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

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. Let r =( 1)ijajb−i Z× . − ∈ (p) If p = 2, (a,b) = r mod p 6 p p r2−1 u−1 v−1 If p = 2, (a,b) =( 1) 8 ( 1) 2 2 . 2 − − Proposition: v be prime or . For a,b Q× ∞ ∈ (1) (a,b)v =(b,a)v

(2) (a,bc)v =(a,b)v(a,c)v (3) (a, a) = 1, If a = 1, then (a, 1 a) = 1. − v 6 − v (4) For p an odd prime, a,b Z× , we have that: ∈ (p) (a,b)p = 1 a mod p (a,pb)p = p (5) a,b Z× , then ∈ (p) 1, if a 1 mod 4 or b 1 mod 4 (a,b)2 = ≡ ≡ ( 1, if a b 1 mod 4 − ≡ ≡− 1, if a 1 mod 8 or a 1 2b mod 8 (a, 2b)2 = ≡ ≡ − ( 1, otherwise − 2.2 p-adic Number Fields

We recall that (a,b) = 1 if and only if x,y R s.t. ax2 + by2 = 1. Using the concept ∞ ∃ ∈ of a p-adic number ﬁeld, we may consider (a,b)p and examine it in a similar manner. We

5 now claim that (a,b) = 1 if and only if x,y Q s.t. ax2 + by2 = 1. After gaining a solid p ∃ ∈ p understanding of the p-adics, we will come back to this claim.

Definition: Qp is called a p-adic number ﬁeld. Elements in Qp are called p-adic numbers.

The p-adic numbers allow us to view Q as a complete ﬁeld, in the sense that every

Cauchy sequence converges. Qp then, may be viewed as the same way we view R, except we endow Q with a different metric in order to obtain Qp. Since we are endowing Qp with a metric, we must have some sense of distance between elements in the field, just as we do in R. Our sense of distance comes from viewing elements in Z modulo pi. For example, for any two numbers a,b Z we may view their congruence modulo p. If a b mod p, we ∈ ≡ may then consider whether or not a b mod p2. Again, if this is true, we may consider ≡ the elements modulo p3 and so on. We say that a,b Z are close in the p-adic sense if ∈ a b mod pn for sufficiently large values of n. We now have a sense of distance for elements ≡ in Qp.

Definition: Let ord (a),a Q be the p-adic valuation. p ∈ We may write a as a = pm u , where m, u, v Z and u,v are not divisible by p. Then, v ∈ ordp(a)= m, where m is the highest power of p that divides a in the usual sense. For the sake of notation, we will let ord (0) = . p ∞

Properties of p-adic valuation:

(1) ordp(ab)= ordp(a)+ ordp(b). (2) ord (a + b) min(ord (a),ord (b)). p ≥ p p

6 (3) If ord (a) = ord (b)= ord (a + b) = min(ord (a),ord (b)). p 6 p ⇒ p p p

We may construct Qp as a completion with respect to the p-adic metric dp(a,b), for a,b Q. The p-adic metric arises from the p-adic absolute value, a = p−ordp(a), which ∈ | |p describes the size of a in the p-adic sense. The smaller the absolute value of a Q is, the ∈ larger a is in the p-adic sense. Then, two rational numbers a and b are p-adically close if ord (a b) is large. There are diﬀerent constructions of Q , with the completion being the p − p analytic one, but we will look at the algebraic construction in depth.

2.3 Qp constructed as an inverse limit

Let us deﬁne Z = a Q ord (a) 0 . Z is a subring of Q where we deﬁne the p { ∈ p| p ≥ } p p elements to be p-adic integers.

n n Definition: lim Z/p Z = (xn)n≥1 fn(xn+1)= xn, n 1 is the inverse limit of Z/p Z. ←n { | ∀ ≥ } Each f : Z/pn+1Z Z/pnZ is given as the projection from Z/pn+1Z to Z/pnZ. n −→

How may we view an element of lim Z/pnZ? Each element is certainly a subset of ←n n Z/p Z, but there is a more intuitive way to look at it. We may consider (xn)n≥1 as nY≥1 a sequence (x ,x ,x ,...) where for x Z, the sequence represents its congruence modulo pn. 1 2 3 ∈

n Claim: lim Z/p Z = Zp. ←n ∼

n n Proof. Let φ be the map from lim Z/p Z Zp and choose (xn)n≥1 lim Z/p Z. ←n −→ ∈ ←n For every i 1, we want to choose y Z such that y x mod pi in Z/piZ. ≥ i ∈ i ≡ i Choose y ﬁrst. Then, y x mod p. 1 1 ≡ 1

