecture and a uniform convexity bound is missing from our

32 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES definition. However, Remark 2.2.14 and 2.2.20 shows that Artin L-functions satisfies Ramanujan- Petersson Conjecture, and so that condition could be dropped from the definition of a family. The uniform convexity bound, on the other hand, will be handled later. We also note the usage of a finite collection instead of a finite set. We wish to point out that it could be the case that for two fields K{F and K1{F , not necessarily Galois but with the same Galois group, their Artin L-functions associated to some representation ρ might be the same. Here is an example, noted by [PTBW17]. Example 3.1.1. Recall that two fields K,K1 are arithmetically equivalent if their Dedekind zeta- 1 functions ζK “ ζK1 . Of course, this does not happen in the Galois setting, i.e. K{Q and K {Q are Galois, which is the setting we will later be in. We still maintain our definition of a family, to allow for such generality. As we are mainly interested in the zero-region of Artin L-functions, we will define a zero-counting function, both on the level of individual Artin L-functions and on the level of families. For an Artin L-function Lps, ρ; K{F q, we define

Zpα, T, ρ; K{F q “ t s P C; Repsq ě α, | Impsq| ď T,Lps, ρ; K{F q “ 0 u and Npα, T, ρ; K{F q “ ord Lps, ρ; K{F q s“s0 s PZpα,T,ρ;K{F q 0 ÿ Note in particular that Npα, T, ρ; K{F q counts the zeroes with multiplicities. On the level of families, we deﬁne NF,α,T to be the function on N deﬁned by

NF,α,T pXq “ Npα, T, ρ; K{F q Lps,ρ;K{F qPFpXq ÿ The following is the major example of this thesis. Example 3.1.2. Let G be a ﬁnite group, and let

FF,G “ t FF,GpXq; X P N u where for each X P N, F FF,GpXq “ ζK psq; K{F Galois, GalpK{F q “ G, NQ p∆K{F q ă X

• For any ζK psq “ Lps, reg G; K{F q P FF,GpXq, we get by the Conductor-Discriminant( Formula that the conductor of ζK psq is

dim regG F f rK:F s F |G| ApregG; K{F q “ |∆F {Q| NQ p pregG; K{F qq “ |∆F {Q| NQ p∆K{F q ď |∆F {Q| X

• For every X P N, we get that Gal |FF,GpXq| ď NF,|G|pXq Gal where NF,n pXq is the number of Galois extensions of F of degree n with absolute discriminant at most X, i.e. Gal F NF,n pXq :“ K{F Galois; rK : F s “ n, NQ p∆K{F q ă X By a well-known result of Schmidtˇ [Sch95], we get that (ˇ ˇ ˇ |G|`2 Gal F 4 |FF,GpXq| ď NF,|G|pXq ď NF,|G|pXq :“ K{F ; rK : F s “ n, NQ p∆K{F q ă X ď CpF, |G|qX where CpF, nq is a constant dependingˇ on F and n. (ˇ ˇ ˇ 33 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

|G| |G|`2 With the constants a “ 1, Ccond “ |∆F {Q| , d “ 4 and Corder “ CpF, |G|q, this makes FF,G a family of Artin L-functions of degree |G|. To conclude, we will deﬁne the notion of a subfamily.

Deﬁnition 3.1.2. Let F be a family of Artin L-functions of degree m. A set

H “ t HpXq; X P N u

is a subfamily of F if for all X P N, we have that

HpXq Ď FpXq

Note that it is clear by deﬁnition that a subfamily is also a family of Artin L-functions of the same degree, with possibly tighter parameters.

3.1.2 Kowalski–Michel Ameanable Families The deﬁnition of family of Artin L-functions is not strong enough to have the analytical properties needed for the application of Kowalski–Michel type estimate of the zeroes of the family. Here, we will deﬁne a special class of families which we will show have the suitable analytical properties to apply Kowalski–Michel type estimates.

Definition 3.1.3. Let F “ tFpXq; X P Nu be a family of Artin L-functions of degree m, and let K be a number field. F is said to be Kowalski–Michel K-ameanable if for each X P N, FpXq is a finite collection of holomorphic Artin L-functions associated to m-dimensional representations such that

• For every Lps, ρ; K{F q P F, we have that K Ď F , with the same rF : Ks. We denote this by rF : Ks.

•F satisfy uniform convexity bound: For every ε ą 0, there exists a constant Cconvexpεq such that for all Lps, ρ; K{F q P F, Lps, ρ; K{F q satisfy the convexity bound

1´Repsq `ε K K K dim ρ 2 |Lps, ρ ; K {Kq| ď Cconvexpεq Apρ; K {Kqp|t| ` 2q ´ ¯ for 0 ď Repsq ď 1. •F is well-behaved under convolution over K, that is

– For Lps, ρ; K{F q,Lps, τ; K1{F 1q P F, their convolution over K

K K 1 L s, ConvKpρ, τq; K K {k ´ ¯ where ConvKpρ, τq is given by

K K 1 GalpK {KqˆGalpK {Kq K ˚K ConvKpρ, τq “ Res K K ρ b τ GalpK K1 {Kq

is entire if and only if K ‰ K1 and ρ ﬂ τ, with the only pole at s “ 1 of the same order when K “ K1, F “ F 1 and ρ – τ.

34 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

1 1 1 – for every ε ą 0, there exists a constant Cconvexpεq such that for all Lps, ρ; K{F q,Lps, τ; K {F q P F, their convolution over K satisfy the convexity bound

1´Repsq K K K K 2 2 2 `ε 1 1 1 m rF:Ks L s, ConvKpρ, τq; K K {K ď Cconvexpεq A ConvKpρ, τq; K K {K p|t| ` 2q ˇ ´ ¯ˇ ´ ´ ¯ ¯ ˇ ˇ ˇfor 0 ď Repsq ď 1. ˇ We also deﬁne the tuple of data attached to such families.

Deﬁnition 3.1.4. Let F be a Kowalski–Michel K-ameanable family of Artin L-function of degree m. Its Kowalski–Michel K-family parameters is the tuple

1 pm, K, a, Ccond, d, Corder,Cconvex,Cconvexq

1 where Cconvex,Cconvex are thought of as functions in ε ą 0. We also denote κpFq to be the order of the K K 1 residue at s “ 1 of any L s, ConvKpρ, ρq; K K {k P F. ´ ¯ Note that the property of well-behavedness under convolution corresponds to the behavior under (unramiﬁed) Rankin-Selberg of automorphic L-function used in [KM02]. Indeed, under the assumption of the Strong Artin Conjecture, the convolution we described is the corresponding Rankin-Selberg of the automorphic L-function on the level of Galois representation, via functoriality of the correspondence. Since [PTBW17] assumes Strong Artin Conjecture, this property of being well-behaved under convolution can be safely dropped. However, we do not assume Strong Artin Conjecture, so we can only produce Kowalski–Michel estimates for families that do have this property. We also note that the uniform convexity bounds was used by [PTBW17] to replace the assumption of [KM02] that all members of the family having the same gamma factor at inﬁnity. [PTBW17] notes that that since the assumption of [KM02] is only used to establish a uniform convexity bound, there is no harm in assuming that directly instead. We will follow the cue of [PTBW17] in this regard. Here, we will highlight a non-example of a Kowalski–Michel ameanable family that is closely related to a family to be introduced later that is Kowalski–Michel ameanable.

Example 3.1.3. Consider the family FF,G in Example 3.1.2. Now, since each ζLpsq P FF,G is not entire, FF,G can not be Kowalski–Michel ameanable. But this is really the only real technical obstruction to applying Kowalski–Michel type estimate. We will introduce a family in Chapter 4 as a ﬁx to this obstruction, and thus have an example of a Kowalski–Michel ameanable family.

We also highlight the following straightforward result.

Proposition 3.1.5. Let F be a Kowalski–Michel K-ameanable family. Then, any subfamily of F is also Kowalski–Michel K-ameanable with the same K-family parameters.

We conclude by proving the following lemma, which will allow us to reduce the results of special cases to the case of Kowalski–Michel Q-ameanable families. Lemma 3.1.6. Let K be a number ﬁeld, and let F be a Kowalski–Michel K-ameanable family of Artin L- 1 function of degree n, with Kowalski–Michel K-family parameters pm, K, a, Ccond, d, Corder,Cconvex,Cconvexq. Then, for any number ﬁeld K Ď K such that K{K is Galois, F is also Kowalski–Michel K-ameanable, 1 rK:Ks with Kowalski–Michel K-family parameters pm, K, a, Ccond, d, Corder,Cconvex, pCconvexq q.

35 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

Proof. The uniform convexity bound condition on each member of F is a direct consequence of inductivity and inﬂation of Artin L-function and its conductor. Thus the only thing we needed to check is that F is 1 rK:Ks well-behaved under convolution over K, with the same convexity bound constant pCconvexq . • For Lps, ρ; K{F q,Lps, τ; K1{F 1q P F, by Proposition 2.2.32 we get that

K K GalpK K1 {Kq ConvKpρ, τq “ rK : KsInd K K ConvKpρ, τq GalpK K1 {Kq

By noting that since K{K is Galois, K K K “ K and by inductivity and additivity of Artin L-function, we get that

K K K K rK:Ks 1 1 L s, ConvKpρ, τq; K K {K “ L s, ConvKpρ, τq, K K {K ´ ¯ ´ ¯ is entire if and only if K ‰ K1 and ρ ﬂ τ, with the only pole at s “ 1 when K “ K1, F “ F 1 and ρ – τ.

1 1 1 • for every ε ą 0, there exists a constant Cconvexpεq such that for all Lps, ρ; K{F q,Lps, τ; K {F q P F, their convolution over K satisfy the convexity bound

K K 1 L s, ConvKpρ, τq; K K {K

ˇ ´ K ¯ˇ K rK:Ks ˇ ˇ1 ˇ “ L s, ConvKpρ, τq, K Kˇ {K 1´Repsq ˇ ´ ¯ˇ 2 `ε K K rK:Ks 2 2 ˇ 1 rK:Ks ˇ 1 m rF:Ks rK:Ks ď Cˇ convexpεq A ConvKpρ, τˇq; K K {K p|t| ` 2q ˆ ˙ ´ ¯ 1´Repsq K K 2 2 2 `ε 1 rK:Ks 1 m rF:Ks ď Cconvexpεq A ConvKpρ, τq; K K {K p|t| ` 2q ´ ´ ¯ ¯ for 0 ď Repsq ď 1. We thus get that F is also Kowalski–Michel K-ameanable, with Kowalski–Michel K-family parameters 1 rK:Ks pm, K, a, Ccond, d, Corder,Cconvex, pCconvexq q.

3.2 Components of Kowalski–Michel Estimate

We now introduce pre-requistes needed to deduce the Kowalski–Michel estimate for the zero-region of Kowalski–Michel Q-ameanable families. In this section, we set F a Kowalski–Michel Q-ameanable family. For every member Lps, ρ; K{F q P F, by inﬂation and inductivity of Artin L-functions we get that

Q Q GalpK {Qq Q Lps, ρ; K{F q “ Lps, ρ ˝ π Q ; K {F q “ L s, Ind ρ ˝ π Q ; K { GalpK {Kq GalpKQ{Kq GalpK {Kq Q ˆ ˙ Q where K is the Galois closure of K over Q. Thus, for simplicity, we think of F as a degree m “ degpFqrF : Qs family, and every member of F will be written as Lps, ρ; F {Qq with F {Q being Galois. We also get that GalpF {QqˆGalpF 1{Qq ˚ Conv pρ, τq “ Res 1 ρ b τ Q GalpFF {Qq

36 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

3.2.1 Dirichlet Series Expansion Let Lps, ρ; F {Qq P F. Consider its Dirichlet series expansion

8 ´s Lps, ρ; F {Qq “ λnpρqn n“1 ÿ For a general Dirichlet series 8 ´s fpsq “ ann n“1 ÿ we denote rnsfpsq :“ an which is a notation adapted from enumerative combinatorics. In particular,

rnsLps, ρ; F {Qq :“ λnpρq

We note that since Lps, ρ; F {Qq is defined via an Euler product, r¨sLps, ρ; F {Qq : N Ñ C is a multiplicative arithmetic function. There are minor issues with rpsLps, ρ; F {Qq for p Apρ; F {Qq, that is the primes which ramifies in F . To overcome these, we define the following.

Definition 3.2.1. The unramified Dirichlet series associated to Lps, ρ; F {Qq is the Dirichlet series fps, ρq such that rnsLps, ρ; F {Qq, if gcdpn, Apρ; F {Qqq “ 1 rnsfps, ρq “ #0, otherwise 1 ˚ GalpF {QqˆGalpF {Qq ˚ We also denote fps, ρ b τ q “ fps, Res 1 ρ b τ q. GalpFF {Qq A direct consequence of the Euler product and the definition of the conductor of Artin L-functions is the following lemma.

Lemma 3.2.2. Let fps, ρq be the unramiﬁed Dirichlet series associated to Lps, ρ; F {Qq. Then

fps, ρq “ Lunramps, ρ; F {Qq As such, Artin L-functions and its unramified Dirichlet series differ only by a finite Euler product which is entire and does not vanish in Repsq ą 0. Hence, we will lose no generality by examining only fps, ρq instead of the full Lps, ρ; F {Qq. In particular,

res fps, ρq “ res Lps, ρ; F {Qq s“1 s“1

Remark 3.2.3: For Lps, ρ, F {Qq, the eigenvalues of ρ˚ are the conjugate of the eigenvalues of ρ, as the latter are roots of unity by Remark 2.2.14. Thus, for any prime p,

˚ ˚ ˚ rpsLps, ρ ; F {Qq “ Trpρ pσpqq “ Trpρpσpqq “ rpsLps, ρ ; F {Qq, if p ﬄ Apρ; F {Qq rpsfps, ρ˚q “ “ rpsfps, ρq #0, otherwise Thus, for squarefree n, by multiplicativity,

rnsfps, ρ˚q “ rnsfps, ρq

37 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

In the following, let us also denote

5 an :“ an “ |µpnq|an n n n ÿ n squarefreeÿ ÿ where µ is the usual Möbiusfunction. We also define, for z ě 1, P pzq :“ p păz ź The following lemma is straighforward. Lemma 3.2.4. Write fps, ρq “ f 5ps, ρqf 7ps, ρq where 5 f 5ps, ρq “ prnsfps, ρqq n´s “ p1 ` prpsfps, ρqqp´sq gcdpn,P pzqq“1 pěz ÿ ź Then, for z ě 1 large enough, we get that for all Lps, ρ; F {Qq P F, f 7ps, ρq is entire and non-vanishing 1 at Repsq ą 2 , and for every ε ą 0, there exists a constant Mpεq ą 0 such that |f 7ps, ρq| ď Mpεq 1 in Repsq ą 2 ` ε. We will now define the final ingredient from this section. We note that the order of the residue at s “ 1 of f 5ps, ρ b ρ˚q is κpFq, and the Laurent expansion of f 5ps, ρ b ρ˚q at s “ 1 is

5 ˚ k f ps, ρ b ρ q “ ckpρqps ´ 1q kě´κpFq ÿ For ϕ a suitable test function, we note that

κpFq κpFq 5 ˚ s k k´1 res f ps ` 1, ρ b ρ qϕpsqx “ c0pρqOp1q ` c´kpρqOplog xq ! plog xq c´kpρqOplog xq s“0 k“1 k“1 ÿ ÿ We therefore denote r κpFq k´1 sϕpρ, xq “ c´kpρqOplog xq k“1 ÿ 3.2.2 Pseudocharacters Kowalski and Michel adapted Selberg’s usage of pseudocharacters to their setting, which they used as a sort of counting device to help deduce their main theorem. We will adapt their results to our setting. We begin by deﬁning pseudocharacters.

