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Tame Symbols and Reciprocity Laws in Number Theory and Geometry Vanessa Radzimski

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Tame Symbols and Reciprocity Laws in Number Theory and Geometry Vanessa Radzimski Florida State University Libraries Honors Theses The Division of Undergraduate Studies 2012 Tame Symbols and Reciprocity Laws in Number Theory and Geometry Vanessa Radzimski Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] Abstract The tame symbol serves many purposes in mathematics, and is of particular value when we use it to evaluate curves over certain number fields. A well known example is that of the Hilbert symbol, which gives us insight into the existence of a rational solution to a conic of the form ax2 + by2 = c for a,b,c Q×. In order to properly examine this symbol, we must ∈ gain a solid understanding into the p-adic completion of the rationals, Qp. We will explore the algebraic construction of the subring of p-adic integers, Zp, whose field of fractions is Qp itself. In general, we may look at a type of tame symbol, which we call a local symbol, that we take over an algebraic curve defined over a field into some abelian group G. The properties of these local symbols correspond directly to those of the Hilbert symbol. We then examine if it is possible to define a type of local symbol over a degree 2 extension of Z, the Gaussian Integers Z[i]. The construction of this symbol is analogous to one for a degree 2 extension of Z which is a Euclidean domain. Central extensions of groups are explored to give a foundation for viewing the tame symbol in terms of determinates as viewed by Parshin. Keywords: Reciprocity, tame symbol, p-adic i THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES TAME SYMBOLS AND RECIPROCITY LAWS IN NUMBER THEORY AND GEOMETRY By VANESSA RADZIMSKI A Thesis submitted to the Department of Mathematics in partial fulfillment of the requirements for graduation with Honors in the Major Degree Awarded: Spring, 2012 The members of the Defense Committee approve the thesis of Vanessa Radzimski defended on April 17, 2012.: Dr. Ettore Aldrovandi Thesis Director Dr. Susan Blessing Outside Committee Member Dr. Wolfgang Heil Committee Member ii TABLE OF CONTENTS 1 Introduction 1 2 Rational Points on Conics 3 2.1 Properties of the Hilbert Symbol . 4 2.2 p-adicNumberFields............................... 5 2.3 Qp constructedasaninverselimit . 7 2.4 The product formula for the Hilbert symbol . 10 3 Local Symbols over an algebraic curve 14 3.1 General construction . 14 3.2 Local symbol into a multiplicative group . 17 4 A Local Symbol for the Gaussian Integers, Z[ı] 18 4.1 Primes in Z[i]................................... 18 4.2 Construction of the local symbol . 20 5 Extensions of Groups 24 5.1 Central Extensions of Groups . 24 5.2 Commutator of an Extension . 26 6 Conclusion 27 iii CHAPTER 1 INTRODUCTION The research described in this paper focuses on reciprocity laws and geometric symbols in number theory. We start out with the concrete example of the Hilbert symbol, where we study certain conics and their solutions in Q. By means of the Law of Quadratic Reciprocity, we develop a symbol, the Hilbert symbol, that serves as the tool to realize the existence of such solutions. Introduced by David Hilbert, the proof of the symbol relies heavily on knowledge of the p-adic numbers. The p-adic numbers, denoted by Qp are a completion of the rationals with respect to a particular metric. The most well-known example of a completion of the rationals, is the real numbers R. The p-adic numbers give us a completely different perspective on the rationals in terms of congruence modulo a prime in Z. There are numerous constructions of the p-adics, but we will explore the algebraic construction in depth. After we obtain a solid understanding of the p-adics, we are able to study particular properties satisfied by the Hilbert symbol and their implications. Next, we investigate the generalization of the Hilbert symbol, a local symbol over an algebraic curve. It is convenient to view a curve as an extension of the field of rational functions of k[t] modulo an irreducible polynomial, where k is a field. This is explicitly 1 denoted by k(t)[x]/(f(t,x)), with f(t,x) irreducible. We find that this is a ramified covering of k(t). Since k(t) is a field, k(t)[x] is a principal ideal