7 For n = 2,y x mod p2. But, x x mod p by construction of the inverse limit. 2 ≡ 2 2 ≡ 1 For n = 3,y x mod p3. But, x x mod p2 x mod p x mod p. 3 ≡ 3 3 ≡ 2 ≡ 2 ≡ 1 This process continues for all values of i and shows us that our chosen yi is contained in the class [x ] for every i. In fact, y is in the class [x ] if i j. When considering (y ) , 1 p i j p ≥ n n≥1 and the observations above, it seems that there must be some sort of convergence of the sequence. Note that y y mod pN if m, n N. m ≡ n ≥ Thus, using the analytical deﬁnition of a p-adic Cauchy sequence, we see that (yn)n≥1 is a p-adic Cauchy sequence itself. Also, since for every value of n, ord (y ) 0, we have that p n ≥ the limit of this sequence must belong to Zp. Thus, φ will send (xn)n≥1 to the limit of the sequence (y ) Z . n n≥1 ∈ p

n Now, consider the map ψ : Zp lim Z/p Z and consider x Zp. If we break φ −→ ←n ∈ down into its coordinate maps, ψ : Z Z/piZ, we may send x to x = x mod pi i p −→ i n under ψi. Then, the combination of the ψi’s induces a map Zp lim Z/p Z where −→ ←n x (ψ (x),ψ (x),ψ (x), ). 7−→ 1 2 3 ···

It is certainly true that for any a Q×, a may be written in the form pnu, where n Z ∈ p ∈ and u Z× = . Thus, there exists an obvious isomorphism between Z . ∈ p U ×U

Let be the group of units of Z and let us deﬁne group homomorphisms ǫ,ω : Z/2Z U 2 U −→ as follows:

u 1 0, if u 1 mod 4 ǫ(u)= − mod 2 = ≡ 2 (1, if u 1 mod 4 ≡−

u2 1 0, if u 1 mod 8 ω(u)= − mod 2 = ≡± 2 (1, if u 5 mod 8 ≡±

8 Deﬁne =1+ pnZ . Un p

Claim: = ker(π : (Z/pnZ)×), where π is the the projection to the nth coordi- Un n U −→ n nate of a in the inverse limit deﬁnition. ∈U

Proof. It is obvious that 1 + pnZ ker(π ). p ⊆ n Let a ker(π ). Then, a 1 mod pn and a 1 mod pm, m n. ∈ n ≡ ≡ ∀ ≤ If we multiply a by pn, pna is an element of Z . But, when considering =1+ pnZ , all p Un p terms will be equal to 1 up to the nth term in the sequence. Thus, a 1+ pnZ = . ∈ p Un = ker(π ) Un n

Thus, we have the following commutative diagram:

π n / (Z/pnZ)× U 9 f j / U Un and / = (Z/pnZ)× by the First Isomorphism Theorem. U Un ∼

By this fact, U = Z× = lim Z/pZ Z/p2Z Z/p3Z . Note that for the group homo- p ←n × × ×··· morphism ǫ, = ker(ǫ). The proof of this equality is a direct consequence of congruences U2 modulo 4.

Claim: Q×/(Q×)2 = (Z/2Z) (Z/2Z) (Z/2Z), where p = 2. p ∼ × ×

9 Let us deﬁne a homomorphism φ : Q× (Z/2Z) (Z/2Z) (Z/2Z), where p −→ × × a = pnu (n mod p,ǫ(u),ω(u)). We claim that ker(φ)=(Q×)2. 7−→ p

Proof. Let a = pnu, u = Z×. ∃ ∈U p Note that ker(φ) = a Q× φ(a) = (0, 0, 0) { ∈ p | } If n 0 mod 2, then n is even. ≡ If ǫ(u)= ω(u) = 0, then u 1 mod 4 and u 1 mod 8. ≡ ≡± Thus, u 1 mod 8, making u a square in Z×. ≡ 2 If u is a square in Z×, then u 1 mod 8 1 mod 4. This then implies that ǫ(u)= ω(u) = 0. 2 ≡ ≡ × × 2 Thus, a is a square in Qp and ker(φ)=(Qp ) .