Definition 3.2.5. We define a function ψρ : N Ñ R associated to Lps, ρ; F {Qq by µpnqn |rnsfps,ρq|2 , if gcdpn, Apρ; L{Qqq “ 1 ψρpnq “ #0, otherwise and we define a pseudocharacter associated to Lps, ρ; F {Qq and integer n to be the function ψρ,n : N Ñ R given by 2 ψρ,nprq “ µprq ψρpgcdpn, rqq

38 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

We note that since ψρpnq is a multiplicative arithmetic function, the ψρ,n are indeed the pseudocharacter as deﬁned in [Jut78]. 1 Following [KM02], we also deﬁne, for 0 ă δ ă 2 , the following subset of integers

´δ Rδ,zpρq “ r P N; r squarefree, gcdpr, Apρ; F {QqP pzqq “ 1, such that p r ùñ rpsfps, ρq| ą p Ď N We will need a result about the abundance of integers in R ρ . ˇ ( δ,zp q ˇ 1 Lemma 3.2.6. Fix z ě 1 large enough and 0 ă δ ă 2 . There exists a constant M such that for Lps, ρ; F {Qq P F, we get that 1 ď Ms pρ, Rq log R |ψ prq| ϕ rďR ρ rPRÿδ,zpρq for R ě 2. Further, let X P N be such that for Lps, ρ; F {Qq P FpXq. For C ą ma, there exists a constant M 1pCq for which we get that 1 ě M 1pCqs pρ, Rq log R |ψ prq| ϕ rďR ρ rPRÿδ,zpρq for R ą XC . Proof. First, note that for r squarefree, 1 |rrsfps, ρq|2 “ |ψρprq| r Also, for primes p, by Remark 3.2.3,

|rpsfps, ρq|2 “ prpsfps, ρqqprpsfps, ρ˚qq “ rpsfps, ρ b ρ˚q Hence, since r¨sfps, ρq are multiplicative arithmetic functions, we get that 1 rrsfps, ρ b ρ˚q “ |ψ prq| r rďR ρ rďR rPRÿδ,zpρq rPRÿδ,zpρq Now, note that 1 rrsfps, ρ b ρ˚q 5 rrsfps, ρ b ρ˚q “ ď |ψ prq| r r rďR ρ rďR rďR rPRÿδ,zpρq rPRÿδ,zpρq ÿ Using a suitable test function ϕ, we estimate using the usual Mellin transform, 1 res f 5ps ` 1, ρ b ρ˚qϕpsqxs ` f 5ps ` 1, ρ b ρ˚qϕpsqxs ds s“z 2πı 1 σ1ăRe zăσ żpσ q ÿ ` ˘ 1 But since F is Kowalski–Michel Q-ameanable,r Lps, ConvQpρ, ρq; FF {Qq has ar pole only at s “ 1 and satisfy a uniform convexity bound, and so does f 5ps, ρ b ρ˚q. In particular, the integral can be bounded independent of Lps, ρ; F {Qq within F. Therefore, with σ1 ă 0, 1 ď s pρ, Rq log R ` Op1q |ψ prq| ϕ rďR ρ rPRÿδ,zpρq

39 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

with the bound on the Op1q factor being independent of individual Lps, ρ; F {Qq within F. This gives us the ﬁrst result. For the second assertion, we note that

1 ` |rpsfps, ρq|2p´s ď 1 ` p´2δ´s

pRRδ,zpρq pěz źpěz ` ˘ pﬄAźpρ;F {Qq ` ˘ pﬄApρ;F {Qq |rpsfps,ρq|ďp´δ

converges for Repsq ą 1 ´ 2δ. So by letting

5 F ps, ρq “ |rrsfps, ρq|2r´s rPR ÿδ,z we get that f 5ps, ρ b ρq “ F ps, ρq 1 ` |rpsfps, ρq|2p´s

pRRδ,zpρq źpěz ` ˘ pﬄApρ;F {Qq Now, for ε ą 0 the uniform convexity bound and Corollary 2.2.31 will put a bound on f 5ps, ρ b ρq on ma 2δ´ε` pC´maqp2δ´εq the line Repsq “ 1 ´ 2δ ` ε of size of magnitude X p ma q “ XCp2δ´εq with implied constant pC´maqp2δ´εq depending on ma . We estimate using Mellin transform to get 1 rrsfps, ρ b ρ˚q “ ě M 1pCqs pρ, Rq log R |ψ prq| r ϕ rďR ρ rďR rPRÿδ,zpρq rPRÿδ,zpρq

as long as R ą XC . The following lemma of [Jut78] about pseudocharacters has been adapted to our setting.

1 ρ,τ Lemma 3.2.7. Let Lps, ρ; F {Qq,Lps, τ; F {Qq P F, and let r P Rδ,zpρq, t P Rδ,zpτq. Deﬁne hr,t to be such that

8 ρ,τ ´s ´s ´s ´s hr,t pdqd “ 1 ` pψρppq ´ 1qp 1 ` pψτ ppq ´ 1qp 1 ` pψρppqψτ ppq ´ 1qp d“1 pr pt pgcdpr,tq ÿ źpﬄt ` ˘ źpﬄr ` ˘ ź ` ˘ Then, for n ě 1, we get that 2 ρ,τ ψρ,rpnqψτ,tpnq “ µpnq hr,t pdq dn ÿ If F “ F 1 and τ – ρ˚, then

8 ρ,ρ˚ 2 ´1 hr,t pdqηρpdq|rdsfps, ρq| d “ δr,t|ψρprq| d“1 ÿ where δ¨,¨ is the usual Kronecker delta function and

2 ´1 ´1 ηρpdq “ 1 ` |rpsfps, ρq| p pd ź ` ˘

40 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

Proof. We ﬁrst note that

8 ρ,τ ´s ζpsq hr,t pdqd d“1 ÿ ´s ´s ´s ´1 1 ` pψ ppq ´ 1qp 1 ` pψ ppq ´ 1qp 1 ` pψ ppqψ ppq ´ 1qp “ 1 ´ p´s ρ τ ρ τ 1 ´ p´s 1 ´ p´s 1 ´ p´s pfflr pr pt pgcdpr,tq źpfflt ` ˘ źpfflt źpfflr ź 8 8 8 ´s ´1 k ´s k ´s k ´s “ 1 ´ p 1 ` ψρppq pp q 1 ` ψτ ppq pp q 1 ` ψρppqψτ ppq pp q pfflr pr ˜ k“1 ¸ pt ˜ k“1 ¸ pgcdpr,tq ˜ k“1 ¸ źpfflt ` ˘ źpfflt ÿ źpfflr ÿ ź ÿ 8 ´s “ ψρppq ψτ ppq n ¨ ˛ ¨ ˛ n“1 pgcdpn,rq pgcdpn,tq ÿ ź ź ˝ ‚˝ ‚ 8 2 2 ρ,τ ´s 2 ρ,τ ùñ ψρ,rpnqψτ,tpnq “ µpnq ψρppq ψτ ppq “ µpnq rns ζpsq hr,t pdqd “ µpnq hr,t pdq pgcdpn,rq pgcdpn,tq ˜ d“1 ¸ dn ź ź ÿ ÿ 1 ˚ ρ,ρ˚ Secondly, when F “ F and τ – ρ , notice that hr,t pdq ‰ 0 only for square-free d and that for primes p, by Remark 3.2.3

´p ´p |rpsfps,ρ˚q|2 “ |rpsfps,ρq|2 , if gcdpn, Apρ; L{Qqq “ 1 ψρ˚ ppq “ “ ψρppq #0, otherwise Thus, at s “ 1,

1 Proposition 3.2.8. Fix z ě 1 large enough and 0 ă δ ă 2 , and let F be a Kowalski–Michel Q-ameanable 2 family. Let N ě R . Then, there exists a function GF pX,N,Rq such that for ε ą 0 satisfying

1 ´ε d` ma 2`2δ N 2 ą X 2 R log R there exists a constant Mpεq such that for an P C,

2 1 1 2 anψρ,rpnqprnsfps, ρqq ď MpεqNGF pX,N,Rq |an| sϕpρ, Rq |ψρprq| ˇ ˇ Lps,ρ;F {QqPFpXq rďR ˇn„N ˇ n„N ÿ rPRÿδ,zpρq ˇ ÿ ˇ ÿ ˇ ˇ ˇ ˇ 41 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

Proof. The proof is essentially as in [KM02], which we will detail here for completeness. By considering the linear operator

ψρ,rpnqprnsfps, ρqq Q : panqn„N ÞÑ an ˜n„N sϕpρ, Rq|ψρprq| ¸ Lps,ρ;F {QqPFpXq ÿ rPRδ,zpρq with rďR a of ﬁnite-dimensional Hilbert spaces, the inequality is equivalent to

}Q}2 ď MpεqN

By general Hilbert space theory, we get that the norm of T is the same as the norm of the conjugate of its adjoint, given by

ψρ,rpnqprnsfps, ρqq pbρ,rq Lps,ρ;F {QqPFpXq ÞÑ ¨ bρ,r ˛ rPR pρq with rďR sϕpρ, Rq|ψρprq| δ,z Lps,ρ;F {QqPFpXq rďR ˚ ÿ rPRÿδ,zpρq ‹ ˚ ‹n„N ˝ a ‚ Thus, the inequality is now equivalent to

ψρ,rpnqprnsfps, ρqq 2 ˇ bρ,r ˇ ď MpεqN |bρ,r| ˇ sϕpρ, Rq|ψρprq| ˇ n„N ˇLps,ρ;F {QqPFpXq rďR ˇ Lps,ρ;F {QqPFpXq rďR ÿ ˇ ÿ rPRÿδ,zpρq ˇ ÿ rPRÿδ,zpρq ˇ ˇ ˇ a ˇ ˇ ˇ By expandingˇ the sum on the left hand side, we get thatˇ 2

ψρ,rpnqprnsfps, ρqq ˇ bρ,r ˇ ˇ sϕpρ, Rq|ψρprq| ˇ n„N ˇLps,ρ;F {QqPFpXq rďR ˇ ÿ ˇ ÿ rPRÿδ,zpρq ˇ ˇ ˇ 8ˇ a ˇ ˇ bρ,rbτ,t ˇ n ď ˇ ˇ ψρ,rpnqψτ,tpnqprnsfps, ρqqprnsfps, τqqϕ sϕpρ, Rqsϕpτ, Rq|ψρprqψτ ptq| N n“1 Lps,ρ;F {QqPFpXq r,tďR 1 ´ ¯ ÿ Lps,τ;Fÿ{QqPFpXq rPRÿδ,zpρq tPRδ,zpτq a b b “ ρ,r τ,t Spρ, τ, r, tq sϕpρ, Rqsϕpτ, Rq|ψρprqψτ ptq| Lps,ρ;F {QqPFpXq r,tďR Lps,τ;Fÿ1{QqPFpXq rPRÿδ,zpρq tPRδ,zpτq a

1 where θ : Rě0 Ñ r0, 1s is a suitable smooth test function with compact support in r 2 , 3s and θpxq “ 1 for x P r1, 2s, and 8 n Spρ, τ, r, tq :“ ψ pnqψ pnqprnsfps, ρqqprnsfps, τqqθ ρ,r τ,t N n“1 ÿ ´ ¯ We note that by deﬁnition, ψρ,rpnq and ψτ,tpnq are zero if n squarefree or pn, P pzqq ‰ 1. Along with

42 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

Lemma 3.2.7 and Remark 3.2.3, we get that

8 n Spρ, τ, r, tq “ ψ pnqψ pnqprnsfps, ρqqprnsfps, τqqθ ρ,r τ,t N n“1 ´ ¯ ÿ 5 n “ ψ pnqψ pnqprnsfps, ρqqprnsfps, τ ˚qqθ ρ,r τ,t N ně1 gcdpn,Pÿpzqq“1 ´ ¯

5 ˚ n “ hρ,τ pdq rnsfps, ρ b τ ˚qθ r,t N ně1 ˜dn ¸ gcdpn,Pÿpzqq“1 ÿ ´ ¯

5 ρ,τ ˚ 5 nd “ h pdqprdsfps, ρqqprdsfps, τ ˚qq ¨ rnsfps, ρ b τ ˚qθ ˛ r,t N dďN gcdpn,dq“1 ˆ ˙ gcdpd,Pÿpzqq“1 ˚gcdpn,Pÿpzqq“1 ‹ ˚ ‹ ˝ ‚ Let us set 5 n T pxq “ rnsfps, ρ b τ ˚qθ x gcdpn,dq“1 gcdpn,Pÿpzqq“1 ´ ¯ To estimate the above, we consider the Dirichlet series

ρ,τ ˚ 5 ˚ ´s 5 ˚ ˚ ´s ´1 gd,z psq “ rnsfps, ρ b τ qn “ f ps, ρ b τ q 1 ` rpsfps, ρ b τ qp gcdpn,dq“1 pd gcdpn,Pÿpzqq“1 źpěz ` ˘ Using Mellin transform, we get that

ρ,τ ˚ s 1 ρ,τ ˚ s T pxq “ res gd,z psqθpsqx ` gd,z psqθpsqx ds s“z 2πı 1 σ1ăRe ză3 pσ q ÿ ´ ¯ ż 1 1 r r ρ,τ ˚ where 2 ă σ ă 1. Since F is Kowalski–Michel Q-ameanable, we get that the only pole of gd,z psq is at s “ 1 if τ – ρ and F “ F 1, and no poles otherwise. In the former case,

ρ,τ ˚ s 5 ˚ ˚ ´s´1 ´1 s res gd,z psqψpsqx “ res f ps ` 1, ρ b τ q 1 ` rpsfps, ρ b τ qp θps ` 1qx x s“1 s“0 ¨ ˛ pd ´ ¯ źpěz ` ˘ r ˚ r ‹ ˝ κpFq ‚ k´1 ď xηρpdq c´kpρqOplog xdq k“1 ÿ where the constants depends on θ and d. We denote

κpFq k´1 gθpρ, d, xq “ c´kpρqOplog xdq k“1 ÿ We have therefore that

1 ρ,τ ˚ s ds T pxq “ δρ,τ δF,F 1 ηρpdqxgθpρ, d, xq ` gd,z psqψpsqx 2πı 1 s żpσ q r 43 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

1 To bound the integral, we use the unifrom convexity bound on Lps, ConvQpρ, τq,FF {Qq on Repsq “ 1 1 σ “ 2 ` ε, giving us a constant Kpεq such that

1 `ε m2 4 |fps, ρ b τ ˚q| ď Kpεq A pρ b τ ˚; FF 1{Qq p|t| ` 2q ´ ¯ By Corollary 2.2.31,

˚ 1 ˚ 1 m 2m 2ma A pρ b τ ; FF {Qq ď pApρ; F {QqApτ ; F {Qqq ď CcondX so we get that a constant K1pεq such that

˚ 1 ma |fps, ρ b τ q| ď K pεqX 2

1 1 7 ˚ 2 Thus, on Repsq “ σ “ 2 ` ε, and bounding f ps, ρ b τ q by Lemma 3.2.4, there is a constant K pεq such that ρ,τ ˚ ma 2 2 |gd,z psq| ď K pεqX So, there exists a constant K3pεq such that

1 ρ,τ ˚ ds ma 1 s 3 2 2 `ε gd,z psqψpsqx ď K pεqX x 2πı 1 s żpσ q Substituting T , and denoting r N gθ pρ, Nq “ max gθ ρ, d, dďN d ˆ ˙ we get that Spρ, τ, r, tq is bounded by

5 ˚ N hρ,τ pdqprdsfps, ρqqprdsfps, τ ˚qqT r,t d dďN ˆ ˙ gcdpd,Pÿpzqq“1

5 ρ,τ ˚ ´1 “ δρ,τ ˚ δF,F 1 Ngθ pρ, Nq hr,t pdqηρpdqprdsfps, ρqqprdsfps, τ qqd ` Err dďN gcdpd,Pÿpzqq“1 where there exists a constant Cpεq such that the error term is at most

ρ,τ ˚ ˚ 1 `ε ma 5 |hr,t pdqprdsfps, ρqqprdsfps, τ qq| Err ď CpεqN 2 X 2 ? dďN d gcdpd,Pÿpzqq“1 ρ,τ ˚ ˚ 1 `ε ma |hr,t pdqprdsfps, ρqqprdsfps, τ qq| ď CpεqN 2 X 2 ? dďN d gcdpd,Pÿpzqq“1

44 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

ρ,τ ˚ By deﬁnition of hr,t pdq and since r P Rδ,zpρq, t P Rδ,zpτq, we get that

˚ |hρ,τ pdqprdsfps, ρqqprdsfps, τ ˚qq| r,t ? dďN d gcdpd,Pÿpzqq“1 ˚ ´s ˚ ˚ ´s ! 1 ` |ψρppq ´ 1||prpsfps, ρqqprpsfps, τ qq|p 1 ` |ψτ ppq ´ 1||prpsfps, ρqqprpsfps, τ qq|p pr pt ź ź ` ˚ ˚ ˘ ` ˚ ˘ ! p|ψρppqprpsfps, ρqqprpsfps, τ qq|q p|ψτ ppqprpsfps, ρqqprpsfps, τ qq|q pr pt ź ź p p ! |rpsfps, τ ˚q| |rpsfps, ρq| ! r1`δt1`δ ď R2`2δ |rpsfps, ρq| |rpsfps, τ ˚q| pr pt ź ˆ ˙ ź ˆ ˙ Thus, there exists a constant C1pεq such that

1 2`2δ 1 `ε ma Err ď C pεqR N 2 X 2

Meanwhile, the main term for Spρ, τ, r, tq only appears when ρ – τ and F “ F 1, and in this case contributes by Lemma 3.2.7

5 ρ,ρ˚ ˚ ´1 Ngθ pρ, Nq hr,t pdqηρpdqprdsfps, ρqqprdsfps, ρ qqd “ Ngθ pρ, Nq δr,t|ψρprq| dďN gcdpd,Pÿpzqq“1 Finally, returning to

b b ρ,r τ,t Spρ, τ, r, tq sϕpρ, Rqsϕpτ, Rq|ψρprqψτ ptq| Lps,ρ;F {QqPFpXq r,tďR Lps,τ;Fÿ1{QqPFpXq rPRÿδ,zpρq tPRδ,zpτq a • The error term of Spρ, τ, r, tq contribute at most

