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Dedekind Zeta Functions and Arithmetical Equivalence: Analytic and Algebraic Approaches

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Dedekind Zeta Functions and Arithmetical Equivalence: Analytic and Algebraic Approaches Corso di Laurea Magistrale in Matematica Dedekind Zeta functions and arithmetical equivalence: analytic and algebraic approaches Relatore: Prof. Giuseppe Molteni Tesi di laurea di: Francesco Battistoni Matricola 863778 Anno Accademico 2015-2016 Contents 1 Recalls of algebraic number theory 6 1.1 Numberfieldsandringsofintegers . 6 1.1.1 Numberfieldsandtheirembeddings . 6 1.1.2 Integralityandringsofintegers . 8 1.2 Dedekind domains, fractional ideals and class group . ... 9 1.2.1 Prime factorization in Dedekind domains . 9 1.2.2 DVR,ramificationandinertiadegrees . 11 1.2.3 Fractionalidealsandclassgroup . 14 1.3 Somealgebraicinvariants. 17 1.3.1 Normandtrace...................... 17 1.3.2 Thediscriminant . 18 1.4 Norm of ideals in K ....................... 20 1.4.1 NormofproperidealsO . 20 1.4.2 Normoffractionalideals . 22 1.5 TheoremsbyMinkowskianDirichlet . 22 1.6 Thedifferentideal ........................ 23 2 The Riemann Zeta Function 26 2.1 Dirichletseries .......................... 26 2.1.1 Convergencetheorems . 26 2.1.2 Arithmetical functions and Euler product. 28 2.2 The Riemann Zeta Function ζ(s) ................ 31 2.3 Thefunctionalequation . 32 3 The Dedekind Zeta Functions 35 3.1 Definitionandfirstproperties . 35 3.2 The functional equation of ζK and the class number formula . 37 3.2.1 Thefunctionalequation . 37 3.2.2 Analyticlemmas ..................... 39 3.2.3 Proof of the functional equation for ζ ( ; s)...... 41 K R 1 3.2.4 Computation of the residue in 1: the Class Number Formula .......................... 47 4 Arithmetical equivalence: the analytic way 49 4.1 Formulationandfirstresults . 49 4.1.1 Theproblem........................ 49 4.1.2 Resultonimmersionsanddegrees . 50 4.2 The Asymptotic formula for the non trivial zeros of ζK .... 51 4.2.1 Arithmetically equivalent fields have same discriminant 51 4.2.2 The non vanishing of ζK (s)overtwolines . 52 4.2.3 Recalls on entire functions and Phragmen-Lindel¨of The- orem............................ 54 4.2.4 Proofoftheasymptoticformula . 55 5 Arithmetical equivalence: the algebraic approach 64 5.1 Decomposition type and Gassmann equivalence . 64 5.1.1 Decomposition type of prime numbers in the rings of integers .......................... 64 5.1.2 Gassmannequivalence . 66 5.1.3 NumberfieldsandGaloisgrouptheory . 69 5.1.4 Arithmetically equivalent number fields share same roots ofunity .......................... 71 5.2 Arithmetically equivalent fields which are not isomorphic... 73 5.2.1 Cohomologicaltools . 73 5.2.2 Anexplicitexample. 77 6 Appendix 79 6.1 Summationformulas . 79 6.2 PropertiesoftheGammafunction. 80 6.3 Groupactions........................... 81 6.4 BasicGaloisTheory ....................... 82 2 Introduction “It’s this simple law, which every writer knows, of taking two opposites and putting them in a room together.” Trey Parker s The Riemann Zeta function ζ(s) := n n− is a typical example of a math- ematical instrument originally related to certain fields of Mathematics and suddenly become fundamental for theP study of other problems. Euler used this function (which for real s is a classic object of study in Analysis) to give a different proof of the existence of infinite prime numbers, while Riemann exploited its meromorphic continuation on the plane C of complex numbers for a deeper study of the distribution of the prime numbers (arriving to the formulation of the Riemann Hypothesis). The results obtained showed how much important can be analytic tools for the investigations of classical Arith- metic and Number Theory. If one is interested in the study of the number fields K and the rings of s integers , then the Dedekind Zeta functions ζ (s) := N(I)− (where OK K I the sum is made over the ideals of K and N(I) is the norm of the ideal I) play a similar role. The behavior ofO these functions is analoP gue to the Rie- mann Zeta function’s one (in fact, ζ(s) is just the Dedekind Zeta function related to the simplest number field, Q) and their analytic properties can be used to investigate the distribution of the prime ideals in K . The Dedekind Zeta functions are useful also to detect the values of theO many algebraic in- variants of K and K , like the discriminant, the class number, the regulator and the number ofO roots of unity contained in the number field. The first aim of this thesis is to give the basic theory of the Dedekind Zeta functions and to proof a bunch of their properties, among which there is the functional equation, which permits to meromorphically extend the Dedekind Zeta function and to find the Class Number formula, an important relation between the invariants of K and . OK 3 At this point one might be tempted to ask: how much the properties of the number field