2.4 The product formula for the Hilbert symbol

Theorem: Product Formula for the Hilbert Symbol Let a,b Q×. Then (a,b) = 1 for all but ﬁnitely many values of v and ∈ v

(a,b) = 1, for v = ,p prime. (2.1) v ∞ v Y Proof. By point (2) in the proposition, we note that the Hilbert symbol is bilinear. Thus, we need only consider the cases in which a or b is equal to 1 or some prime number p. −

Case 1: a = 1= b − By deﬁnition, ( 1, 1) = 1=( 1, 1) . − − ∞ − − − 2 For p an odd prime, ( 1 1)p = 1, thus, (a,b)v = 1. − − v Q

10 Case 2: a = 1 and b = q, where q is prime. − If q = 2, then ( 1, 2) = 1, for all values of v. − v If q = 2, then ( 1,q) = 1, for v = 2,q and ( 1,q) =( 1,q) =( 1)ǫ(q). 6 − v 6 − 2 − q − ǫ(q) ǫ(q) Thus, (a,b)v = 1 ( 1) ( 1) = 1. v · − · − Q Case 3: a = q, b = r, where q and r are primes. If q = r, then (q,r) = ( 1,q) , which we have already proven to satisfy the necessary v − v property in case 2. 2 If q = r and r = 2, (q, 2) = 1, v = 2,q. (q, 2) =( 1)ωq and (q, 2) = =( 1)ω(q). 6 v ∀ 6 2 − q q − So, (q, 2)v = 1. v If q Q= r = 2, then (q,r)v = 1 for all v = r,q, 2. 6 6 r 6 q Then we have that (q,r) = , (q,r) = , and (q,r) =( 1)ǫ(q)ǫ(r). q q r r 2 − q r By the Law of Quadratic Reciprocity, = ( 1)ǫ(q)ǫ(r). r q − = (q,r)v = 1 as desired. ⇒ v Q

Theorem: Let a.b Q×. The following are equivalent statements: ∈ 2 2 (1) ax + by = 1 has a solution (x0,y0) in Q (2) ax2 + by2 = 1 has a solution (x ,y ) in Q , where v is a prime, 0 0 v ∞

Proof. (1) (2) This direction is trivial since Q Q , for all v prime or (Recall that ⇒ ⊆ v ∞ Qv is the completion of Q under the appropriate metric). So, if we have a solution to the equation in Q, then we may take that pair (x ,y ) to be the solution in Q . Thus, (2) 0 0 v ⇒ (1) is the important direction.

11 Let a,b Q× and suppose ax2 + by2 = 1 has a solution in Q , for every v prime or . ∈ v ∞ Also, we deﬁne a′ = ar and b′ = bs,r,s (Q×)2. ∈ When we consider the equation a′x2 + b′y2 = 1, we note that the existence of a solution remains the same. Thus, we may conveniently assume that a and b are square free integers. That is, if a = a′u where u (Q×)2, then we need only consider a′ Z. We will complete ∈ ∈ this proof by method of induction on the maximum value of a and b . | | | | If either a or b is equal to 1, we ﬁnd that x2 + by2 = 1 has the guaranteed solution x = 1,y = 0. But, if we have that max( a , b ) = 1, then either a > 0 or b > 0 since we | | | | must have a solution in R by assumption (R = Q∞). So, we note that either a = 1 or b = 1, which then gives us a solution in Q. If max( a , b ) > 1, suppose that, without loss of generality, that a < b . Since b is a square | | | | | | | | free integer by assumption, we may view b as a product of distinct prime numbers p p p . 1 1 ··· k

Claim: a mod b (Z/bZ)2 ∈ Proof: Suppose not. Then a mod p (Z/p Z)2, for some p where p is a divisor of b. i 6∈ i i i a If pi = 2, then (a,b)p = = 1, which by deﬁnition of the Hilbert symbol im- 6 i pi − 2 2 plies that ax + by = 1 does not have a solution in Qpi . This is a contradiction to our initial assumption that ax2 +by2 = 1 has a solution in all Q . Thus, a mod b (Z/bZ)2. v ∈

Since a mod b (Z/bZ)2, we may ﬁnd some integer r, where 0 r |b| and r2 a mod b. ∈ ≤ ≤ 2 ≡ This is a general consequence of quadratic residues, and it follows that r2 a = bk, k Z. − ∃ ∈ If k = 0, then r2 = a. Setting x equal to 1 and y = 0, we now have a solution to ◦ r ax2 + by2 =1 in Q.

2 2 if k = 0, k = r −a r + a r2 + 1 |b| + 1 b ◦ 6 | | | b |≤| b | | b |≤ ≤ 4 ≤| | Since r2 a = bk, we may reduce to the existence of solutions to ax2 + ky2 = 1. −

12 If a < b , by the inductive hypothesis and the fact that k < b (as shown above), ◦ | | | | | | | | we have the existence of a solution in Q to ax2 + ky2 = 1, which in turn implies the existence of a solution in Q to ax2 + by2 = 1. If a = b , since we have that k < b , we may again only consider the instance for ◦ | | | | | | | | which a < b . By the bullet above, we have a solution in Q. | | | |

Using the properties of the Hilbert symbol mentioned earlier, we can solve interesting problems involving conics, such as the following example:

Example: There exist x,y Q satisfying p = x2+26y2 if and only if p 1, 3 mod 8 and p ∈ ≡ ≡ 1, 3, 4, 9, 10, 12 mod 13.