1 2`2δ 1 `ε ma bρ,rbτ,t C pεqR N 2 X 2 sϕpρ, Rqsϕpτ, Rq|ψρprqψτ ptq| Lps,ρ;F {QqPFpXq r,tďR Lps,τ;Fÿ1{QqPFpXq rPRÿδ,zpρq tPRδ,zpτq a 2

1 2`2δ 1 `ε ma bρ,r “ C pεqR N 2 X 2 ˇ ˇ ˇ sϕpρ, Rq|ψρprq|ˇ ˇLps,ρ;F {QqPFpXq rďR ˇ ˇ ÿ rPRÿδ,zpρq ˇ ˇ ˇ ˇ a ˇ ˇ ˇ ˇ ˇ 1 2`2δ 1 `ε ma 2 1 1 ď C pεqR N 2 X 2 ¨ |bρ,r| ˛ ¨ ˛ sϕpρ, Rq |ψρprq| Lps,ρ;F {QqPFpXq rďR Lps,ρ;F {QqPFpXq rďR ˚ ÿ rPRÿδ,zpρq ‹ ˚ ÿ rPRÿδ,zpρq ‹ ˚ ‹ ˚ ‹ ˝ ‚˝ ‚ where the last inequality as aﬀorded by Cauchy-Schwarz inequality. By Lemma 3.2.6, we get that 1 1 ! |FpXq| log R ! Xd log R sϕpρ, Rq |ψρprq| Lps,ρ;F {QqPFpXq rďR ÿ rPRÿδ,zpρq

45 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

so there is a constant Epεq such that the error term contribute at most

1 `ε ma `d 2`2δ 2 EpεqN 2 X 2 R log R ¨ |bρ,r| ˛ Lps,ρ;F {QqPFpXq rďR ˚ ÿ rPRÿδ,zpρq ‹ ˚ ‹ ˝ ‚ • The main term of Spρ, τ, r, tq contribute only when ρ – τ and r “ t, and in that case gives the contribution gθ pρ, Nq 2 N |bρ,r| sϕpρ, Rq Lps,ρ;F {QqPFpXq rďR ÿ rPRÿδ,zpρq Since FpXq is ﬁnite, we can bound

gθ pρ, Nq ď GF pX,N,Rq sϕpρ, Rq

for some function GF pX,N,Rq. Combining both analysis, we get a constant Mpεq such that 2

ψρ,rpnqprnsfps, ρqq ˇ bρ,r ˇ ˇ sϕpρ, Rq|ψρprq| ˇ n„N ˇLps,ρ;F {QqPFpXq rďR ˇ ÿ ˇ ÿ rPRÿδ,zpρq ˇ ˇ ˇ ˇ a ˇ ˇ bρ,rbτ,t ˇ ď ˇ ˇ Spρ, τ, r, tq sϕpρ, Rqsϕpτ, Rq|ψρprqψτ ptq| Lps,ρ;F {QqPFpXq r,tďR Lps,τ;Fÿ1{QqPFpXq rPRÿδ,zpρq tPRδ,zpτq a 1 `ε ma `d 2`2δ 2 ď Mpεq NGF pX,N,Rq ` N 2 X 2 R log R |bρ,r| Lps,ρ;F {QqPFpXq rďR ´ ¯ ÿ rPRÿδ,zpρq

1 ´ε d` ma 2`2δ The assumption that N 2 ą X 2 R log R ﬁnally gives the result.

Fix appropriate test functions ϕ, θ. Let us focus on this function GF pX,N,Rq that was deﬁned in the proof above. In the case where κpFq “ 1, that is pole at s “ 1 of f 5ps, ρ b ρ˚q for all members Lps, ρbq P F is simple, then GF pX,N,Rq ! 1, and thus we retrieve the result of Kowalski–Michel. In fact, we conjecture that this is true in general. Conjecture 3.2.9. Fix z ě 1 large enough, and let F be a Kowalski–Michel Q-ameanable family. Let 2 N ě R . Then, the function GF pX,N,Rq in Proposition 3.2.8 satisfy

GF pX,N,Rq ! 1 Under the veracity of this conjecture, we note that we will get the corresponding Kowalski–Michel estimate for Kowalski–Michel Q-ameanable family. In other words, we have replaced the reliance of Strong Artin Conjecture with Conjecture 3.2.9. Unfortunately, we could not prove Conjecture 3.2.9 at this moment. Instead, we will assume Conjecture 3.2.9 henceforth. Let us now deﬁne Mpα, T q “ t z P C; Repzq ě α, | Impzq| ď T u and consider for each Lps, ρ; F {Qq P F a ﬁnite subset Jpρq Ď Mpα, T q.

46 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

Deﬁnition 3.2.10. Jpρq is q-well-spaced if for all s0 ‰ s1 P Jρ, 1 | Imps ´ s q| ě 0 1 log q Then, proceeding as in [Mon06, §7], with Proposition 3.2.8 replacing the role of Theorem 2.5 in [Mon06], we get the following corollary of Proposition 3.2.8.

Corollary 3.2.11. For each Lps, ρ; F {Qq P FpXq, let Jpρq be a X-well-spaced ﬁnite subset of Mpα, T q. 1 Fix z ě 1 large enough and 0 ă δ ă 2 . Suppose also that F is Kowalski–Michel Q-ameanable family, and assume Conjecture 3.2.9. Let N ě R2, and suppose there exists ε ą 0 such that

1 ´ε d` ma 2`2δ N 2 ą X 2 R log R

Then, there exists constants Mpεq and B ą 0 such that for an P C, 2 1 1 ´s0 anψρ,rpnqprnsfps, ρqqn sϕpρ, Rq |ψρprq| ˇ ˇ Lps,ρ;F {QqPFpXq s0PJpρq rďR ˇn„N ˇ ÿ ÿ rPRÿδ,zpρq ˇ ÿ ˇ ˇ ˇ B log 2N ˇ 2 1 2α ˇ ď MpεqT logpXNq 1 ` log |a | n ´ log 2R n n„N ˆ ˙ ÿ 3.2.3 A Zero-Detector For each Lps, ρ; F {Qq P F, we denote the set of zeroes of Lps, ρ; F {Qq in Mpα, T q by

Zρpα, T q :“ t s P Mpα, T q; Lps, ρ; F {Qq “ 0 u

Now, suppose we are given for each Lps, ρ; F {Qq P FpXq, a ﬁnite subset Jρpα, T q Ď Zρpα, T q which is X-well-spaced. Then, we wish to bound the size of the set

JX pα, T q :“ Jρpα, T q Lps,ρ;F { qPFpXq ďQ We then have the following lemma regarding the size of the set JX pα, T q. Lemma 3.2.12. Let R ą XC , where C ą ma. Then, we have that there exists a constant MpCq such that 1 1 |JX pα, T q| log R ď MpCq sϕpρ, Rq |ψρprq| Lps,ρ;F {QqPFpXq s0PJρpα,T q rďR ÿ ÿ rPRÿδ,zpρq |J pα, T q| “ MpCq ρ sϕpρ, Rq|ψρprq| Lps,ρ;F {QqPFpXq rďR ÿ rPRÿδ,zpρq Proof. By Lemma 3.2.6, we get that there exists a constant MpCq such that 1 1 1 ě MpCq sϕpρ, Rq log R sϕpρ, Rq |ψρprq| sϕpρ, Rq Lps,ρ;F {QqPFpXq s0PJρpα,T q rďR Lps,ρ;F {QqPFpXq s0PJρpα,T q ÿ ÿ rPRÿδ,zpρq ÿ ÿ

“ MpCq|JX pα, T q| log R

47 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

Now, we wish to leverage Corollary 3.2.11 to bound the sum in the previous lemma, which will in turn help us bound |JX pα, T q|. We will therefore need a sort of function which can “detect” zeroes of Lps, ρ; F {Qq. We begin by deﬁning the following.

Deﬁnition 3.2.13. Let 1 ď z1 ă z2. We deﬁne

µpdq, if d ď z1 log d z2 λz ,z pdq “ $µpdq , if z1 ă d ď z2 1 2 ’ log´ z1 ¯ ’ z2 &0, ´ ¯ otherwise ’ We also denote %’

∆z1,z2 pnq “ λz1,z2 pdq dn ÿ We quote the following result of Graham in [Gra81].

1 Lemma 3.2.14. For 2 ă α ă 1, we get that there exists a constant C such that

log x 2 1´2α z1 2´2α ∆z1,z2 pnq n ď C x ´ z2 ¯ z1ănďx log ÿ z1 ´ ¯ We now deﬁne a condition on a pair of parameters.

1 Deﬁnition 3.2.15. Let X P N, 0 ă δ ă 2 , z, R, x, T ě 1 and 1 ď z1 ă z2. We say that a pair of 1 real numbers pε, αq, with ε ą 0 and 2 ď α ď 1, is pX, δ, z, R, z1, z2, x, T q-suitable if they satisfy the conditions

1 `ε 1 m 1`4δ 2α´1 x " z2X 2 T 2 R log x ď log´ XT ¯

1 1 plog Xq 2 ď log R ď log x 2

We also deﬁne, given pX, δ, z, R, z1, z2, x, T q, a zero-detector, which helps us in bounding each Jρpα, T q.

1 Deﬁnition 3.2.16. Let X P N, 0 ă δ ă 2 , z, R, x, T ě 1 and 1 ď z1 ă z2. For Lps, ρ; F {Qq P FpXq and r P Rδ,zpρq with r ď R, we deﬁne a pX, δ, z, R, z1, z2, x, T q-zero-detector as the function

5 n ´ x ´s zpX,δ,z,R,z1,z2,x,T qps, ρ, rq “ ∆z1,z2 pnqψρ,rpnqprnsfps, ρqqe n 1ănăx gcdpn,Pÿpzqq“1

We now wish to give a lower bound to these zero-detector at the zeroes of Lps, ρ; F {Qq. This involves bounds on the Dirichlet series

5 ´s ∆z1,z2 pnqψρ,rpnqprnsfps, ρqqn gcdpn,P pzqq“1 ÿ The following lemma gives a factorisation of the Dirichlet series.

48 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

Lemma 3.2.17. We get the factorisation

5 ´s 5 z1,z2 ∆z1,z2 pnqψρ,rpnqprnsfps, ρqqn “ f ps, ρqMr ps, ρq gcdpn,P pzqq“1 ÿ where ´s p r p1 ` ψρppqprpsfps, ρqqp q M z1,z2 ps, ρq “ gcdpr,nq λ pnqψ pnqprnsfps, ρqqn´s r p1 ` prpsfps, ρqqp´sq z1,z2 ρ,r gcdpn,P pzqq“1 ś prn ÿ ś Proof. This is rather straightforward: By mutliplicativity of ψρ,r and r¨sfps, ρq, we get that

5 ´s ∆z1,z2 pnqψρ,rpnqprnsfps, ρqqn gcdpn,P pzqq“1 ÿ 5 ´s “ λz1,z2 pdq ψρ,rpnqprnsfps, ρqqn gcdpn,P pzqq“1 ˜dn ¸ ÿ ÿ 5 ´s 5 ´s “ λz1,z2 pdqψρ,rpdqprdsfps, ρqqd ψρ,rpnqprnsfps, ρqqn gcdpd,P pzqq“1 gcdpn,dP pzqq“1 ÿ ÿ ´s ´s “ λz1,z2 pdqψρ,rpdqprdsfps, ρqqd 1 ` ψρ,rppqprpsfps, ρqqp gcdpd,P pzqq“1 gcdpp,dP pzqq“1 ÿ ź ` ˘ (as ψρ,rppq “ 1 for p ﬄ r and ψρ,rppq “ ψρppq for p r) ´s ´s ´s “ λz1,z2 pdqψρ,rpdqprdsfps, ρqqd 1 ` ψρppqprpsfps, ρqqp 1 ` prpsfps, ρqqp gcdpd,P pzqq“1 pěz pěz p r p rd ÿ gcdźpr,dq ` ˘ źﬄ ` ˘ (p r ùñ p ě z) ´s p r p1 ` ψρppqprpsfps, ρqqp q “ f 5ps, ρq gcdpr,dq λ pdqψ pdqprdsfps, ρqqd´s p1 ` prpsfps, ρqqp´sq z1,z2 ρ,r gcdpd,P pzqq“1 ś prd ÿ 5 z1,z2 “ f ps, ρqMr ps, ρq ś

z1,z2 The following lemma will provide a bound on Mr ps, ρq. 1 Lemma 3.2.18. Fix z ě 1 large enough and 0 ă δ ă 2 . Let Lps, ρ; F {Qq P FpXq, r P Rδ,zpρq and 1 ď z1 ă z2. Then, for ε ą 0, there exists a constant Cpεq such that for all s P C with 0 ă Repsq ď 1,

z1,z2 1`2δ´Repsq`ε 1´Repsq`ε |Mr ps, ρq| ď Cpεqr z2 Proof. First, we note that for n not squarefree,

ψρ,rpnq “ 0

Since for n ď z2,

|λz1,z2 pnq| ď 1 and since for n ą z2, λz1,z2 pnq “ 0, we get that

´s 5 p r p1 ` ψρppqprpsfps, ρqqp q M z1,z2 ps, ρq “ gcdpr,nq λ pnqψ pnqprnsfps, ρqqn´s r p1 ` prpsfps, ρqqp´sq z1,z2 ρ,r 1ďnďz2 ś prn gcdpn,Pÿpzqq“1 ś 49 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

´s 5 p r p1 ` ψρppqprpsfps, ρqqp q z1,z2 gcdpr,nq ´ Repsq ùñ |Mr ps, ρq| ď |ψρ,rpnqprnsfps, ρqq| n ˇś ´s ˇ 1ďnďz2 ˇ prn p1 ` prpsfps, ρqqp q ˇ gcdpn,Pÿpzqq“1 ˇ ˇ ˇ ˇ We now analyse the bound of each pieceˇś of the summand. For that,ˇ we recall the familiar arithmetic ˇ ˇ functions τpnq “ 1 and σpnq “ |t p prime; p n u| dn ÿ • Note that for each prime p,

p1´Repsq 1 ` ψ ppqprpsfps, ρqqp´s ď 1 ` ρ |rpsfps, ρq| ˇ ˇ ˇ r ˇ Since r P Rδ,zpρq, for all p gcdpr,nq , 1 p r ùñ ă pδ |rpsfps, ρq| Thus, we get that

´s 1`δ´Repsq 1`δ´Repsq 1`δ´Repsq 1 ` ψρppqprpsfps, ρqqp ď 1 ` p “ d ď τprqr p r prn d r gcdźpr,nq ˇ ˇ ź ` ˘ gcdÿpr,nq ˇ ˇ • Recall that for each (unramiﬁed) prime p, |rpsfps, ρq| is a sum of roots of unity by Remark 2.2.14, 1 hence |rpsfps, ρq| ď dim ρ. Thus, for z large enough, say z ě p2 dim ρq Repsq , we get that for p ě z,

pRepsq ´ dim ρ 1 pRepsq ě 2 dim ρ ùñ ě pRepsq 2

dim ρ pRepsq ´ dim ρ 1 ùñ 1 ` prpsfps, ρqqp´s ě 1 ´ |rpsfps, ρq|p´ Repsq ě 1 ´ “ ě pRepsq pRepsq 2 ˇ ˇ Hence, we getˇ that ˇ 1 n ď 2 “ τprnq “ τprqτ ´s gcdpr, nq p1 ` prpsfps, ρqqp q prn prn ź ˆ ˙ ˇ ˇ ˇś ˇ • Now, note thatˇ if n is squarefree, then ˇ

50 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

Thus, we get for n squarefree that

σ n gcdpr, nq |ψ pnqprnsfps, ρqq| ď pdim ρq p gcdpr,nq q ρ,r |rgcdpr, nqsfps, ρq| τ n 1 δ τ n δ ď pdim ρq p gcdpr,nq q gcdpr, nq ` ď pdim ρq p gcdpr,nq qr gcdpr, nq

ε Combining all these and using the fact that τpnq !ε n , we get that there exists a constant Cpεq such that

´s 5 p r p1 ` ψρppqprpsfps, ρqqp q z1,z2 gcdpr,nq ´ Repsq |Mr ps, ρq| ď |ψρ,rpnqprnsfps, ρqq| n ˇś ´s ˇ 1ďnďz2 ˇ prn p1 ` prpsfps, ρqqp q ˇ gcdpn,Pÿpzqq“1 ˇ ˇ ˇ ˇ 5 ˇś1`2δ´Repsq`ε ´ Repsq`ε ˇ ď Cpεq rˇ n ˇ 1ďnďz2 gcdpn,Pÿpzqq“1 ď Cpεqr1`2δ´Repsq`ε n´ Repsq`ε 1ďnďz ÿ 2 The result follows from the fact that

´ Repsq`ε 1´Repsq`ε n ! z2 1ďnďz ÿ 2 With this, we can prove the following lemma about these zero-detector, which explains the naming choice. 1 Lemma 3.2.19. Let X P N, 0 ă δ ă 2 , z, R, x, T ě 1 and 1 ď z1 ă z2. Let F be Kowalski–Michel Q-ameanable. Then, for each pX, δ, z, R, z1, z2, x, T q-suitable pair pε, αq, there is a constant Cpεq such that for Lps, ρ; F {Qq P FpXq and r P Rδ,zpρq with r ď R, for s0 P Mpα, T q with

Lps0, ρ; F {Qq “ 0 we get that

Cpεq ď zpX,δ,z,R,z1,z2,x,T qps0, ρ, rq

Proof. Let pε, αq be pX, δ, z, R, z1, z2, x, T q-suitable. First, we note that Γpsq is the Mellin transform of e´s. Thus, using Mellin transform, we get

´ 1 ´s ´ n 1 5 z ,z s e x ` ∆ pnqψ pnqprnsfps, ρqqn 0 e x “ f ps ` s , ρqM 1 2 ps ` s , ρqΓpsqx ds z1,z2 ρ,r 2πı 0 r 0 ną1 p3q gcdpn,Pÿpzqq“1 ż 1 1 Let σ “ 2 ´ α ` ε, where ε ą 0, so that σ ` α ą 2 . Since Γpsq has a simple pole only at s “ 0, 5 5 which is cancelled out by the zero of f ps ` s0, ρq at s “ 0, and f ps, ρq has no poles due to F being Kowalski–Michel Q-ameanable, we can shift the contour to the line Repsq “ σ, giving

´ 1 ´s ´ n 1 5 z ,z s e x ` ∆ pnqψ pnqprnsfps, ρqqn 0 e x “ f ps ` s , ρqM 1 2 ps ` s , ρqΓpsqx ds z1,z2 ρ,r 2πı 0 r 0 ną1 pσq gcdpn,Pÿpzqq“1 ż

Since pε, αq is pX, δ, z, R, z1, z2, x, T q-suitable, using the uniform convexity bound of the family, along 5 1 with Lemma 3.2.4, applied on f ps ` s0, ρq on the line σ ` Repsq ě σ ` α “ 2 ` ε, and using the bound z1,z2 in Lemma 3.2.18 on Mr , we get that the integral is bounded by a constant Cpεq. The result then follows.