domain, which implies that irreducible elements are primes. We may do this for every prime ℘ in k(t)[x]. There is a strong parallel between this geometric case of curves and an arithmetic case of number fields, that is, an extension of Z. The construction of a local differs a bit since Z is not a field, but the general properties between the geometric and analytic case are analogous. When considering a degree two extension of Z, call it Z(√ d), we note that − this is isomorphic to Z[t]/(t2 + d). We may consider prime elements of the extension and whether or not they are prime when projected into Z. By considering the primes in Z(√ d), − and viewing them as a curve themselves, we get a ramified covering of the primes of Z. The localization at each prime gives us a residue field k that we may work in for convenience. By considering completions for each prime and the field of quotients of the completion, which themselves are isomorphic to k[[t]] and k((t)) respectively, we come up similar calculations to that of the geometric case. We explicitly calculate a local symbol for the degree two extension of the integers, the Gaussian Integers Z[i], which is especially nice to work with since it is a Euclidean domain. The tame symbol we work with is defined over the 1-dimensional field k((t)), which satisfies the product formula for the symbol taken at two rational functions over all points of the curve, P1. We examine the commutator of an extension of groups in order to understand the current realm of research in this area of mathematics, where we view the symbol in terms of determinates and commensurable subspaces of infinite dimensional vector spaces. The current work of Parshin and Osipov focuses on the study of a multidimensional tame symbol, in particular the 2-dimensional case k((t1,t2)). 2 CHAPTER 2 RATIONAL POINTS ON CONICS Let a,b,c Q. If ax2 + by2 = c has a point (x ,y ) Q, we have a method to obtain all ∈ 0 0 ∈ rational points in the conic. Consider (x ,y ) Q and any line P in R2 that intersects 0 0 ∈ (x0,y0) in which P has rational slope. If we are given a rational point, finding others within the conic proves not to be difficult. The question is, given a conic of the form ax2 + by2 = 1 with a,b Q×, how can we determine the existence of rational points? The Hilbert Symbol ∈ and Quadratic Reciprocity law will play key roles in answering this question. Definition: We define the quadratic residue symbol for a,p Z⋆ with p an odd prime as: ∈ a 1, if x s.t. x2 a mod p = ∃ 2 ≡ p ( 1, if ∄x s.t. x a mod p − ≡ a This symbol is multiplicative with the assumption that gcd( b ,p) = 1. That is, ab a b = p p p Theorems: (1) Law of Quadratic Reciprocity: If q is an odd prime not equal to p, then q p−1 q−1 p =( 1) 2 2 p − q 3 1 p−1 1, if p 1 mod 4 (2) First Supplementary Law: − =( 1) 2 = ≡ p − ( 1, if p 3 mod 4 − ≡ 2 p2−1 1, if p 1, 7 mod 8 (2) Second Supplementary Law: =( 1) 8 = ≡ p − ( 1, if p 3, 5 mod 8 − ≡ Definition: For any a,b Q×, let us define the Hilbert Symbol (a,b) = 1 , for ∈ v {± } 1, if a> 0 or b> 0 v = or p prime. For v = , (a,b)∞ = . Thus, (a,b)∞ =1 if ∞ ∞ ( 1, if a< 0 and b< 0 − and only if x,y R s.t. ax2 + by2 = 1. ∃ ∈ The Hilbert symbol serves as a geometric version of the Legendre symbol used in the Law of Quadratic Reciprocity. As with the Legendre symbol, we obtain the following theorem that is a powerful tool for finding the existence of rational solutions of a conic. The proof of it requires knowledge of the p-adic numbers which we will examine in Section 2.2. Theorem: For a,b Q×, x,y Q satisfying ax2 + by2 = 1 if and only if (a,b) = 1, for ∈ ∃ ∈ v all v = and p prime. ∞ 2.1 Properties of the Hilbert Symbol Let us define the following subring of Q: Z = a a,b Z,p b . (p) { b | ∈ 6| } Z(p) is not only a subring of Q, but is in fact a discrete valuation ring. We then note that Z× = a a,b Z,p a and p b is then the group of units of Z . Using the (p) { b | ∈ 6 | 6 | } (p) properties of Z , we note that for any x Q∗, x may be uniquely written as pmu, where (p) ∈ m Z and u Z× ∈ ∈ (p) 4 . This construction will allow us to handle the properties of the Hilbert symbol better since we may write each rational in terms of the prime in which we take the Hilbert symbol over. Defintion: For a,b Q×, p prime, we define (a,b) as follows: Let a = piu,b = pjv ∈ p constructed in the same way as above.
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