We ﬁrst note that (p, 26) = (p, 13) (p, 2) . For the place at inﬁnity, (p,v) = 1. − v − v v ∞ For v = 2, 13,p :(p, 26) = 1, for p = 13. 6 − v 6 1, p 1, 3 mod 8 For v =2:(p, 26)2 =(p, 2( 13))2 = ≡ − − ( 1, p 5, 7 mod 8 − ≡ For v = 13 : (p, 26)v = (p, 13)v(p, 2)v. The symbol (p, 2)v = 1,p = 13. (p, 13)v = − p − 6 − (p, 1) (p, 13) = , when p = 13. − v v 13 6

For the ﬁnal result, we want (p, 26) = 1. Certainly, (p, 26) = 1 when − v − v v v pY Y p 1, 3 mod 8. For the case of , we note that this equals one if there exists x such ≡ 13 that x2 p mod 13. The squares inZ/13Z are 1,4,9,10, and 12, leading us to the desired ≡ result.

13 CHAPTER 3

LOCAL SYMBOLS OVER AN ALGEBRAIC CURVE

3.1 General construction

Let X be an algebraic curve. For every point P S, we assign n Z and deﬁne ∈ P ∈ >0 this assignment as a modulus. For n P , we associate the modulus which we will p M P ∈S deﬁne later. X

Let g k(X). We say that g 1 mod if the valuation v (1 g) n , P S. If ∈ ≡ M P − ≥ P ∀ ∈ the inequality holds for only particular values of P S, we state that g 1 mod at P . ∈ ≡ M We now consider a map f : X S G, where G is some abelian group. We will \ −→ later evaluate the case for which G is an abelian multiplicative group. For now, let us denote the identity by 0. The map f may be viewed as a homomorphism if we consider only the divisors of the form (g) = vP (g)P which are prime to S. Note that if P ∈X g 1 mod , then (g) is certainly primeX to S. This follows since if g 1 mod , we ≡ M ≡ M know that v (1 g) n , P S. This tells us that g itself must contain 1 since v (g) is P − ≥ P ∀ ∈ P non-zero. This implies that g is non-vanishing for every P S, thus prime with respect to ∈ S itself.

14 Definition: is a modulus for the map f if f((g)) = 0, g k(X) such that g M ∀ ∈ ≡ 1 mod . M

Definition: For a modulus on S, and a fixed f : X S G, we define the local M \ −→ symbol (f,g) as the value in G for each g k(X)∗ and P X, which satisfies the following P ∈ ∈ four properties: ′ ′ (1) (f,gg )P =(f,g)P +(f,g )P (2) (f,g) = 0, if P S and g 1 mod . P ∈ ≡ M (3) (f,g) = v (g)f(P ), if P X S. P P ∈ \ (4) (f,g)P = 0. PX∈X

Proposition: is a modulus for the map f : X S G if and only if there exists a M \ −→ unique local symbol associated to f and . M

Proof. ( ) Suppose there exists a local symbol (f,g) for some map f and modulus . ⇐ P M In particular, it exists for g k(X), where g 1 mod . Then, ∈ ≡ M

f((g)) = vP (g)f(P ) PX6∈S = (f,g)P PX6∈S = (f,g) − P PX∈S = 0

So that is a modulus for the map f. M

15 ( ) Now, let be a modulus for the map f : X S G such that f((g)) = 0 for ⇒ M \ −→ every g k(X) with g 1 mod . Our goal is to construct and define a local symbol ∈ ≡ M (f,g)P . If P S, then then we automatically have a local symbol defined by v (g)f(P ). This 6∈ P is property (3) listed above. So, suppose that P S. For convenience, we will define a ∈ g function gP k(X) such that gP 1 mod at Q S P and 1 mod at P . ∈ ≡ M ∈ \ gP ≡ M Define the local symbol as:

(f,g)P = vQ(gP )f(Q) (3.1) QX6∈S Now, we need only verify that the above deﬁnition for the local symbol satisﬁes the four properties necessary.