51 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

A consequence of the lemma, along with Corollary 3.2.11, is the following.

Corollary 3.2.20. Let F be Kowalski–Michel Q-ameanable, and assume Conjecture 3.2.9. Fix z ě 1 1 large enough and 0 ă δ ă 2 . For each Lps, ρ; F {Qq P FpXq, let Jρpα, T q Ď Zρpα, T q be a X-well-spaced ﬁnite set. Then, there exists a constant Cpεq such that if pε, αq is pX, δ, z, R, z1, z2, x, T q-suitable, with

1 ´ε d` ma 2`2δ x 2 ą X 2 R log R and there exists a constant k such that x z log ď k log 2 z z ˆ 1 ˙ ˆ 1 ˙ we have that |J pα, T q| ρ ď CpεqT Bplog Xqx2p1´αq sϕpρ, Rq|ψρprq| Lps,ρ;F {QqPFpXq rďR ÿ rPRÿδ,zpρq Proof. By Lemma 3.2.19, we get a constant C1pεq such that

2 |J pα, T q| 1 |z X,δ,z,R,z ,z ,x,T ps0, ρ, rq| ρ ď C1pεq p 1 2 q sϕpρ, Rq|ψρprq| sϕpρ, Rq |ψρprq| Lps,ρ;F {QqPFpXq rďR Lps,ρ;F {QqPFpXq s0PJρpα,T q rďR ÿ rPRÿδ,zpρq ÿ ÿ rPRÿδ,zpρq

2 By Corollary 3.2.11 and since pε, αq is pX, δ, z, R, z1, z2, x, T q-suitable, there exists a constant C pεq such that the right hand side is bounded by

n 2 B ´ x 2 1´2α 2 B 2 1´2α C pεqT log X |∆z1,z2 pnqe | n ď C pεqT log X ∆z1,z2 pnq n n„x z ănăx ÿ 1 ÿ n ´ x where the second inequality is due to the fact that |e | ď 1 and ∆z1,z2 pnq “ 0 for 1 ă n ď z1 by

deﬁnition of ∆z1,z2 pnq. Finally, by Lemma 3.2.14, along with the fact that the assumption implies that

log x z1 ď k log ´ z2 ¯ z1 ´ ¯ we get a constant Cpεq such that

|J pα, T q| ρ ď CpεqT Bplog Xqx2´2α sϕpρ, Rq|ψρprq| Lps,ρ;F {QqPFpXq rďR ÿ rPRÿδ,zpρq

3.3 Zero-Region of Kowalski–Michel Q-Ameanable Families

Using what we developed in the previous section, we can now proceed as in [KM02] to give a Kowalski– Michel estimate of the zero-region of Kowalski–Michel Q-ameanable families, which is itself based on [Jut78].

52 CHAPTER 3. KOWALSKI–MICHEL AMEANABLE FAMILIES

3 Consider Mpα, T q, where 4 ď α ď 1 and T ě 2. For each k P Z, we denote k k ` 1 M pα, T q “ s P Mpα, T q; Repsq ě α, ď Impsq ă k log X log X " * We partition Mpα, T q as rT log Xs

Mpα, T q “ Mkpα, T q k“t´T log Xu ğ We will state the following density lemma of Linnik.

Lemma 3.3.1. Let t P R. The number of zeroes of Lps, ρ; K{F q in the square 1 s P ; α ď Repsq ď 1, |Impsq ´ t| ď p1 ´ αq C 2 " * is ! p1 ´ αq logpqp|t| ` 2qq ` 1 With the result of Lemma 3.2.12 and Corollary 3.2.20, we essentially have the result of [KM02] for Kowalski–Michel Q-ameanable families. All it remains now is to choose suitable parameters to invoke those results, which we will demonstrate how below.

3 Theorem 3.3.2. Let F be Kowalski–Michel Q-ameanable, and assume Conjecture 3.2.9. Let α ě 4 and T ě 2. Then, there exists a constant B ą 0, depending on the family parameters, such that for all c ą 10ma ` 4d, there exists a constant Mpcq such that

B c 1´α NF,α,T pXq ď MpcqT X 2α´1

for all X P N. Proof. Let ma ω R “ X , z1 “ X, z2 “ X where δ ą 0 and ω ą 0 are such that 1 4ma ` 2d ą ` ω ` 4maδ 2

For c ą 5ma ` 2d, let ε ą 0 be such that pε, αq is pX, δ, z, R, z1, z2, x, T q-suitable with

p 1 ´εq c d` ma 2`2δ X 2 2α´1 ą X 2 R log R

c By letting x “ X 2α´1 , the result will then follow by Corollary 3.2.20, Lemma 3.2.12 and Lemma 3.3.1. Remark 3.3.3: As a concluding remark, we will note the difference between Theorem 3.3.2 and [KM02], specifically the difference between our c and the c in [KM02]. In reality, we believe that our c is the correct lower bound even in [KM02] using their arguments, which we mirrored here, based on our understanding of their argument. In effect, there should be no difference in the result between our setting and that of Kowalski–Michel, under the assumption of Conjecture 3.2.9.

53 Episode IV

A New Family

In this chapter, we describe and study a particular family of Artin L-functions. These Artin L-functions will be associated to a special type of representation, which we will describe and study as well. In particular, we will prove variants of Aramata–Brauer Theorem for these representations. This in turn will allow us to show that the family is Kowalski–Michel ameanable, and hence we can apply results of the previous chapters regarding the zero-region of the family.

4.1 ν-Regular Representations

In this section, we will describe and study a special representation of a group. For this section, we will set the following: Let G be a ﬁnite non-trivial group, and let Gˆ be the collection of all irreducible

representations of G. Let regG be the regular representation of G, i.e.

regG “ pdim ρqρ ρPGˆ à 4.1.1 Socle of a Group We begin by reviewing some basic group theoretic facts that we will need in the succeeding sections. For more in-depth discussions on minimal normal subgroups and socle of a group, we refer the interested readers to [Rot12]. We start with a deﬁnition.

Deﬁnition 4.1.1. A normal subgroup M E G of G is a minimal normal subgroup of G if M is non-trivial and the only normal subgroups of G contained in N is teu or N. We denote the set of all minimal normal subgroups of G by MpGq. We also denote the set of all abelian minimal normal subgroups of G by ApGq, and the set of all non-abelian minimal normal subgroups of G by T pGq. In particular, MpGq “ ApGq \ T pGq We will give a few examples.

Example 4.1.1. Consider the group Sn, where n ě 5. Then, the alternating group An C Sn is the only non-trivial normal subgroup of Sn, and thus An is the unique minimal normal subgroup of Sn. We recall the following properties of minimal normal subgroups.

Proposition 4.1.2. Let M EG be a minimal normal subgroup of G, and let N EG be a normal subgroup of G. Then, either M ď N or xM,Ny “ M ˆ N.

Proposition 4.1.3. For T P T pGq distinct and N EG such that T ę N, we have that the only non-trivial normal subgroups of G contained in xT,Ny “ T ˆ N are of form teu ˆ M, T ˆ teu or T ˆ M, where M ď N is non-trivial and normal in G.

54 EPISODE IV. A NEW FAMILY

Proof. Let K E G be non-trivial and such that K ď T ˆ N. There are two cases. Case 1: Suppose K X T “ T . Since K X N is a normal subgroup of G which does not contain T , by Proposition 4.1.2, we get that pK X Nq ˆ T “ xK X N,T y Ď K If there exists pn, tq P KzpKXNqˆT , then in particular n R KXN. Now, pe, t´1q P pKXNqˆT Ď K, and that pn, eq “ pe, t´1qpn, tq P K ùñ n P K X N which is a contradiction. Hence, K “ pK X Nq ˆ T . Case 2: Suppose K X T “ teu. Then, by Proposition 4.1.2, K centralises T . Since T is non-abelian, and in fact ZpT q “ teu, every element of K should be of the form pn, eq. Thus, K Ď N. The result follows. Minimal normal subgroups are particularly nice, as demonstrated by the following proposition. Proposition 4.1.4. A minimal normal subgroup of G is either simple or a direct product of isomorphic simple groups. We can now deﬁne the following. Deﬁnition 4.1.5. A socle of G is the subgroup SocpGq generated by all the minimal normal subgroups of G, i.e. SocpGq “ xMpGqy Here are some examples, corresponding to the examples given earlier.

Example 4.1.2. Consider the group Sn, where n ě 5. Since An is the unique minimal normal subgroup of Sn, we get that SocpSnq “ An. The following proposition shows the decomposition of the socle of a ﬁnite group.

Proposition 4.1.6. There exists distinct minimal normal subgroups M1, ¨ ¨ ¨ ,Mk P MpGq of G such that SocpGq “ M1 ˆ ¨ ¨ ¨ ˆ Mk Moreoever, we get that T pGq Ď tM1, ¨ ¨ ¨ ,Mku Proof. Consider the poset

P “ S Ď MpGq; xSy “ M # MPS + ź with S1 ď S2 in P if S1 Ď S2. Since G is ﬁnite, there exists a subset

S “ t M1, ¨ ¨ ¨ ,Mk u Ď MpGq such that it is maximal in P . Then, for all minimal normal subgroup M of G, by Proposition 4.1.2 and by maximality of S we get that M Ď xSy. Thus, we get that

SocpGq “ xSy “ M1 ˆ ¨ ¨ ¨ ˆ Mk For the last assertion, suppose M P MpGq is such that M R S. Then, since M centralises each element of S by Proposition 4.1.2, we get that M Ď ZpSocpGqq, which implies that M is abelian. Hence, every non-abelian minimal normal subgroup of G must be contained in S.

55 EPISODE IV. A NEW FAMILY

As an immediate consequence, we get the following corollary.

Corollary 4.1.7. SocpGq is a direct product of simple groups. In light of Proposition 4.1.6, we write

SocpGq “ AbpSocpGqq ˆ T pSocpGqq where AbpSocpGqq “ SocpGq T pSocpGqq and M T pSocpGqq “ T T PT pGq ź We note that AbpSocpGqq is abelian. Now, by iterated applications of Proposition 4.1.3, we get the following.

Corollary 4.1.8. The normal subgroups of G contained in SocpGq is of the form

A ˆ T ˜T PS ¸ ź where A is a normal subgroup of G contained AbpSocpGqq and S Ď T pGq.

4.1.2 Faithful and Relatively Faithful Representations We will now introduce the concept of relatively faithful representations, which will become important to us in the later part of this chapter. The following is a basic deﬁnition in representation theory.

Deﬁnition 4.1.9. A representation ρ : G Ñ GLpV q of G is faithful if kerpρq “ teu. We denote

Gˆν “ ρ P Gˆ; kerpρq “ teu ! ) We note here that Gˆν may be empty.

Example 4.1.3. For any group, recall that the regular representation of the group is faithful. This is a major example of a faithful representation.

Example 4.1.4. For any simple group G, recall that every non-trivial irreducible representation of G is ˆν ˆ faithful. Thus, G “ Gzt1Gu.

Example 4.1.5. Consider the following subgroup of order 18 of S6

G “ xp123q, p456q, p23qp56qy

One can check that G has no faithful irreducible representation, i.e. Gˆν “H. Faithful representations are less studied in classical representation theory, unlike say irreducible representations. For our purposes, for representation ρ : G Ñ GLpV q of G, we get the diagram

56 EPISODE IV. A NEW FAMILY

πkerpρq 1 kerpρq G G kerpρq 1 M ρ ρ˜

GLpV q where, as before, the map G πN : G N is the usual quotient map. Thus, since Artin L-functionL satisﬁes inﬂation, we view the Artin L-function associated to ρ as “originating” from ρ˜, a faithful representation of G kerpρq, and hence we descend down to the level where ρ is faithful. To be more precise, we have the followingM proposition. Proposition 4.1.10. We have that ν ˆ G G “ ρ ˝ πN ; ρ P N NEG ! ) ğ {L Proof. For any irreducible representation ρ : G Ñ GLpV q of G, we get the diagram

πkerpρq 1 kerpρq G G kerpρq 1 M ρ ρ˜

GLpV q where now,ρ ˜ is clearly faithful. Now, if W ď V is G kerpρq-invariant, then for all g P G, M ρpgqW “ ρ˜ ˝ πkerpρqpgqW “ ρ˜pg kerpρqqW Ď W ν and by irreduciblity of ρ, we get that W is either t0u or V . Thus,ρ ˜ P G kerpρq . G Conversely, for an irreducible faithful representation ρ : N Ñ GLM{pV q of G, then for W ď V which is G-invariant under ρ ˝ π , for all gN P G N N L

ρpgNqW “ ρL˝ πN pgqW Ď W and by irreducibility of ρ, we get that W is either t0u or V . Thus, ρ ˝ πN P G. The result follows. Here, we collect a few cute facts regarding faithful representations. p Proposition 4.1.11. Let ρ P Gˆν, then ZpGq is cyclic. Proof. For z P ZpGq, we get that ρpzq : V Ñ V such that ρpzq ‰ 0. Since ZpGq is abelian, ρpzq is a(n ˆ endo)morphism of irreducible representations, so by Schur’s Lemma, we get that there exists cz P C ˆ such that ρpzq “ czI. Consider the group homomorphism ϕ : ZpGq Ñ C given by

ϕpzq “ cz Since ρ is faithful, we get that ϕ is a injective group homomorphism, i.e. ZpGq is isomorphic to a ﬁnite abelian subgroup of Cˆ via ϕ. Hence, we get that ZpGq is cyclic.

57 EPISODE IV. A NEW FAMILY

Theorem 4.1.12. Let ρ : G Ñ GLpV q be a faithful representation of G. Then, for τ P Gˆ, where τ : G Ñ GLpW q, we get that W is a subrepresentation of SymnpV q. Proof. We now introduce the notion of relatively faithful representations.

Deﬁnition 4.1.13. Let N ď G be a subgroup. A representation ρ : N Ñ GLpV q of N is G-relatively faithful if kerpρq is core-free in G, i.e.

kerpρqg “ teu gPG č We denote Nˆ νpGq “ ρ P Nˆ; ρ is G-relatively faithful ! ) We note that for representation ρ : G Ñ GLpV q of G, since kerpρq is normal, the normal core of kerpρq is

kerpρqg “ kerpρq gPG č Hence, ρ is G-relatively faithful if and only if ρ is faithful. We see, therefore, that the notion of G-relative faithfulness is an extension of the notion of faithfulness. The following proposition is immediate.