K depend on Dedekind Zeta function ζK (s)? If we are 0 given two number fields K and K which share the same Dedekind Zeta function, do they have something in common in their algebraic structure? Fields of this kind are said to be “arithmetically equivalent” and the prob- lem of “arithmetical equivalence” is the main object of study of the rest of the thesis. It will be shown that two fields with same Dedekind Zeta function have the same number of real and complex embeddings in C (hence same de- gree over Q), and same discriminant; all these results can be achieved just by exploiting the analytic properties of the functions ζK (s). The attempt to study further the arithmetical equivalence reveals however that one cannot discover something new just with the help of Analysis, and the way to get out from the impasse is to come back to Algebra. In fact, using concepts typical of Group Theory, it can be shown that two number fields arithmetically equivalent have also the same roots of unity. Moreover it is the theory of Group Cohomology (or at least its basic version) which permits to construct explicit examples of number fields with same Dedekind Zeta functions but which are not isomorphic, showing finally that the arith- metical equivalence is a relation weaker than the isomorphism. Here is how this work is organized. In Chapter 1 we introduce the algebraic concepts necessary for the definition of Dedekind Zeta functions: number fields, rings of integers, class group, dis- criminant, embeddings, group of unities, regulator, norm of ideals. Chapter 2 deals with the basic analytic properties of the Dirichlet series (of which the Dedekind Zeta functions are particular examples). Furthermore there is a brief exposition of the Riemann Zeta function and of its properties (like the functional equation and the meromorphic continuation). Dedekind Zeta functions and their analytic properties are the objects of Chapter 3. In this part of the work we focus on the convergence proper- ties and on the proof (the more detailed possible) of the functional equation, necessary to obtain a meromorphic continuation for ζK (s). Then it is com- puted the residue of these functions in 1, which gives the Class Number formula. In Chapter 4 we introduce the problem of arithmetical equivalence and we expose the main results achievable with an analytic approach. It is showed that two number fields with same Dedekind Zeta functions have same number of real embeddings and of complex embeddings in C, and that the degree of the two fields over Q is the same. Later it is proved that two arithmetically equivalent number fields have same discriminant: this follows by the proof of 4 an asymptotic formula for the function counting of the zeros of ζK (s) with real part between 0 and 1. In Chapter 5 it is proved that two number fields with same Dedekind Zeta functions induce the same kind of decomposition of the integer prime num- bers in their rings of integers (this is called the splitting type). Later it is introduced the concept of Gassmann equivalence, a group theoretical rela- tion which will be shown to be strictly related to the arithmetical equivalence; this instrument is the fundamental algebraic key to prove that arithmetically equivalent number fields have the same roots of unity and that two Galois extensions of Q have same Dedekind Zeta functions if and only if they are isomorphic. Finally we present a way to construct arithmetically equivalent number fields which are not isomorphic, and an explicit example is given. Chapter 6 is an appendix, which recalls results and concepts on summation formulas, the Gamma function, group actions and basic Galois Theory. 5 Chapter 1 Recalls of algebraic number theory 1.1 Number fields and rings of integers 1.1.1 Number fields and their embeddings A number field K is a field which is a finite extension of Q, meaning it is a finite dimensional Q-vector space (the dimension is called degree and denoted with [K : Q]). Remark: Being Q a field of characteristic 0, each of its finite field extensions K is separable, which means that for every α K its minimum polynomial ∈ fα(x) splits without multiple roots when is seen as a polynomial with complex coefficients. Theorem 1. (Primitive element theorem): Let K be a number field. Then there exists an element α K such that there is the ring isomorphism ∈ Q[x] K Q[α] ' ' (fα(x)) where fα(x) is the minimum polynomial of α over Q. Proof. The claim is true for the wider class of separable extensions of a generic field L. See Chapter 5 of [Mil] for the proof. Every number field is a subfield of the field Q of the algebraic numbers (where an algebraic number is a root of a polynomal with rational coefficients) 6 and so it can be embedded in the field C of complex numbers, but there is no canonical way to see this embedding. Theorem 2. (Artin’s Lemma): Let K be a number field of degree n. Then there is an isomorphism of C-algebras: n Ψ: K Q C C .
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