′ (1) Let gP ,gP be functions known to satisfy the necessary properties (auxiliary functions) ′ ′ ′ for g,g respectively. Then, gP gP is an auxiliary function for gg . So,

(f,gg′) = v (g g′ )f(Q) P − Q P P QX6∈S = (v (g )+ v (g′ ))f(Q) − Q P Q P QX6∈S = v (g )f(Q) v (g′ )f(Q) − Q P − Q P QX6∈S QX6∈S ′ =(f,g)P +(f,g )P

(2) If g 1 mod at P , then g 1 mod . Since is a modulus for f, f((g )) = 0. ≡ M P ≡ M M P So, v (g )f(Q)= f((g )) = 0, as needed. − Q p − P QX6∈S (3) If P X S, then P S implies that (f,g) = v (g )f(P ). ∈ \ 6∈ P P P

16 (4) (f,g) = v (g )f(Q) = v (h)f(Q), with h = g . P −  Q P  − Q P P ∈S P ∈S Q6∈S Q6∈S P ∈S X g X X X Y If we deﬁne k = , sinceg,h 1 mod , we must have that k 1 mod . Thus, h ≡ M ≡ M

(f,g) = v (h)f(Q) P − Q PX∈S QX6∈S g = v ( )f(Q) − Q k QX6∈S = v (g)f(Q)+ v (k)f(Q) − Q Q QX6∈S QX6∈S = v (g)f(Q) − Q QX6∈S = (f,g) , by (3) − Q QX6∈S (f,g)P = (f,g)P + (f,g)Q = 0 PX∈X PX∈S QX6∈S

3.2 Local symbol into a multiplicative group

For a some multiplicative group G, we define a map f : X S G, where S is the set \ −→ consisting of the zeros and poles of f. We define the corresponding local symbol as follows: f n (f,g) =( 1)mn (P ), with n = v (g), m = v (f). (3.2) P − gm P P f n We note that for h = ,v (h) = 0, so that h has a non-zero value at the point P . This gm P symbol satisfies the product formula with respect to the multiplicitive identity (f,g)P = P ∈X 1. The proof of this fact follows by considering the curve X as a totally ramifiedY covering of the projective line P and using the norm to project from X to P.

17 CHAPTER 4

A LOCAL SYMBOL FOR THE GAUSSIAN INTEGERS, Z[ı]

The Gaussian integers, denoted by Z[i], is the set of all values a + bi, for a,b Z and ∈ i = √ 1. The Gaussian integers are actually a very nice ring in terms of it’s properties − and share many of the same properties as that of Z. Z[i] is a Euclidean Domain with respect to the valuation N : Z[i] Z , which we deﬁne as the norm of an element in −→ ≥0 Z[i]. We set N(a + bi)=(a + bi)(a bi)= a2 + b2. The set of units of the Gaussian integers, − Z[i]× = 1, i . This is since 1, i all have norm equal to 1. Since Z[i] is a Euclidean {± ± } ± ± domain (In fact, it need only be a PID), we may deﬁne the local ring of Z[i] at some prime ideal (℘). In order to do this though, we must have a good handle on what the primes of Z[i] actually are.

4.1 Primes in Z[i]

One’s intuition would think that since Z Z[i], that all prime elements in Z, denoted ⊆ by Spec(Z), would be in Spec(Z[i]). We ﬁnd out quickly that this is not the case. Consider 5 Z Z[i]. 5 is prime in Z, but if we look at 5 in Z[i], we note that ∈ ⊆ 5=4+1=(2+ i)(2 i). Since 2+ i, 2 i do not divide 5, we ﬁnd that 5 is not prime in − −

18 Z[i]. In this case, we say that 5 splits in Z[i]. In general, any prime in Z which is congruent to 1 mod 4 is not prime in Z[i]. This follows from the fact that a prime number p may be written as the sum of two squares if and only if p 1 mod 4. This is equivalent to considering the norm function deﬁned above. ≡ For a prime element q Z, if q 3 mod 4, then q may not be written as the sum of two ∈ ≡ squares, which shows us that q remains prime in Z[i].

Claim: ℘ = a + bi Z[i] is prime if and only if: ∈ (1) a or b = 0, in which case the non-zero value is congruent to 3 mod 4. Or... (2) a2 + b2 = p, for some prime p Z, p 3 mod 4. ∈ 6≡

p2, if p does not split. This is equivalent to N(℘)= (p, if p splits.

If we view Z[i] as a curve, we see that it is a ramiﬁed covering of Z. We may relate each prime a + bi to a,b, or a2 + b2, dependent on what form the prime is in. We view our “curve” as discrete points. Note that 2 (as usual) is the odd ball case since it is the square of the prime 1+ i (times a unit) in Z[i]. It is apparent then, that every prime other than 1+ i has a norm congruent to 1 mod 4. We will call these primes odd.

Our goal is to define a local symbol over of Z[i] in the same manner as the method defined above. A problem arises because our curve is not defined over a field, but rather a Euclidean domain. We have already encountered such a local symbol though, the Hilbert symbol. We will go about the construction of this new symbol in a similar fashion as we did for the Hilbert symbol by considering the local ring, residue field, completion, and field of quotients.