Proposition 4.1.14. For any M P MpGq, we get that

ˆ ν ˆ M pGq “ Mzt1M u

Proof. For any ρ P Mˆ , note that the normal core in G

kerpρqg ď M gPG č of kerpρq is normal in G. Since M is a minimal normal subgroup, this implies that the normal core of kerpρq is either teu or M. The only irreducible representation of M with the normal core in G of its kernel being M is the trivial one, so all other irreducible representations of M is therefore G-relatively faithful. We also prove the following result, which explains the motivation for the naming choice.

ˆ ν G Proposition 4.1.15. Let H ď G, and let ρ P H pGq. Then, the representation IndH ρ of G is faithful. G Proof. This is immediate by recalling that the kernel of IndH ρ is

kerpρqg gPG č We will also need the following result.

Proposition 4.1.16. For T P T pGq distinct and N E G such that T ę N, we get that ν ν ν pN ˆ T q pGq “ ρ b τ; ρ1 P N pGq, ρ2 P T pGq ! ) { p p 58 EPISODE IV. A NEW FAMILY

Proof. Every representation in N ˆ T is of the form ρ b τ, where ρ P N and τ P T . Recall that a subgroup is core-free in G if and only if it does not contain any non-trivial normal subgroup of G. By Proposition 4.1.3, we get that ker{pρ1 b ρ2q is core-free in G if and only if itp does not containp either T or the normal subgroup of G contained in N. Now, for any normal group N 1 of G contained in N,

N 1 Ď kerpρ b τq ðñ @n P N 1,I “ ρ b τpnq “ ρpnq ðñ N 1 Ď kerpρq

Also, T Ď kerpρ b τq ðñ @t P T,I “ ρ b τptq “ τptq ðñ T Ď kerpτq

Hence, kerpρ1 b ρ2q is core-free in G if and only if both kerpρq and kerpτq are core-free in G. As a direct consequence, we get the following result. Corollary 4.1.17. We get in particular that

ν ν ν SocpGq pGq “ ρ b τT ; ρ P AbpSocpGqq pGq, τT P T pGq # ˜T PT pGq ¸ + â { { Proof. This is just an iteration application of Proposition 4.1.16. p

4.1.3 ν-Regular and Relative ν-Regular Representations We will now introduce a new representation, which will mirror regular representation for faithful and relatively faithful irreducible representations. We start with a series of definitions. Definition 4.1.18. For finite group G such that Gˆν ‰H, the ν-regular representation of G is the representation ν regG “ pdim ρqρ ρPGˆν à

Deﬁnition 4.1.19. Let H ď G be a subgroup such that Hˆ νpGq ‰ H. The G-relative ν-regular representation of H is the representation

ν regH pGq “ pdim ρqρ ρPHˆ ν pGq à We provide a few examples.

ˆν ˆ Example 4.1.6. For any simple group G, recall that G “ Gzt1Gu, and so we get that

ν regG “ regG a 1G

Example 4.1.7. Consider S4. One can compute and ﬁnd that ν S4 “ t ρ, ρ b sgn u

where ρ is the standard representation of Sx4 and sgn is the sign representation. Both faithful irreducible representations of S4 are of degree 3. Thus

ν regS4 “ p3ρq ‘ p3ρ b sgnq

59 EPISODE IV. A NEW FAMILY

2 Consider also the socle of S4, SocpS4q – Z 2 . Since SocpS4q is an abelian group which is not ν Z cyclic, we get that SocpS4q “H. However,` oneL can˘ compute and ﬁnd that ν { SocpS4q pS4q “ SocpS4qzt1SocpS4qu ùñ regν pS q “ ρ ‘ ρ ‘ ρ { SocpS4q 4{ 1 2 3 where ρi are the three non-trivial irreducible representations of SocpS4q. By Proposition 4.1.10, we get a decomposition of regular representations to ν-regular representations, as demonstrated below. Proposition 4.1.20. We get that

reg regν π G “ G ˝ N NEG N à N A very direct consequence of the deﬁnition and Corollary 4.1.17 is the following proposition. Proposition 4.1.21. We get that

ν ν ν ν regSocpGqpGq “ regAbpSocpGqqpGq b regT1 pGq b ¨ ¨ ¨ b regTm pGq We prove the following lemma that relates the ν-regular representation of G with the G-relative ν-regular representation of SocpGq. Lemma 4.1.22. We have that ν G ν regG “ IndSocpGqregSocpGqpGq ν Proof. First, for ρ P SocpGqzSocpGq pGq, note that

G g { {ker IndSocpGqρ “ kerpρq Ľ teu gPG č G ˆν G and so xχ, IndSocpGqρy “ 0 for χ P G . Since rG “ IndSocpGqrSocpGq, we get that

dim χ “ xχ, regGy G “ xχ, IndSocpGqregSocpGqy G ν G G ν “ xχ, IndSocpGqregSocpGqpGqy ` dim ρxχ, IndSocpGqρy “ xχ, IndSocpGqregSocpGqpGqy ν ρPSocpGqzÿSocpGq pGq ˆ ˆν { { G Let ρ P SocpGq. Suppose χ P GzG is such that xχ, IndSocpGqρy ‰ 0, and let N Ď ker χ be a minimal normal subgroup of G. By Frobenius Reciprocity,

{ G G xResSocpGqχ, ρy “ xχ, IndSocpGqρy ‰ 0 ν and this implies that N Ď ker ρ. Thus, ker ρ is not core-free in G, and hence by deﬁnition, ρ R SocpGq pGq. ν G ˆ ˆν Thus, for ρ P SocpGq pGq, we get that xχ, IndSocpGqρy “ 0 for all χ P GzG . This implies that { G ν ˆν { xχ, IndSocpGqregSocpGqpGqy ‰ 0 ùñ χ P G

ν G ν and we get that regG “ IndSocpGqregSocpGqpGq.

60 EPISODE IV. A NEW FAMILY

Here is an example showing Lemma 4.1.22 in action.

Example 4.1.8. Consider S4. We already showed in Example 4.1.7 that

ν regSocpS4qpS4q “ ρ1 ‘ ρ2 ‘ ρ3

2 where ρi are the three non-trivial irreducible representations of SocpS4q – Z 2Z . By computing (using GAP, for example), one can also see that for i “ 1, 2, 3, ` L ˘ IndS4 ρ “ ρ ‘ pρ b sgnq SocpS4q i

where ρ is the standard representation of S4 and sgn is the sign representation. Thus, we have that

IndS4 regν pS q “ 3ρ ‘ 3pρ b sgnq “ regν SocpS4q SocpS4q 4 S4 as predicted by Lemma 4.1.22.

The following property is needed for studying the convolution over F of two ν-regular representations.

Proposition 4.1.23. We get that ν ˚ ν pregGq “ regG Proof. Recall that a (ﬁnite-dimensional) representation of ρ : G Ñ GLpV q is irreducible if and only if its dual is. Further,

g P kerpρq ðñ ρpgq “ I ðñ @φ P V ˚, ρ˚pgqfpvq “ fpρpgq´1vq “ fpvq, @v P V ðñ ρ˚pgq “ I ðñ g P kerpρ˚q

The result now follows by deﬁnition of ν-regular representation.

4.1.4 Variations on a Theorem of Aramata–Brauer In this section, we consider the Artin L-functions associated to ν-regular representations of G. Let us set our notation: Let K{F be a Galois extension of number ﬁelds. We denote the following.

ν Deﬁnition 4.1.24. We denote the Artin L-function associated to ν-regular representation regGalpK{F q as

ν ν ζK{F psq “ Lps, regGalpK{F q; K{F q

ν ν We also denote the Artin L-function associated to ConvF pregGalpK{F q, regGalpK1{F qq as

ν ν ν ν 1 pζK{F , ζK1{F qpsq “ Lps, ConvF pregGalpK{F q, regGalpK1{F qq; KK {F q

ν We now prove an extension of Aramata–Brauer Theorem, for ζK{F psq.

ν Theorem 4.1.25. Let K{F be a Galois extension. The Artin L-function ζK{F psq is entire.

61 EPISODE IV. A NEW FAMILY

Proof. By Lemma 4.1.22, we get that

ν G ν |G|regG “ rG : SocpGqs|SocpGq|IndSocpGqregSocpGqpGq G ν ν ν “ rG : SocpGqs|A|IndSocpGqregApGq b |T1|regT1 pGq b ¨ ¨ ¨ b |Tm|regTm pGq By Proposition 4.1.14 and by Lemma 2.3.5, each T regν G can be written as | i| Ti p q

|T |regν pGq “ |T |preg a 1 q “ m IndTi ψ i Ti i Ti Ti Ci,ψCi Ci Ci Ci ψ PC z1 à Ci ài Ci where the sum is taken over all cyclic subgroups of T and eachx m P . Thus, we get that |G|regν i Ci,ψCi N G is a positive integer combination of representations of the form

G IndAˆC1ˆ¨¨¨ˆCm χ b ψC1 b ¨ ¨ ¨ b ψCm

where χ b ψC1 b ¨ ¨ ¨ b ψCm is a non-trivial 1-dimensional representation of A ˆ C1 ˆ ¨ ¨ ¨ ˆ Cm. The result now follows easily similar to the original Aramata–Brauer theorem.

G G ´1 Remark 4.1.26: Note that for each term IndAˆC1ˆ¨¨¨ˆCm χ b ψC1 b ¨ ¨ ¨ b ψCm the term IndAˆC1ˆ¨¨¨ˆCm χ b ψ´1 ψ´1 also appears as a summand. C1 b ¨ ¨ ¨ b Cm ν ν We also prove an extension of Aramata–Brauer Theorem, for pζK{F , ζL{F qpsq. Theorem 4.1.27. Let K,K1{F be Galois extensions such that GalpK{F q – G and GalpK1{F q – G. Then, ν ν pζK{F , ζK1{F qpsq is entire if and only if K ‰ K1. If K “ K1, then it has a pole only at s “ 1. Proof. First, by Proposition 4.1.3, we get that

ν ν GalpK{F qˆGalpK1{F q ν ν ConvF pregGalpK{F q, regGalpK1{F qq “ ResGalpKK1{F q regGalpK{F q b regGalpK1{F q

2 ν ν By Theorem 4.1.25, consider each summand of |G| regGalpK{F q b regGalpK1{F q as

GalpK{F q GalpK1{F q GalpK{F qˆGalpK1{F q IndS χ b IndR ρ “ IndSˆR χ b ρ ´ ¯ ´ ¯ where χ and ρ are non-trivial 1-dimensional representations on subgroup S ď SocpGalpK{F qq and R ď SocpGalpK1{F qq respectively. By Mackey Decomposition Theorem, we get that

GalpK{F qˆGalpK1{F q GalpK{F qˆGalpK1{F q GalpKK1{F q pSˆRqσ σ ResGalpKK1{F q IndSˆR χbρ “ IndGalpKK1{F qXpSˆRqσ ResGalpKK1{F qXpSˆRqσ pχbρq GalpKK1{F qσpSˆRq à 1 1 • Suppose that K ‰ K . Now, for all σ “ pσ1, σ2q P GalpK{F q ˆ GalpK {F q

1 σ σ GalpKK {F q X pS ˆ Rq “ t φ “ pφ1, φ2q P pS ˆ Rq ; φ1|KXK1 “ φ2|KXK1 u

1 1 Let NK E GalpK{F q be associated to K X K . Since K ‰ K X K , NK ‰ teu, so they must contain TK a minimal normal subgroups of GalpK{F q. In particular, every φ P TK is such that φ|KXK1 is identity, and thus

σ1 σ 1 σ pS X TK q ˆ teu “ ppS X TK q ˆ teuq Ď GalpKK {F q X pS ˆ Rq

62 EPISODE IV. A NEW FAMILY

σ σ1 and by the deﬁnition of χ the pχ b ρq is non-trivial on pS X TK q ˆ teu, and thus

pSˆRqσ σ ResGalpKK1{F qXpSˆRqσ pχ b ρq is a non-trivial 1-dimensional representation. It then follows that each summand of

2 ν ν GalpK{F qˆGalpK1{F q 2 ν ν |G| ConvF pregGalpK{F q, regGalpK1{F qq “ ResGalpKK1{F q |G| regGalpK{F q b regGalpK1{F q is a direct sum of induced non-trivial 1-dimensional representations, and thus it follows that

ν ν |G|2 pζK{F , ζK1{F qpsq

ν ν is entire. Since pζK{F , ζK1{F qpsq is meromorphic, we get that it is therefore entire. • Suppose that K “ K1. Consider the case where S “ R. Now, for σ “ e,

GalpK{F q X pS ˆ Sq “ t pφ, φq; φ P S u

By Remark 4.1.26 we can choose ρ “ χ´1, in which case since χ b χ´1 is 1-dimensional, for all φ P S χpφq b χ´1pφq “ I This implies that SˆS ´1 ResGalpK{F qXpSˆSqχ b χ ν ν is therefore trivial. It then follows that pζK{F , ζK1{F qpsq must have a pole at s “ 1.

4.2 Family of Artin L-Functions Associated to ν-Regular Representation

We now turn our attention to defining a new family of Artin L-functions which is ameanable to the analytical methods of Kowalski–Michel, as established in the previous chapter. We set the following notation: Let G be a finite group, and let F be a number field.

4.2.1 A New Family and Its Kowalski–Michel Ameanability We will now introduce a new family of Artin L-functions.

ν Definition 4.2.1. Let G be a finite group, and let F be a number field. Let FF,G be defined as

ν ν FF,G “ FF,GpXq; X P N where for each X P N, (

ν ν F FF,GpXq “ ζK{F psq; K{F Galois, GalpK{F q – G, NQ p∆K{F q ă X ν ( We prove that FF,G is indeed a family of Artin L-functions.

ν ν Lemma 4.2.2. FF,G is a family of Artin L-function of degree dim regG.

ν Proof. We prove that FF,G satisﬁes the conditions of being a family.

63 EPISODE IV. A NEW FAMILY

ν ν • For any ζK{F psq P FF,GpXq, by Proposition 4.1.20, additivity of Artin conductors and the Conductor- ν Discriminant Formula we get that the conductor of ζK{F psq is

ν dim regν F ν GalpK{F q f ApregGalpK{F q; K{F q “ |∆F {Q| NQ p pregGalpK{F q; K{F qq dim regν F dim regν GalpK{F q f GalpK{F q ď |∆F {Q| NQ p pregG; K{F qq ď |∆F {Q| X

• For every X P N, we get that ν Gal |FF,GpXq| ď NF,|G|pXq Gal where NF,n pXq is the number of Galois extensions of F of degree n with absolute discriminant at most X, i.e. Gal F NF,n pXq :“ K{F Galois; rK : F s “ n, NQ p∆K{F q ă X By a well-known result of Schmidtˇ [Sch95], we get that (ˇ ˇ ˇ |G|`2 ν Gal F 4 |FF,GpXq| ď NF,|G|pXq ď NF,|G|pXq :“ K{F ; rK : F s “ n, NQ p∆K{F q ă X ď CpF, |G|qX where CpF, nq is a constant dependingˇ on F and n. (ˇ ˇ ˇ ν dim regGalpK{F q |G|`2 ν With the constants a “ 1, Ccond “ |∆F {Q| , d “ 4 and Corder “ CpF, |G|q, this makes FF,G ν a family of Artin L-functions of degree dim regG. ν More importantly for us, we will prove that FF,G is Kowalski–Michel ameanable, and thus is ameanable to the results in the previous chapter.

ν Theorem 4.2.3. FF,G is Kowalski–Michel F -ameanable, with (one possible) F -family parameter

ν dim regν |G| ` 2 dim reg , F, a “ 1,C “ |∆ | GalpK{F q , d “ ,C “ CpF, |G|q,C ,C1 G cond F {Q 4 order convex convex ˆ ˙ ν ν Proof. Every member of ζK{F P FF,G is entire by Theorem 4.1.25 and satisfy the same uniform convexity bound by Lemma 2.5.2, as they are all Artin L-functions of the same degree and associated to the ν same representation. Further, by Theorem 4.1.27, FF,G satisfy the convolution property, and that each ν ν pζK{F , ζL{F qpsq satisfy the same uniform convexity bound again by Lemma 2.5.2, as they are all Artin L-functions of the same degree. In light of Lemma 3.1.6, even more is true.

ν Corollary 4.2.4. For K Ď F with F {K being Galois, we have that FF,G is Kowalski–Michel K-ameanable, with K-family parameter

ν dim regν |G| ` 2 rF :Ks dim reg , K, a “ 1,C “ |∆ | GalpK{F q , d “ ,C “ CpF, |G|q,C , pC1 q G cond F {Q 4 order convex convex ˆ ˙ To conclude, we relate the new family to the family of Dedekind zeta-functions. We now deﬁne the following family. Deﬁnition 4.2.5. Let 1 1 FF,G “ FF,GpXq; X P N where for each X P N, ( ζ psq F 1 pXq “ K ; K{F Galois, GalpK{F q “ G, N F p∆ q ă X F,G ζ psq Q K{F " F *

64 EPISODE IV. A NEW FAMILY

The following proposition is straightforward.