19 4.2 Construction of the local symbol

Definition: Let F be a field with discrete valuation v and let be the unique Mv maximal ideal in the valuation ring. Define the residue field to be k(v). Then the tame symbol, F ∗ F ∗ k(v)∗, satisfies the properties of being a local symbol and is defined × −→ to be: v(g) v(f)v(g) f (f,g)=( 1) ( v) (4.1) − gv(f) M

Our local symbol for Z[i] must be defined in the following manner. Let K = Q(i), which is the field of quotients of = Z[i], and consider: O (,) K K K( ˆ ) K( ˆ ) ℘ k(℘) × −→ O℘ × O℘ −−→ where k(℘) represents the residue field. That is, k(℘)= /℘ = ( /℘) = K( /℘). O℘ O℘ ∼ O (0) O n Since (℘) is maximal in ,K( /℘)= /℘. We may view the completion ˆ℘ = lim ℘/℘ ℘ = O O O O ∼ ←n O O ∼ k[[t]], that is, the ring of power series with coefficients in the residue field k. Then, n K( ˆ℘) = K(lim ℘/℘ ℘) = k((t)). O ∼ ←n O O ∼

So, in order to define our local symbol, we must first realize what the residue field is for each prime ℘ Z[i]. ∈

Claim Z F : Let ℘ be prime in [i]. Then, k(℘) ∼= N(℘).

Proof. We recall that for some prime ℘ Z[i], N(℘) = p or p2, where p is prime in Z. ∈ 2 Z Suppose ℘ does not split. That is, N(℘)= p . We recall that k(℘) ∼= [i]/℘ and the fact Z Z 2 that [i] ∼= [t]/(t + 1). Then,

20 (Z[t]/(p)) Z/pZ Z[i]/℘ = = = F 2 ∼ (t2+1) ∼ (t2+1) ∼ p

F 2 So, k(℘) ∼= p . Now, suppose that ℘ = a + bi splits in Z[i] such that p = a2 + b2. Z Z 2 Note that [i]/(℘) ∼= [t]/(a + bt,t + 1). We observe the following:

(a bt)(a + bt)= a2 b2t − − = a2 + b2 b2(t2 + 1) − = p b2(t2 1) − − p =(a bt)(a + bt)+ b2(t2 + 1) −

Thus, p (a + bt,t2 + 1) = I. Now, we want to express t in terms of the ideal I. We ∈ note that t I, so we may write t as: 6∈

t = x +(c + dt)(a + bt)+(u + vt)(t2 + 1)

= x + ca + cbt + dat + dbt2 + ut2 + u + vt3 + vt

=(x + ca + u)+(cb + da)t +(db + u)t2 + vt(t2 + 1)

x + ca + u = 0 Thus, we have the system of equations cb + da = 1 . But since gcd(b,a) = 1, db + u = 0 there exists a solution in Z to cb + da = 1. Thus, x Z and after more involved calculations  ∈ we ﬁnd that (a + bt,t2 + 1) = (t x,p) as ideals, so: −

Z[t]/(a + bt,t2 + 1) = Z[t]/(t x,p) = Z/pZ. ∼ − ∼

21 (,) We have found that our local symbol will take us from K( ˆ ) K( ˆ ) ℘ F , O℘ × O℘ −−→ N(℘) where N(℘)= p or p2. If we view Z[i] as a curve, we may take equation (4) as our local symbol, but as mentioned before, our curve is not defined over a field, so we will have to take particular care in the construction. At this point, our symbol constructed in the F× F× manner of the tame symbol will land in the residue field of either p or p2 . We want our symbol to land in the group group of units of Z[i], that is, 1, i . So, as in the case for {± ± } the Hilbert symbol, we need a group homomorphism from the residue field into the units of Z[i]. This homomorphism is defined to be a character.

Proposition: For any x Z[i]/(℘)×, xN(℘)−1 1 mod ℘. ∈ ≡ Proof. This proof is analogous to a proof of Fermat’s Little Theorem for Z[i]/(℘). We have proved that Z[i]/(℘) = F = N(℘). So, if we write out the set of classes of Z[i]/(℘), | | N(℘) we have 0= q ,q ,...,q . Since Z[i]/(℘) is a ﬁeld and we are assuming that x is { 0 1 N(℘)−1 } non-zero, we note that x must be a unit. Thus, 0 = xq ,q ,...,xq includes all { 0 1 N(℘)−1} elements of Z[i]/(℘) as well. So, we may observe the following congruence:

q ...q (xq ) ... (xq ) mod ℘ 1 N(℘)−1 ≡ 1 N(℘)−1 xN(℘)−1(q ...q ) mod ℘ ≡ 1 N(℘)−1 xN(℘)−1 1 mod ℘ ≡

Since every prime ℘ in Z[i] besides 1+ i is such that N(℘) 1 mod 4, we may consider N(℘) 1 ≡ the ratio − and claim the following statement. 4