1 Proposition 4.2.6. FF,G is a family of Artin L-functions of degree |G| ´ 1. 1 Proof. We prove that FF,G satisﬁes the conditions of being a family. • For any ζK psq P F 1 pXq, by additivity of conductors and the Conductor-Discriminant Formula we ζF psq F,G ν get that the conductor of ζK{F psq is

ApregGalpK{F q a 1GalpK{F q; K{F q ď ApregGalpK{F q; K{F q |G| F f |G| “ |∆F {Q| NQ p pregG; K{F qq ď |∆F {Q| X

• For every X P N, we get by a well-known result of Schmidt [Sch95] that

|G|`2 1 Gal F 4 |FF,GpXq| ď NF,|G|pXq ď NF,|G|pXq :“ K{F ; rK : F s “ n, NQ p∆K{F q ă X ď CpF, |G|qX ˇ (ˇ where CpF, nq is a constant dependingˇ on F and n. ˇ |G| |G|`2 1 With the constants a “ 1, Ccond “ |∆F {Q| , d “ 4 and Corder “ CpF, |G|q, this makes FF,G a family of Artin L-functions of degree |G| ´ 1.

1 We denote the a and d that makes FF,G a family by aF,G and dF,G respectively. We now factorise ζK psq courtesy of Proposition 4.1.20. Proposition 4.2.7. We get that

ν ν ζK psq “ ζKN {F psq “ ζF psq ζKN {F psq N G N G źE źC In light of Proposition 4.2.7, we can think of each ζK psq P F 1 as ζF psq F,G

ν ν N ν ζ N psq “ Lps, reg G ; K {F q P F G K {F {N F, {N NCG ´ ¯ 4.2.2 Zero-Free Region For Subfamilies of Dedekind Zeta-Functions We now wish to leverage Kowalski–Michel zero-region estimate to subfamilies of Dedekind zeta-functions. 1 1 For a subfamily H of FF,G, we denote the a and d that makes FF,G a family by aF,GpHq and dF,GpHq respectively. We will assume Conjecture 3.2.9. Let us deﬁne the following.

1 Deﬁnition 4.2.8. Let H be a subfamily of FF,G. We deﬁne

ν ν F G pHq “ F G pHqpXq; X P F, {N F, {N N ! ) where ν ν ν ζK psq F G pHqpXq “ ζKN F psq P F G pXq; P HpXq F, {N { F, {N ζ psq " F * ν ν ν It is clear that F G pHq is a subfamily of F G , so by Proposition 3.1.5, F G pHq is Kowalski– F, {N F, {N F, {N Michel F -ameanable. We will describe the following condition of Pierce–Turnage-Butterbaugh–Wood on subfamilies of 1 FF,G, which they denoted as inequality (6.6) of Condition 6.4 in [PTBW17].

65 EPISODE IV. A NEW FAMILY

1 Deﬁnition 4.2.9. Let H be a subfamily of FF,G. We say H is sterile if for each normal N C G (note that N ‰ G), there exists 0 ď τN ă dF,GpHq and constant MN , depending on the family parameters of ν ν 1 ν F G , such that for all X P N, for Lps, reg G ; K {F q P F G pXq F, {N {N F, {N ζ s K p q ν ν 1 τN P HpXq; ζKN F psq “ Lps, reg G ; K {F q ď MN X ζ psq { {N ˇ" F *ˇ ˇ ˇ In this case, the contaminationˇ index of H is ˇ ˇ ˇ

τ “ max τN NCG ν We will remark that the sterility condition only depends on members of F G pHq, rather than the full F, {N ν F G , to satisfy the required inequality. F, {N One way to think of the sterility condition is the following: When trying to bring Kowalski–Michel ν zero-region estimate for each F G up to the level of H, we needed to ensure that each “bad” Artin F, {N L-functions, the ones which are not zero-free, do not “contaminate” too many members H. Propogation of such “contamination” is eﬀectively controlled by the inequality condition in the deﬁnition of sterility. Then, proceeding as in [PTBW17], we get the average zero-free region result of Pierce–Turnage- Butterbaugh–Wood which is unconditional, in the case where F {Q is Galois. Theorem 4.2.10. Let F {Q be Galois, and assume Conjecture 3.2.9. Let H be a sterile subfamily of 1 τ 1 F with contamination index τ “ maxN G τN . Then, for any 0 ă ∆ ă 1 ´ and η ă , there F,G C dF,GpHq 4 ν exists B (depending on the Kowalski–Michel family parameter of F G pHq for each N G), a constant F, {N C 1 MpG, aF,GpHq, dF,GpHq, ∆, τq and 0 ă δ ď 4 , given by ε δ “ 20mrF : QsaQ,GpHq ` 8dQ,GpHq ` 4ε ν with m “ maxNCG dim reg G and {N τ ε ∆ “ 1 ´ ´ dF,GpHq 2dF,GpHq

p1´p1´ηq∆qdF,GpHq such that for all X P N, at most MpG, aF,GpHq, dF,GpHq, ∆, τqX members of HpXq that can have a zero in the region

η∆dF,GpHq η∆dF,GpHq r1 ´ δ, 1s ˆ ´X B ,X B „  Proof. Let ε ą 0 be such that τ ε ∆ “ 1 ´ ´ dF,GpHq 2dF,GpHq ν As F {Q is Galois, for each N G, by Corollary 4.2.4, F G pHq is Kowalski–Michel Q-ameanable, so C F, {N let 1 pnN , Q, aN ,Ccond,N , d, Corder,N ,Cconvex,N ,Cconvex,N q ν be the Kowalski–Michel Q-family parameters of F G pHq. We deﬁne F, {N

cN “ 10nN rF : QsaF,GpHq ` 4dF,GpHq ` ε

66 EPISODE IV. A NEW FAMILY

Note that we can choose the aN and dN such that aF,GpHq “ maxNCG aN , and dF,G “ maxNCG dN , so for simplicity we will assume that. Thus

cN “ 10nN rF : QsaF,GpHq ` 4dF,GpHq ` ε ą 10nN aN ` 4dN

Let us deﬁne, for each N C G, ε cN ` p1 ´ ∆qdF,GpHq ´ τ cN ` 2 ε 3 αN “ “ “ 1 ´ ě cN ` 2pp1 ´ ∆qdF,GpHq ´ τq cN ` ε 2cN ` 2ε 4 Note that by simple calculation,

1 ´ αN ε cN “ “ p1 ´ ∆qdF,GpHq ´ τ 2αN ´ 1 2 By Theorem 3.3.2, we get that there exists a constant B ą 0, depending on the family parameters, there exists a constant MN (which depends on nN , aF,GpHq, dF,GpHq, ∆ and τ) such that for all TN ě 2,

c 1´αN BN N 2α ´1 BN p1´∆qdF,GpHq´τ NF ν H ,α ,T pXq ď MN T X N “ MN T X G p q N N F, {N

η∆dF,GpHq B We ﬁnally set TN “ X N , and thus we get that

η∆dF,GpHq`p1´∆qd´τ p1´p1´ηq∆qdF,GpHq´τ NF ν H ,α ,T pXq ď MN X “ MN X G p q N N F, {N Let ε ε α “ max αN “ 1 ´ “ 1 ´ N G C 2rF : Qs maxNCG cN ` 2ε 20rF : QspmaxNCG nN qaF,GpHq ` 8dF,GpHq ` 4ε which is therefore dependent on G, aF,GpHq, dF,GpHq, ∆ and τ. Further, set

η∆dF,GpHq max B T “ min TN “ X NCG N and M “ max MN NCG NCG Now, combining all the above and by sterility of H, we get that

τ p1´p1´ηq∆qd pHq N ν F,G NH,α,T pXq ď X NF pHq,αN ,TN pXq ď MX |t N C G u| F, G N G {N ÿC By noting that M is therefore dependent on G, aF,GpHq, dF,GpHq, ∆ and τ, we get that there exists at most MXp1´p1´ηq∆qdF,GpHq members of HpXq that can have a zero in the region

η∆dF,GpHq η∆dF,GpHq rα, 1s ˆ ´X B ,X B „  where B “ maxNCG BN . Finally, we note here the dependency of B on the Kowalski–Michel family ν parameter of F G pHq for each N G. F, {N C We wish to point out that for general number ﬁelds F , given a Kowalski–Michel type zero-region estimate for Kowalski–Michel F -ameanable families, one can proceed exactly as in the proof of Theorem 1 4.2.10 to get a similar unconditional average zero-free region of members of sterile subfamilies of FF,G. To conclude, let us deﬁne the following.

67 EPISODE IV. A NEW FAMILY

1 Deﬁnition 4.2.11. Let 0 ă δ ă 2 . A degree n G-extension of FK{F , i.e. extension K{F with F ζ F psq rK : F s “ n and GalpK {F q “ G, is a δ-exceptional ﬁeld if K has a zero in the region ζF psq

2 2 δ δ 1 δ, 1 log ∆ F , log ∆ F r ´ s ˆ ´ K {F K {F „ ´ ˇ ˇ¯ ´ ˇ ˇ¯  ˇ ˇ ˇ ˇ We note that under the veracity of Generalisedˇ Riemannˇ Hypothesis,ˇ thereˇ exists no δ-exceptional field 1 for any δ ă 2 . We also define the following. Definition 4.2.12. For F Z Ď K{F ; rK : F s “ n, GalpK {F q “ G we denote ! ) HpZq “ t HpZqpXq; X P N u where F ζK psq H Z X ; K F Z, ∆ F X p qp q “ { P K {F ă ζF psq " ˇ ˇ * with each H Z X a multi-set. ˇ ˇ p qp q ˇ ˇ A consequence of Theorem 4.2.10 is the following.

Corollary 4.2.13. Let F {Q be Galois, and assume Conjecture 3.2.9. Let

F Z Ď K{F ; rK : F s “ n, GalpK {F q “ G ! ) 1 1 If HpZq is a sterile subfamily of FF,G with contamination index τ, then for any 0 ă ε ă 2 with τ ε ∆ “ 1 ´ ´ ą 0 dF,GpHpZqq 2dF,GpHpZqq there exists a constant DpZ, τ, εq such that for all X P N, there are at most DpZ, τ, εqXτ`ε δ-exceptional ﬁelds in

K F Z; ∆ F X { P K {F ă where ! ˇ ˇ ) ˇ ε ˇ δ “ ˇ ˇ 20mrF : QsaF,GpHpZqq ` 8dF,GpHpZqq ` 4ε ν with m “ maxNCG dim reg G . {N Proof. Deﬁne τ ε ∆ “ 1 ´ ´ ą 0 dF,GpHpZqq 2dF,GpHpZqq and let η “ ε . Note that 2dF,GpHpZqq ε p1 ´ p1 ´ ηq∆qd pHpZqq “ τ ` p1 ` ∆q ď τ ` ε F,G 2

ν Then, by Theorem 4.2.10 and noting that the Kowalski–Michel family parameter of F G pHpZqq F, {N depends on Z, we get that there exists a B depending on Z and MpZ, τ, εq such that for all X P N, at

68 EPISODE IV. A NEW FAMILY

ζ F psq τ`ε I K most MpZ, τ, εqX ﬁelds K{F P Z pF,Gq with ∆ F ă X have the property that has a zero n K {F ζF psq in ˇ ˇ ˇ ε∆ˇ ε∆ r1 ´ δ, 1s ˆˇ ´X 2Bˇ,X 2B Let D1pZ, τ, εq be a constant such that for all y ě ”D1pZ, τ, εq, weı have that

2 ε∆ plog yq d ď y 2B

Then, the only possible δ-exceptional ﬁeld that is not eliminated via the count above are the ones whose absolute discriminant is less than D1pZ, τ, εq. Now,

I 1 1 K F Z F,G ; ∆ F D Z, τ, ε H Z D Z, τ, ε { P n p q K {F ă p q “ | p qp p qq| ˇ! ˇ ˇ )ˇ 1 dF,GpHpZqq ˇ ˇ ˇ ˇ ď KD pZ, τ, εq ˇ ˇ ˇ ˇ for some constant K since HpZq is a family of Artin L-functions. We therefore get the result.

F 1 4.2.3 Sterile Subfamilies of Q,G Although we developed no additional result towards the study of sterile subfamilies in this thesis, we will present what is known so far for completeness. We will now focus on the case of F “ Q, and assume Conjecture 3.2.9. In this case, we can convert the sterility condition to a field-counting condition. To make this transition rigorously, we cite the following lemma of Pierce–Turnage-Butterbaugh–Wood, which they refined from a result of Klünersand Nicolae in [KN16].

Lemma 4.2.14. Let χ be a chracter of G Ď Sn, a ﬁxed transitive subgroup. For any Galois extensions K{Q, K1{Q with GalpK{Qq – GalpK{Qq – G, we have that

Lps, χ; K{Qq “ Lps, χ; K1{Qq if and only if Kkerpχq “ pK1qkerpχq. For a proof, we refer the interested readers to [PTBW17] for the reduction to the case χ is faithful, and to [KN16] for the proof in that case. We will, however, stress that the lemma critically relies on the fact that we are working over Q. Kl¨uners and Nicolae did prove a relative verstion of the lemma, which Pierce–Turnage-Butterbaugh–Wood have remarked that it may be possible to obtain an analogous version of the lemma for certain cases of F ‰ Q and certain choices of G. With this lemma, the following proposition is straightforward. Proposition 4.2.15. Let H be a subfamily of F 1 . Then, H is sterile if for each normal N G Q,G C (note that N ‰ G), there exists 0 ď τN ă dF,G and constant MN , depending on the family parameters ν 1 1 G of F G , such that for all X P N, for any Galois extension K {Q with GalpK {Qq – N and F, {N |∆K1{Q| ă X, L ζK psq P HpXq; KN “ K1 ď M XτN ζ psq N ˇ" Q *ˇ ˇ ˇ In this case, the contaminationˇ index of H is ˇ ˇ ˇ τ “ max τN NCG

69 EPISODE IV. A NEW FAMILY

We will now briefly summarise the result of Pierce–Turnage-Butterbaugh–Wood which gives a list of sterile subfamilies of F 1 , as field-counting is beyond the scope of this thesis. We will define the Q,G following.

Deﬁnition 4.2.16. Let n P N, and let G Ď Sn be a transitive subgroup. Let

k I “ Ci i“1 ď where each Ci is a conjugacy class of G. We say that a degree n G-extension of Q K{Q, i.e. extension Q K{Q with rK : Qs “ n and GalpK {Qq “ G, has tame-ramiﬁcation type I if for all rational prime p which is tamely-ramiﬁed in K, the inertia group of primes of OK lying over p is generated by an element of I . We denote

I Q Zn pQ,Gq “ K{Q; rK : Qs “ n, GalpK {Qq “ G, K{Q has tame-ramification type I ! ) Note that the terminology “tame-ramification type” is not a common terminology used in the literature, and it is defined here only for ease of reference.

Deﬁnition 4.2.17. Let n P N, and let k I “ Ci i“1 ď where each C is a conjugacy class of G. We deﬁne a subfamily HI ,G of F 1 to be i n pQ q Q,G

I I Hn pQ,Gq “ Hn pQ,GqpXq; X P N where ( ζ s I KQ p q I H p ,GqpXq “ ; K{ P Z p ,Gq, ∆ Q ă X n Q Q n Q K {Q ζQpsq " ˇ ˇ * I ˇ ˇ As aQ,G ď 1, we can choose aQ,GpHn pQ,Gqq “ 1. ˇ ˇ The ﬁeld-counting result of Pierce–Turnage-Butterbaugh–Wood can be summarised as follows.

Proposition 4.2.18. HI ,G is a sterile subfamily of F 1 if G, is one of the following: n pQ q Q,G I • G simple and any I , with contamination index τ “ 0.

• G is cyclic and I the set of all generators of G, with contamination index τ “ 0.

• G “ Sn with 3 ď n ď 5 and I is the conjugacy class of transpositions, with contamination index 1 1 τ “ 3 when n “ 3, τ “ 2 when n “ 4 and τ ă 1 for n “ 5.

• G “ D2p, a dihedral group of order 2p, with p an odd prime and I is the conjugacy class of p`1 p`3 1 p2 pq ¨ ¨ ¨ p 2 2 q, with contamination index τ “ p´1 .

• G “ A4 and I is the union of the conjugacy class of p123q and p132q, with contamination index τ “ 0.2784.....

70 EPISODE IV. A NEW FAMILY

We refer interested readers to [PTBW17]. Instead, we conclude with the following corollary, with the additional observation that in the case listed in Proposition 4.2.18, as long as

I d ,GpHn pQ,Gqq ε ă Q 4 we have that τ ε ∆ “ 1 ´ ´ ą 0 I I dQ,GpHn pQ,Gqq 2dQ,GpHn pQ,Gqq Corollary 4.2.19. For G, I being one of those listed in Proposition 4.2.18 with the corresponding I 1 dQ,GpHn pQ,Gqq contamination index τ, we have that for any 0 ă ε ă minp 2 , 4 q, there exists a constant Dpn, G, I , τ, εq such that for all X P N, there are at most Dpn, G, I , τ, εqXτ`ε δ-exceptional ﬁelds in

I K{ P Z p ,Gq; ∆ Q ă X Q n Q K {Q ! ˇ ˇ ) where ˇ ˇ ε ˇ ˇ δ “ I 20m ` 8dQ,GpHn pQ,Gqq ` 4ε ν with m “ maxNCG dim reg G . {N We note that the corollary diﬀers slightly from the result in [PTBW17], since we consider ﬁelds

according to the absolute discriminant ∆ Q of the Galois closure of K rather than the absolute K {Q discriminant ∆ of K itself. We remarkˇ thatˇ we can obtain the result in [PTBW17] by tweaking K{Q ˇ ˇ ˇ ˇ ˇ ˇ ζ F psq ˇ ˇ HpZqpXq “ K ; K{F P Z, ∆ ă X ζ psq K{F " F * ˇ ˇ and making the appropriate tweaks for each result as required.ˇ ˇ

71 Chapter 5

An Application: Effective Chebotarev Den- sity Theorem

In this chapter, we will give a brief overview of the work of Pierce, Turnage-Butterbaugh and Wood on a new eﬀective Chebotarev density result. This will demonstrate an application of our result on the average zero-free region of Kowalski–Michel Q-ameanable families developed in the previous chapter. Readers are encouraged to refer to [PTBW17] for a more detailed account of their work.

5.1 Previously Known Chebotarev Density Results

Let F be a number ﬁeld, and consider a normal extension K{F with Galois group GalpK{F q. For C Ď GalpK{F q a conjugacy class, consider the set

PpC,K{F q “ t p C OF ; p is unramified in K, σp “ C u Chebotarev density result, first proved by Chebotarev in [Tsc26], describes the density of the set PpC,K{F q. Chebotarev Density Theorem. Let F be a number field, and consider a normal extension K{F with Galois group GalpK{F q. Define PpF q “ t p C OF u For C Ď GalpK{F q a conjugacy class, the set

PpC,K{F q “ t p C OF ; p is unramiﬁed in K, σp “ C u has density p P PpC,K{F q; N F ppq ď x |C| lim Q “ xÑ8 p P PpF q; N F ppq ď x |GalpK{F q| ˇ Q (ˇ ˇ ˇ Chebotarev Density Theoremˇ is a celebrated result in(ˇ analytic number theory, being the principal tool in deducing many analytic propertyˇ of subset of primes.ˇ It is also a generalisation of Dirichlet’s theorem on primes in arithmetic progressions. We deﬁne the following prime counting function in OF , given by p F πpx, C,K{F q “ P PpC,K{F q; NQ ppq ď x One can rephrase the Chebotarev Density Theoremˇ as (ˇ ˇ ˇ |C| |C| x πpx, C,K{F q „ Lipxq „ |GalpK{F q| |GalpK{F q| log x where we denote x 1 Lipxq “ dt log t ż2 72 CHAPTER 5. AN APPLICATION: EFFECTIVE CHEBOTAREV DENSITY THEOREM

Chebotarev’s original result, while powerful, gave no information regarding the error terms of πpx, C,K{F q, which is deficient in deducing more delicate results in prime counting. We are interested in an effective version of Chebotarev Density Theorem, i.e. one with more explicit result regarding the error terms of πpx, C,K{F q. Lagarias and Odlyzko proved two versions of effective Chebotarev Density Theorems in [LO77], which we will quote here. The first version is a version which is conditional on the Generalised Riemann Hypothesis, which has been refined further by Serre in [Ser81].

Theorem 5.1.1. Let F be a number ﬁeld, and consider a normal extension K{F with Galois group

GalpK{F q. Let the absolute discriminant of K be |∆K{Q|. If the Generalised Riemann Hypothesis holds for ζK psq, then there exists a constant C such that for C Ď GalpK{F q a conjugacy class, and for x ě 2, we have that

|C| |C| 1 rK: s πpx, C,K{F q ´ Lipxq ď C x 2 logp|∆ |x Q q |GalpK{F q| |GalpK{F q| K{Q ˇ ˇ ˇ ˇ ˇ ˇ This versionˇ of eﬀective Chebotarev densityˇ result has been useful in many applications, but is unfortunately dependent on the veracity of Generalised Riemann Hypothesis. However, Lagarias and Odlyzko proved an unconditional version, which we state below.

Theorem 5.1.2. Let F be a number ﬁeld, and consider a normal extension K{F with Galois group

GalpK{F q. Let the absolute discriminant of K be |∆K{Q|. If rK : Qs ą 1, then ζK psq has at most one zero β0 in the region

1 1 s P C; Repsq ě 1 ´ , Impsq ď # 4 log ∆K{Q 4 log ∆K{Q + ˇ ˇ ˇ ˇ The exceptional zero β0, if it exists, is real andˇ simple.ˇ Further, thereˇ existsˇ eﬀectively computable 2 10rK:Qsplog|∆K{ |q constants C1 and C2 such that for all x ě e Q ,

1 C C log x 2 | | | | β0 ´C2p q πpx, C,K{F q ´ Lipxq ď Lipx q ` C xe rK:Qs |GalpK{F q| |GalpK{F q| 1 ˇ ˇ ˇ ˇ β0 ˇ ˇ The x term isˇ present only when β0 exists. ˇ However, this version is not useful in a lot of application, simply because of the requirement on x is too large and restrictive.

5.2 Effective Chebotarev Density Theorem of Pierce–Turnage-Butterbaugh–Wood

In [PTBW17], Pierce, Turnage-Butterbaugh and Wood obtained a new eﬀective Chebotarev density δ result which removes the term corresponding to the exceptional zero and to hold for x as small as |∆K{Q| for any ﬁxed δ ą 0. Here, we will give a brief sketch of the proof of their result. For a detailed account of the proof, refer to [PTBW17].

73 CHAPTER 5. AN APPLICATION: EFFECTIVE CHEBOTAREV DENSITY THEOREM

5.2.1 Statement of Effective Chebotarev Density Theorem of Pierce–Turnage-Butterbaugh–Wood The idea is to reach a compromise between the two versions of effective Chebotarev density result of Lagarias–Odlyzko. Specifically, we wish to prove an effective Chebotarev density result which is conditional on a weaker zero-free condition on the Dedekind zeta-function ζK psq, in the hopes of trading a more powerful result. In particular, instead of a zero-free condition on ζK psq in the form of Generalised Riemann Hypothesis, we instead have a zero-free condition on ζK psq , which is a much weaker condition ζF psq that is accomodating to phenomenon like the possible existence of an exceptional zero of ζF psq in its standard zero-free region. Due to the essential weakening of the Generalised Riemann Hypothesis condition, one expects to pay a cost for such a trade. In light of our result in the previous chapter and the assumptions in [PTBW17], the cost manifests as a result which is true for “almost all” fields instead of a result for all fields. We first recall the following zero-free region results for ζF psq for a number field F . C.f. [IK04] for more details and proof of the following.

Lemma 5.2.1. Let F {Q be a number ﬁeld with absolute discriminant |∆F {Q|. There exists an absolute constant cF ą 0 such that ζF psq has no zeroes in the region c s P ; Repsq ě 1 ´ F C 2 rF :Qs # rF : Qs log ∆F {Q p|Impsq| ` 3q + pF q `ˇ ˇ ˘ except for a pssibly simple real zero β0 ă 1. ˇ ˇ In the case where F “ Q, we have a stronger result, due to Vinogradov [Vin58] and Korobov [Kor58], which is the current best known result in this direction.

Lemma 5.2.2. There exists an absolute constant cQ ą 0 such that ζQpsq has no zeroes in the region

cQ s P C; Repsq ě 1 ´ 2 1 # plog p|t| ` 2qq 3 plog log p|t| ` 3qq 3 +

With the above zero-free region on ζF psq, we can state the effective Chebotarev Density Theorem of Pierce–Turnage-Butterbaugh–Wood. We first state it for general number field F .

Theorem 5.2.3. Let F {Q be a number ﬁeld with absolute discriminant |∆F {Q|. Let cF be such that ζF psq is zero-free in the region c s P ; Repsq ě 1 ´ F C 2 rF :Qs # rF : Qs log ∆F {Q p|Impsq| ` 3q + pF q `ˇ ˇ ˘ and let β0 ă 1 be the exceptional real zero of ζF psqˇin thisˇ region if it exists. Let A ě 2 and let 1 0 ă δ ď 2A . Let G be a transitive subgroup of Sn, where n P N with rF : Qs|G| ě 2. Then, there exists eﬀectively computable absolute constants C1,C2,C3,C4, and a constant pF q Dpδ, A, |G|, cF , β0 , rF : Qs,C1,C2,C3q pF q such that for any Galois extension K{F with GalpK{F q – G, |∆K{Q| ą Dpδ, A, |G|, cF , β0 , rF : ζK psq s,C1,C2,C3q and being zero-free in the region Q ζF psq

2 2 δ δ r1 ´ δ, 1s ˆ ´ log ∆K{Q , log ∆K{Q ” ` ˇ ˇ˘ ` ˇ ˇ˘ ı ˇ ˇ ˇ ˇ 74 CHAPTER 5. AN APPLICATION: EFFECTIVE CHEBOTAREV DENSITY THEOREM there exists constants

pF q ki “ kipδ, A, |G|, cF , β0 , |∆F {Q|, rF : Qs, rK : Qs,C3,C4q

k 2 k2plog log |∆ | 3 q for i “ 1, 2, 3 such that for any conjugacy class C Ď G, we have for all x ě k1e K{Q , |C| |C| x πpx, C,K{F q ´ Lipxq ď |G| |G| plog xqA ˇ ˇ ˇ ˇ We further state the resultˇ specialised to the case ofˇ F . ˇ ˇ “ Q

Theorem 5.2.4. Let cQ be such that ζQpsq is zero-free in the region

cQ s P C; Repsq ě 1 ´ 2 1 # plog p|t| ` 2qq 3 plog log p|t| ` 3qq 3 +

1 Let A ě 2 and let 0 ă δ ď 2A . Let G be a transitive subgroup of Sn, where n P N with |G| ě 2. Then, there exists eﬀectively computable absolute constants C1,C2,C3,C4, and a constant

Dpδ, A, |G|, cQ,C1,C2,C3q

such that for any Galois extension K{Q with GalpK{Qq – G, |∆K{Q| ą Dpδ, A, |G|, cQ,C1,C2,C3q and ζK psq being zero-free in the region ζQpsq

2 2 δ δ r1 ´ δ, 1s ˆ ´ log ∆K{Q , log ∆K{Q ” ` ˇ ˇ˘ ` ˇ ˇ˘ ı there exists constants ˇ ˇ ˇ ˇ ki “ kipδ, A, |G|, cQ, rK : Qs,C3,C4q 5 1 k 2 k2plog log |∆ | 3 q 3 plog log log |∆ | q 3 for i “ 1, 2, 3 such that for any conjugacy class C Ď G, we have for all x ě k1e K{Q K{Q , |C| |C| x πpx, C,K{Qq ´ Lipxq ď |G| |G| plog xqA ˇ ˇ ˇ ˇ ˇ ˇ ˇ5.2.2 Sketch ofˇ the Proof The proof of both versions of the eﬀective Chebotarev Density Theorem of Pierce–Turnage-Butterbaugh– Wood, Theorem 5.2.3 and Theorem 5.2.4, can be done simulatenously with the appropriate constants, and so we will proceed by focusing on Theorem 5.2.3, with appropriate changes noted for Theorem 5.2.4. First, we will deduce explicitly the zero-free region of ζK psq based on the assumption of Theorem pF q pF q pF q 5.2.3. Let δ0pδ, β0 q ď δ be such that 1 ´ δ0pδ, β0 q ą β0 , with the understanding that δ0pδq “ δ if pF q β0 do not exists for F . We now consider the region

2 2 pF q δ δ r1 ´ δ0pδ, β0 q, 1s ˆ ´ log ∆K{Q , log ∆K{Q ” ` ˇ ˇ˘ ` ˇ ˇ˘ ı which is still a zero-free region of ζK psq by assumptionˇ of theˇ theorem,ˇ andˇ note that ζK psq is entire by ζF psq ζF psq Aramata–Brauer Theorem. By considering the intersection of this region with the standard zero-free region of ζF psq aﬀorded by Lemma 5.2.1, we get that ζF psq is zero-free in the region

t s P C; Repsq ě 1 ´ TF psq u

75 CHAPTER 5. AN APPLICATION: EFFECTIVE CHEBOTAREV DENSITY THEOREM where we deﬁne pF q δ0pδ, β0 q, if | Impsq| ď T0 2 cF δ , if T0 ď | Impsq| ď log ∆K and F ‰ TF psq “ $ F : 2 log ∆ Im s 3 rF :Qs {Q Q r Qs p| F {Q|p| p q|` q q ’ 2 ’ cQ δ & 2 1 , if T0 ď | Impsq| ď `log ˇ∆K{Qˇ˘ and F “ Q plogp|t|`2qq 3 plog logp|t|`3qq 3 ˇ ˇ ’ ` ˇ ˇ˘ pF q and T0 is the height’ where the standard zero-free region of ζF psq intersectsˇ the lineˇ Repsq “ 1´δ0pδ, β0 q. % 2 δ For simplicity, we assume that T0 ď log ∆K{Q . A tedious computation shows that it is suﬃcient instead to assume |∆ | ě D pc , δq K{Q 0 F ` ˇ ˇ˘ ˇ cFˇ ee , if F ‰ Q D0pcF , δq “ c 2 pexppexpp Q qqq δ #e δ , if F “ Q The remaining part of the proof can be roughly divided into two major parts, which we sketch below.

Proof for Large x We now wish to leverage Theorem 5.1.2. Note that 1 pF q pF q 1 4δ0pδ,β0 q |∆K{Q| ě e ùñ 1 ´ δ0pδ, β0 q ă 1 ´ 4 log ∆K{Q 1 pF q 4δ0pδ,β0 q ˇ ˇ Then, for Galois extensions K{F with |∆K{Q| ě max e , 4 , we haveˇ thatˇ the region ˆ ˙ 1 1 s P C; Repsq ě 1 ´ , Impsq ď # 4 log ∆K{Q 4 log ∆K{Q + contains no exceptional zero of ζK psq. Thus, byˇ Theoremˇ 5.1.2, thereˇ existsˇ effectively computable ˇ ˇ 2 ˇ ˇ 10rK:Qsplog|∆K{ |q constants C1 and C2 such that for all x ě e Q , 1 C log x 2 | | Ć2p q πpx, C,K{F q ´ Lipxq ď C xe rK:Qs |G| 1 ˇ ˇ ˇ ˇ 10rK: splog ∆ q2 It now remains to exhibit a constantˇ D1 such that for |∆ˇ | ě D1, we get for x ě e Q | K{Q| that ˇ ˇ K{Q 1 log x 2 C x Ć2p q | | C xe rK:Qs ď 1 |G| plog xqA Pierce–Turnage-Butterbaugh–Wood verified and calculated in Appendix C of [PTBW17] the conditions 1 1 1 on D , which depends on A, rF : Qs,C1,C2, and so we define D “ D pA, rF : Qs,C1,C2q. Therefore, by letting 1 F pF q p q 4δ0pδ,β q 1 D1pδ, A, β0 , rF : Qs,C1,C2q “ max 4, e 0 ,D pA, rF : Qs,C1,C2q ˆ ˙ pF q we get that for Galois extensions K{F with |∆K{Q| ě D1pδ, A, β0 , rF : Qs,C1,C2q, for all x ě 10rK: splog ∆ q2 e Q | K{Q| , |C| |C| x πpx, C,K{F q ´ Lipxq ď |G| |G| plog xqA ˇ ˇ ˇ 10rK:ˇ splog ∆ q2 This concludes the proof of Theoremˇ 5.2.3 for x ě e ˇQ | K{Q| . ˇ ˇ 76 CHAPTER 5. AN APPLICATION: EFFECTIVE CHEBOTAREV DENSITY THEOREM

Proof for Small x The main difficulty of Theorem 5.2.3 lies in the proof of the result for the remaining region of x. Pierce–Turnage-Butterbaugh–Wood first define an auxilary weighted prime-counting function

F p ψpx, C,K{F q “ logpNQ q p OF mP C F mN p unramiﬁedÿ in OK N ÿp ďx Q m pσpq “C

m m where pσpq denotes the conjugacy class of σP for any PCOK lying over p. Note that this is well-deﬁned, since p is unramiﬁed in OK . Pierce–Turnage-Butterbaugh–Wood then proved the following auxilary proposition.