22 Claim: N(℘)−1 4 is a character that takes us from the residue ﬁeld to the roots of unity of Z[i]. That is, for any u Z[i]/(℘), where ℘ is an odd Gaussian prime, then, ∈ N(℘)−1 u 4 1, i mod ℘. ≡± ±

N(℘)−1 N(℘)−1 Proof. Certainly, we may observe that u 2 1 u 2 + 1 0 mod ℘ as a conse- − N(℘)−1 ≡ quence of the above proposition. This shows that u 2 1 mod ℘. Then, by taking ≡± the square root of both sides, we obtain the desired result of:

N(℘)−1 u 4 1, i mod (℘) ≡± ±

Writing a = ℘v℘(a)u = ij(1+ i)va∗ and b = ℘v℘(b)v = ik(1+ i)wa∗, such that p u,v and 6| a ,b 1mod1+ i3, we deﬁne our local symbol for every prime ℘ Z[i] as: ∗ ∗≡ ∈ v℘(b) v℘(a)v℘(b) a (a,b)℘ =( 1) (℘), for ℘ =1+ i (4.2) − bv℘(a) 6 2vw+jw−kv (a,b)1+i = i , for ℘ =1+ i (4.3)

Since each symbol lands in a diﬀerent residue ﬁeld, we need a character to take us into the roots of unity. We have found this character above and claim that:

N(℘)−1 4 (a,b)λ (a,b)℘ = 1 (4.4) ℘Y6=λ From now on, we deﬁne λ =1+ i for notational ease. It is obvious that this symbol satisﬁes the same conditions (2), and (3) that the Hilbert symbol did. With these observations, the proof of this global product formula is reduced to proving it for a or b being equal to i, i + 1, or a prime number.

23 CHAPTER 5

EXTENSIONS OF GROUPS

5.1 Central Extensions of Groups

Let G and H be groups and consider a surjective morphism π between them. Denote ker(π) by N so that we have the following diagram:

N ı G π H. −→ −→ We then say that G is an extension of H by N. If we require that N be abelian, we obtain an action of G on N in the following manner: Let g G and n N, so that we may consider the product g(ı(n))g−1. ∈ ∈

π(g(ı(n))g−1)= π(g)π(ı(n))π(g−1)

= π(g)e π(g)−1, since ı(n) ker(π) H ∈

= eH

Thus, g(ı(n))g−1 N. For convenience, we denote this element by gn. Our action will be a ∈ map from G N N will send (g,n) g n. In fact, this action is only dependent on × −→ 7−→ values in H.

24 Consider g′ G such that π(g)= π(g′). Then, g′ = g(ı(m)). So, for any n N, ∈ ∈ g′(ı(n))g′−1 = gı(m)ı(n)ı(m)−1g−1

= g(ı(n))g−1

′ Thus, gn = g n, if π(g) = π(g′). We note that the action is actually H N N, × −→ (h,n) gn, where g is such that π(g)= h. This works since π is surjective. 7−→

Definition: An extension of this type is central if the action is trivial. That is, gn = n.

Now, consider the following central extension of groups:

N ֒ G ։ H → Let σ : H G be a set map. We want to know how far σ is from being a group −→ homomorphism. For any h ,h H, π(σ(h h )) = h h = π(σ(h )σ(h )). Thus, σ(h h ) and σ(h )σ(h ) 1 2 ∈ 1 2 1 2 1 2 1 2 1 2 must diﬀer by something in the kernel of π. So we must have that σ(h1h2)ı(f(h1,h2)) = σ(h )σ(h ), with f : H H N being a set map. 1 2 × −→

Suppose that h ,h ,h H, then: 1 2 3 ∈

σ(h1)σ(h2)σ(h3)= σ(h1)σ(h2h3)ı(f(h2,h3))

= σ(h1h2h3)ı(f(h1,h2h3)f(h2,h3))

σ(h1)σ(h2)σ(h3)= σ(h1h2)ı(f(h1,h2))σ(h3)

= σ(h1h2h3)ı(f(h1,h2)f(h1h2,h3))

25 Since ı is a monomorphism, we have that:

f(h1,h2h3)f(h2,h3)= f(h1,h2)f(h1h2,h3) (5.1)

Definition: A set map f which satisﬁes property (2) we will call a 2-Cocycle on H with values in N.