Proposition 5.2.5. Let F {Q be a number ﬁeld with absolute discriminant |∆F {Q|. Let cF be such that ζF psq is zero-free in the region

c s P ; Repsq ě 1 ´ F C 2 rF :Qs # rF : Qs log ∆F {Q p|Impsq| ` 3q +

pF q `ˇ ˇ ˘ and let β0 ă 1 be the exceptional real zero of ζF psqˇin thisˇ region if it exists. Let A ě 2 and let 1 0 ă δ ď 2A . Let G be a transitive subgroup of Sn, where n P N with rF : Qs|G| ě 2. Then, there exists eﬀectively computable absolute constants C3,C4 such that for any 0 ă c0 ď 1, there exists a parameter D2pc0, δ, A, |G|, rF : Qs,C3q such that for any Galois extension K{Q with

1 pF q 4δ0pδ,β0 q |∆K{Q| ě max 4, e ,D0pcF , δq,D2pc0, δ, A, |G|, rF : Qs,C3q ˆ ˙ and ζK psq being zero-free in the region ζF psq

2 2 δ δ r1 ´ δ, 1s ˆ ´ log ∆K{Q , log ∆K{Q ” ` ˇ ˇ˘ ` ˇ ˇ˘ ı there exists constants ˇ ˇ ˇ ˇ

1 1 pF q ki “ kipc0, δ, A, |G|, cF , β0 , |∆F {Q|, rF : Qs, rK : Qs,C3,C4q

for i “ 1, 2, 3 such that for any conjugacy class C Ď G, for all

1 1 k3 2 10 K: log ∆ 2 1 k2plog log |∆K{ | q r Qsp | K{Q|q k1e Q ď x ď e

we have |C| |C| x ψpx, C,K{F q ´ x ď c |G| 0 |G| plog xqA´1 ˇ ˇ ˇ ˇ The following is the versionˇ specialised to the caseˇ of F “ . ˇ ˇ Q

Proposition 5.2.6. Let cQ be such that ζQpsq is zero-free in the region

cQ s P C; Repsq ě 1 ´ 2 1 # plog p|t| ` 2qq 3 plog log p|t| ` 3qq 3 +

77 CHAPTER 5. AN APPLICATION: EFFECTIVE CHEBOTAREV DENSITY THEOREM

1 Let A ě 2 and let 0 ă δ ď 2A . Let G be a transitive subgroup of Sn, where n P N with |G| ě 2. Then, there exists eﬀectively computable absolute constants C3,C4 such that for any 0 ă c0 ď 1, there exists a parameter D2pc0, δ, A, |G|,C3q such that for any Galois extension K{Q with

1 4δ |∆K{Q| ě max 4, e ,D0pcF , δq,D2pc0, δ, A, |G|,C3q ´ ¯ and ζK psq being zero-free in the region ζQpsq

2 2 δ δ r1 ´ δ, 1s ˆ ´ log ∆K{Q , log ∆K{Q ” ` ˇ ˇ˘ ` ˇ ˇ˘ ı there exists constants ˇ ˇ ˇ ˇ 1 1 ki “ kipc0, δ, A, |G|, cQ, rK : Qs,C3,C4q for i “ 1, 2, 3 such that for any conjugacy class C Ď G, for all

1 5 1 1 k3 3 2 3 10 K: log ∆ 2 1 k2plog log |∆K{ | q plog log log |∆K{ | q r Qsp | K{Q|q k1e Q Q ď x ď e we have |C| |C| x ψpx, C,K{F q ´ x ď c |G| 0 |G| plog xqA´1 ˇ ˇ ˇ ˇ We will not provide a proofˇ of this, instead referingˇ interested readers to [PTBW17]. In particular, ˇ ˇ 1 D2pc0, δ, A, |G|, rK : Qs,C3q is described in Lemma 5.7 of [PTBW17], while the ki in both propositions are described in their proof of the propositions. On the other hand, C3 is derived from an application of Theorem 7.1 of [LO77], and C4 is derived from bounding the Spx, T q term that appears in the theorem. Pierce–Turnage-Butterbaugh–Wood then deﬁne the function

F p θpx, C,K{F q “ logpNQ q pCOF p unramiﬁedÿ in OK N F pďx Q pσpq“C

The goal now is to approximate this sum by πpx, C,K{F q. Pierce–Turnage-Butterbaugh–Wood computed that 3 1 |ψpx, C,K{F q ´ θpx, C,K{F q| ď rK : sx 2 log x 2 log 2 Q Under the assumption of Proposition 5.2.5, the x in the region

1 1 k3 2 10 K: log ∆ 2 1 k2plog log |∆K{ | q r Qsp | K{Q|q k1e Q ď x ď e satisﬁes the inequality 3 1 |C| x rK : Qsx 2 log x ď c0 2 log 2 |G| plog xqA´1 Similarly in the case F “ Q, the assumption of Proposition 5.2.6 forces x in the region

1 5 1 1 k3 3 2 3 10 K: log ∆ 2 1 k2plog log |∆K{ | q plog log log |∆K{ | q r Qsp | K{Q|q k1e Q Q ď x ď e to satisfy the same inequality

3 1 |C| x rK : Qsx 2 log x ď c0 2 log 2 |G| plog xqA´1

78 CHAPTER 5. AN APPLICATION: EFFECTIVE CHEBOTAREV DENSITY THEOREM

This allows us to conclude that |C| |C| x θpx, C,K{F q ´ x ď 2c |G| 0 |G| plog xqA´1 ˇ ˇ ˇ ˇ ˇ ˇ under the same assumptions ofˇ Proposition 5.2.5 andˇ Proposition 5.2.6 respectively. For simplicity, we let

1 1 k3 2 1 k2plog log |∆K{ | q k1e Q , if F ‰ Q x0pF q “ 1 5 1 1 k3 3 2 3 1 k2plog log |∆K{ | q plog log log |∆K{ | q #k1e Q Q , if F “ Q

10 K: log ∆ 2 1 r Qsp | K{Q|q with the appropriate ki. By partial summation, we get that for x0pF q ď x ď e , x θpt, C,K{F q θpx, C,K{F q πpx, C,K{F q “ dt ` t log2 t log x żλ x0pF q θpt, C,K{F q x θpt, C,K{F q θpx, C,K{F q “ dt ` dt ` t log2 t t log2 t log x żλ żx0pF q where λ is the minimum among N F p running over p O . With the trivial bound of θ t, C,K F Q p q C F p { q ď rK : Qst log t for λ ď t ď x0pF q, we have that

x0pF q θpt, C,K{F q dt ď rK : QsLipx0pF qq t log2 t żλ Integration by parts give that

x 1 x x B 1 x x pF q t dt ` “ t ´ dt ` “ Lipxq ´ Lipx pF qq ´ 0 t log2 t log x Bt log t log x 0 log x pF q żx0pF q żx0pF q ˆ ˙ ˆ 0 ˙ In light of the our approximation of θpx, C,K{F q, we have that in summary that

|C| πpx, C,K{F q “ Lipxq ` E |G| where

x x0pF q |C| 1 |C| x |E| ď rK : QsLipx0pF qq ` Lipx0pF qq ´ ` 2c0 dt ` 2c0 log x pF q |G| plog tqA`1 |G| plog xqA ˇ 0 ˇ żx0pF q ˇ ? ˇ ˇ |C| x ˇ 1 x 1 |C| x ď prK : Qs ` 2qLipx0pFˇ qq ` 2c0 ˇ dt ` dt ` 2c0 |G| plog tqA`1 ? plog tqA`1 |G| plog xqA ˜żx0pF q ż x ¸ x A`1 |C| 1 2 |C| x ď prK : Qs ` 2qLipx0pF qq ` 2c0 x 2 ` dt ` 2c0 |G| ? plog xqA`1 |G| plog xqA ˆ ż x ˙ A`1 |C| 1 2 x |C| x ď prK : Qs ` 2qLipx0pF qq ` 2c0 x 2 ` ` 2c0 |G| plog xqA`1 |G| plog xqA ˆ ˙ |C| 1 A`2 |C| x ď prK : Qs ` 2qLipx0pF qq ` 2c0 x 2 ` p2 ` 2qc0 |G| |G| plog xqA

79 CHAPTER 5. AN APPLICATION: EFFECTIVE CHEBOTAREV DENSITY THEOREM

By noting that 10rK: splog ∆ q2 1 x x pF q ď x ď e Q | K{Q| ùñ x 2 ď 0 plog xqA we have that A`2 |C| x |E| ď prK : Qs ` 2qLipx0pF qq ` p2 ` 4qc0 |G| plog xqA 1 Lastly, by enlarging the respective ki to the appropriate ki, we can get that in the range x0pF q ď x ď 10rK: splog ∆ q2 e Q | K{Q| , A`2 |C| x prK : Qs ` 2qLipx0pF qq ď p2 ` 4qc0 |G| plog xqA |C| x ùñ |E| ď p2A`3 ` 8qc 0 |G| plog xqA 1 In conclusion, by setting c0 “ 2A`2`8 and combining with the case of large x, we now have that

|C| |C| x πpx, C,K{F q ´ Lipxq ď |G| |G| plog xqA ˇ ˇ ˇ ˇ ˇ ˇ as desired in the required rangeˇ of x. For completion, weˇ will record the resulting ki and threshold D for the absolute discriminant below:

1 A`3 A´1 A pF q A 3 72C3C4p2 ` 8q10 |G|rK : Qs δ0pδ,β0 q k “ p2 ` ` 8q 1 δ2 ˆ ˙ 2A 4ArF : Qs3 k “ max , ` 2A 2 pF q c δ ˜δ0pδ, β0 q F ¸ 1 A`3 A`3 1 6p12C3p2 ` 8qq 2A`1 p12C3C4p2 ` 8qq 2A |∆F { |rK : Qs k “ Q 3 δA pF q D “ max D0pcF , δq,D1pδ, A, β0 , rF : Qs,C1,C2q,D2pc0, δ, A, |G|, rF : Qs,C3q ´ ¯ 5.3 Removal of Strong Artin Conjecture

As a consequence of what we developed in the previous chapter, we will now demonstrate the following theorem, which specialises to a strengthening of Theorem 7.1 in [PTBW17].

Theorem 5.3.1. Let F {Q be Galois, and assume Conjecture 3.2.9. Let

F Z Ď K{F ; rK : F s “ n, GalpK {F q “ G ! ) 1 Suppose HpZq is a sterile subfamily of FF,G with contamination index τ. Let A ě 2. Then, for every 0 ă ε ă 1 with 2 τ ε ∆ “ 1 ´ ´ ą 0 dF,GpHpZqq 2dF,GpHpZqq such that ε 1 δ “ ď 20mrF : QsaF,GpHq ` 8dF,GpHq ` 4ε 2A

80 CHAPTER 5. AN APPLICATION: EFFECTIVE CHEBOTAREV DENSITY THEOREM

there exists eﬀectively computable absolute constants C1,C2,C3,C4, a constant

pF q D “ DpZ, τ, ε, A, |G|, cF , β0 , rF : Qs,C1,C2,C3q

such that for X P N, all but DXτ`ε ﬁelds K{F in

K F Z; ∆ F X { P K {F ă ! ˇ ˇ ) ˇ ˇ has that there exists constants ˇ ˇ pF q F ki “ kipδ, A, |G|, cF , β0 , |∆F {Q|, rF : Qs, rK : Qs,C3,C4q

k 2 k2plog log |∆ F | 3 q for i “ 1, 2, 3 with for any conjugacy class C Ď G, we have for all x ě k1e K {Q ,

F |C| |C| x πpx, C, K {F q ´ Lipxq ď |G| |G| plog xqA ˇ ˇ ˇ ˇ ˇ ˇ Proof. By Theorem 5.2.3, we getˇ eﬀectively computable absoluteˇ constants C1,C2,C3,C4, and a constant pF q Dpδ, A, |G|, cF , β0 , rF : Qs,C1,C2,C3q

pF q such that for any K{F P Z with GalpK{F q – G, |∆ F | ą Dpδ, A, |G|, cF , β , rF : s,C1,C2,C3q K {Q 0 Q which is not δ-exceptional, there exists constants

pF q F ki “ kipδ, A, |G|, cF , β0 , |∆F {Q|, rF : Qs, rK : Qs,C3,C4q

k 2 k2plog log |∆ F | 3 q for i “ 1, 2, 3 such that for any conjugacy class C Ď G, we have for all x ě k1e K {Q ,

F |C| |C| x πpx, C, K {F q ´ Lipxq ď |G| |G| plog xqA ˇ ˇ ˇ ˇ ˇ ˇ pF q We proceed to count ﬁelds whichˇ either are δ-exceptionalˇ or with |∆ F | ď Dpδ, A, |G|, cF , β , rF : K {Q 0 Qs,C1,C2,C3q as in the proof of Corollary 4.2.13, with using

1 pF q D “ maxpD pZ, τ, εq,Dpδ, A, |G|, cF , β0 , rF : Qs,C1,C2,C3qq and obtain the result.

Specialising to F “ Q, and noting that in the case listed in Proposition 4.2.18, as long as

I d ,GpHn pQ,Gqq ε ă Q 4 we have that τ ε ∆ “ 1 ´ ´ ą 0 I I dQ,GpHn pQ,Gqq 2dQ,GpHn pQ,Gqq we get the following.

81 CHAPTER 5. AN APPLICATION: EFFECTIVE CHEBOTAREV DENSITY THEOREM

Corollary 5.3.2. Let A ě 2, and assume Conjecture 3.2.9. For G, I being one of those listed in Proposition 4.2.18 with the corresponding contamination index τ, we have that for any 0 ă ε ă I 1 dQ,GpHn pQ,Gqq minp 2 , 4 q with ε 1 δ “ ď I 20m ` 8dQ,GpHn pQ,Gqq ` 4ε 2A there exists eﬀectively computable absolute constants C1,C2,C3,C4, a constant

D “ Dpn, G, I , τ, ε, A, cQ,C1,C2,C3q

such that for X P N, all but DXτ`ε ﬁelds K{F in

I K{F P Z p ,Gq; ∆ Q ă X n Q K {Q ! ˇ ˇ ) ˇ ˇ has that there exists constants ˇ ˇ Q ki “ kipδ, A, |G|, cQ, rK : Qs,C3,C4q

5 1 k3 3 2 3 k2plog log |∆ Q | q plog log log |∆ Q | q for i “ 1, 2, 3 with for any conjugacy class C Ď G, we have for all x ě k1e K {Q K {Q ,

Q |C| |C| x πpx, C, K {Qq ´ Lipxq ď |G| |G| plog xqA ˇ ˇ ˇ ˇ ˇ ˇ We note that the corollaryˇ diﬀers from Theorem 7.1ˇ of [PTBW17] in that it does not require the Strong Artin Conjecture for any case considered. We also consider ﬁelds according to the absolute

discriminant ∆ Q of the Galois closure of K rather than the absolute discriminant ∆K{ of K itself. K {Q Q Again, we remarkˇ thatˇ we can obtain the result in [PTBW17] by tweaking ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ζ F psq HpZqpXq “ K ; K{F P Z, ∆ ă X ζ psq K{F " F * ˇ ˇ and making the appropriate tweaks for each result as required.ˇ ˇ

82 Chapter 6

Conclusion

We will highlight here a few open questions that we have yet to resolve in this thesis. The main open problem is the analytical issue with non-simple poles of L-functions in our setting, encapsulated in Conjecture 3.2.9. Note that since the possible poles of Rankin-Selberg of cuspidal automorphic L-functions are simple, this issue does not manifest itself in [KM02]. The solution to this conjecture will truly make the result of Pierce–Turnage-Butterbaugh–Wood unconditional. The other major open is ﬁnding the relation between ConvK and ConvK in the case where K{K ν is not Galois. A solution towards this will possibly allow us to prove that FF,G is Kowalski–Michel ν Q-ameanable for any number ﬁeld F , resulting in a Kowalski–Michel estimate of zero-regions of FF,G. This will allow us to remove the condition of F {Q being Galois in all subsequent results we proved in this thesis. Towards this end, we suspect that ν-regular representation can be written as a direct sum of induced representations, which will allow us to prove a relation we seek. In particular, we conjecture the following. K K Conjecture 6.0.3. For any extension K{K, there exists a representation ρ of GalpK {K q such that

K ν K GalpK {Kq reg ˝ π K K “ Ind K K ρ GalpK{F q GalpK {K q GalpK {K q The following Hasse diagram gives a picture of the conjecture.

K K

K K p?q ρ

K K K regν GalpK{F q K regν GalpK{F q F

Figure 6.0.1: Hasse Diagram involved in Conjecture 6.0.3 We also wish to point out the paper of Thorner and Zaman [TZ18], which improves upon the result of Kowalski and Michel. We did not yet consider how our work might be aﬀected, but we suspect that with a suitable tweak of our methods in this thesis, we might be able to combine with the methods of Thorner and Zaman to produce an improved result on the zero-region estimate of families of Artin L-functions.

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