5.2 Commutator of an Extension

Consider the central extension with A being an abelian group:

A֒ G ։ K 1 1 −→ → −→ For g,h G, take [g,h]= ghg−1h−1. Also, require g,h G are such that π(g), π(h) B, ∈ ∈ ∈ where B is an abelian subgroup of K. Note that π(ghg−1h−1)= π([g,h])=[π(g), π(h)] = 1, since B is abelian. For b ,b B, g ,g G such that π(g ) = b for i = 1, 2. So, 1 2 ∈ ∃ 1 2 ∈ i i [g ,g ] A. 1 2 ∈ ′ ′ ′ These g1,g2 are unique. Choose gi such that π(gi)= bi = π(gi). Then, gi = giai, with a A, so that: i ∈ ′ ′ ′ ′ ′−1 ′−1 [g1,g2]= g1g2g1 g2 −1 −1 = g1a1g2a2(g1a1) (g2a2)

−1 −1 −1 −1 = g1g2g1 g2 a1a1 a2a2 , since A is abelian.

=[g1,g2]

This construction deﬁnes a map B B A, in which (b ,b ) [g ,g ], with g such × −→ 1 2 7−→ 1 2 i that π(g )= b ,i = 1, 2. This map, which we will denote as a symbol b ,b , satisﬁes the i i { 1 2}c property of being a 2-Cocycle since the commutator is commutative and bimultiplicitive.

26 CHAPTER 6

CONCLUSION

Through the course of study, we started out with an explicit formula for the study of curve over another. Fulling understanding the Hilbert symbol required us to gain knowledge in localization of integral domains and using this localization to construct completions. With the study of completions, we constructed the p-adic numbers which gave us a full-bodied view of Hilbert symbol and its consequences. Using our knowledge of the Hilbert symbol and its behavior, we were able to generalize the concept into that of a local symbol over a particular algebraic curve. We found that the tame symbol is a particular instance of a local symbol into a multiplicative group. Finally, putting the concepts learned from both the Hilbert symbol and local symbols over curves together, we were able to construct a symbol of our own. During the process of this construction though, we were diverted to better understand some of the nice properties of the Gaussian integers. As a vista of research in the future, we hope to understand the Parshin symbol as a generalization of the tame symbol to a surface S, while current research focuses on extending the notion to higher dimensions.

27 BIBLIOGRAPHY

[1] Ettore Aldrovandi, Hermitian-holomorphic (2)-gerbes and tame symbols, J. Pure Appl. Algebra 200 (2005), no. 1-2, 97–135, DOI 10.1016/j.jpaa.2004.12.023. MR2142353 (2006e:18014)

[2] P. Deligne, Le symbole mod´er´e, Inst. Hautes Etudes´ Sci. Publ. Math. 73 (1991), 147–181 (French). MR1114212 (93i:14030)

[3] Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito, Number theory. 1, Translations of Mathematical Monographs, vol. 186, American Mathematical Society, Providence, RI, 2000. Fermat’s dream; Translated from the 1996 Japanese original by Masato Kuwata; Iwanami Series in Modern Mathematics. MR1728620 (2000i:11002)

[4] Kazuya and Kurokawa Kato Nobushige and Saito, Number theory 2, Translations of Mathematical Monographs, vol. 240, American Mathematical Society, Providence, RI, 2011. Introduction to class ﬁeld theory, Translated from the 1998 Japanese original by Masato Kuwata and Katsumi Nomizu, Iwanami Series in Modern Mathematics.

[5] John Milnor, Introduction to algebraic K-theory, Princeton University Press, Princeton, N.J., 1971. Annals of Mathematics Studies, No. 72. MR0349811 (50 #2304)

[6] Denis Osipov, To the multidimensional tame symbol (2011), available at arXiv:1105.1362v1[math.AG].

[7] Fernando Pablos Romo, Cohomological characterization of the Hilbert symbol over Qp, Journal of the Australian Mathematical Society 79 (2005), no. 3, 361–368, DOI 10.1017/S1446788700010958.

[8] A. N. Parshin, Representations of Higher Adelic Groups and Arithmetic, Steklov Mathematical In- stitute, Moscow, 2010. http://www.mathnet.ru/php/presentation.phtml?presentid=3508&option_ lang=eng.

[9] , Representations of Higher Adelic Groups and Arithmetic (2010), available at arXiv:1012. 0486v2[math.NT]. Talk at the 2010 ICM.

28 [10] Dinakar Ramakrishnan, A regulator for curves via the Heisenberg group, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 191–195, DOI 10.1090/S0273-0979-1981-14942-9. MR621888 (82m:12008)

[11] J.-P. Serre, A Course in Arithmetic, Springer-Verlag, New York, 1973. Translated from the French; Graduate Texts in Mathematics, No. 7. MR0344216 (49 #8956)

[12] Jean-Pierre Serre, Algebraic groups and class ﬁelds, Graduate Texts in Mathematics, vol. 117, Springer- Verlag, New York, 1988. Translated from the French.

[13] , Local ﬁelds, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York, 1979. Trans- lated from the French by Marvin Jay Greenberg. MR554237 (82e:12016)