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1074_(GTM232)An Introduction to Number Theory

Definition 4.12

Definition 4.12. A nonzero ideal \( I \neq R \) in a commutative ring \( R \) is called maximal if for any ideal \( J, J \mid I \) implies that \( J = I \) . An ideal \( P \) is prime if \( P \mid {IJ} \) implies that \( P \mid I \) or \( P \mid J \) . Exercise 4.12. In a commutative ring \( R \), let \( M \) and \( P \) denote ideals. (a) Show that \( M \) is maximal if and only if the quotient ring \( R/M \) is a field. (b) Show that \( P \) is prime if and only if \( R/P \) is an integral domain (that is, in \( R/P \) the equation \( {ab} = 0 \) forces either \( a \) or \( b \) to be 0 ). (c) Deduce that every maximal ideal is prime. Theorem 4.13. [Fundamenta Theorem of Arithmetic for Ideals] Any nonzero proper ideal in \( {O}_{\mathbb{K}} \) can be written as a product of prime ideals, and that factorization is unique up to order. Proof. If \( I \) is not maximal, it can be written as a product of two nontrivial ideals. Comparing norms shows these ideals must have norms smaller than \( I \) . Keep going: The sequence of norms is descending, so it must terminate, resulting in a finite factorization of \( I \) . By Exercise 4.12, every maximal ideal is prime, so all that remains is to demonstrate that the resulting factorization is unique. This uniqueness follows from Corollary 4.9, which allows cancellation of nonzero ideals common to two products. ## 4.4 The Ideal Class Group In this section, we are going to see how the nineteenth-century mathematicians interpreted Exercise 3.32 on p. 81 in terms of quadratic fields. The major result we will present is that ideals in \( {O}_{\mathbb{K}} \), for a quadratic field \( \mathbb{K} \), can be described using a finite list of representatives \( {I}_{1},\ldots ,{I}_{h} \) ; any nontrivial ideal \( I \) can be written \( {I}_{i}P \), where \( 1 \leq i \leq h \) and \( P \) is a principal ideal. Thus \( h \), known as the class number, measures the extent to which \( {O}_{\mathbb{K}} \) fails to be a principal ideal domain. This statement was proved for arbitrary algebraic number fields and proved to be influential in the way number theory developed in the twentieth century. Given two ideals \( I \) and \( J \) in \( {O}_{\mathbb{K}} \), define a relation \( \sim \) by \[ I \sim J\text{if and only if}I = {\lambda J}\text{for some}\lambda \in {\mathbb{K}}^{ * }\text{.} \] Exercise 4.13. Show that \( \sim \) is an equivalence relation. We are going to outline a proof of the following important theorem. Theorem 4.14. There are only finitely many equivalence classes of ideals in \( {O}_{\mathbb{K}} \) under \( \sim \) . One class is easy to spot - namely the one consisting of all principal ideals. Of course, \( {O}_{\mathbb{K}} \) is a principal ideal domain if and only if there is only one class under the relation. One can define a multiplication on classes: If \( \left\lbrack I\right\rbrack \) denotes the class containing \( I \), then one can show that the multiplication defined by \[ \left\lbrack I\right\rbrack \left\lbrack J\right\rbrack = \left\lbrack {IJ}\right\rbrack \] (4.2) is independent of the representatives chosen. Corollary 4.15. The set of classes under \( \sim \) forms a finite Abelian group. The group in Corollary 4.15 is known as the ideal class group of \( \mathbb{K} \) (or just the class group). Proof of Corollary 4.15. In the class group, associativity of multiplication is inherited from \( {O}_{\mathbb{K}} \) . The element \( \left\lbrack {O}_{\mathbb{K}}\right\rbrack \) acts as the identity. Finally, given any nonzero ideal \( I \), the relation \( I{I}^{ * } = \left( {N\left( I\right) }\right) \) shows that the inverse of the class \( \left\lbrack I\right\rbrack \) is \( \left\lbrack {I}^{ * }\right\rbrack \) . Lemma 4.16. Given a square-free integer \( d \neq 1 \), there is a constant \( {C}_{d} \) that depends upon \( d \) only such that for any nonzero ideal \( I \) of \( {O}_{\mathbb{K}},\mathbb{K} = \mathbb{Q}\left( \sqrt{d}\right) \) , there is a nonzero element \( \alpha \in I \) with \( \left| {N\left( \alpha \right) }\right| \leq {C}_{d}N\left( I\right) \) . Exercise 4.14. *Prove Lemma 4.16. The basic idea is a technique similar to that used in the proof of Theorem 3.21 showing that a lattice point must exist in a region constrained by various inequalities. Since the original proof, considerable efforts have gone into decreasing the constant \( {C}_{d} \) for practical application. The best techniques use the geometry of numbers, a theory initiated by Minkowski. Proof of Theorem 4.14. First show that every class contains an ideal whose norm is bounded by \( {C}_{d} \) . Given a class \( \left\lbrack I\right\rbrack \), apply Lemma 4.16 with \( {I}^{ * } \) replacing \( I \) . Now \( \left( \alpha \right) \subseteq {I}^{ * } \), so we can write \( \left( \alpha \right) = {I}^{ * }J \) for some ideal \( J \) . However, this gives a relation \( \left\lbrack {I}^{ * }\right\rbrack \left\lbrack J\right\rbrack = \left\lbrack \left( \alpha \right) \right\rbrack \) in the class group. This means that \( \left\lbrack J\right\rbrack \) is the inverse of \( \left\lbrack {I}^{ * }\right\rbrack \) . However, we remarked earlier that \( \left\lbrack I\right\rbrack \) and \( \left\lbrack {I}^{ * }\right\rbrack \) are mutual inverses in the class group. Hence \( \left\lbrack I\right\rbrack = \left\lbrack J\right\rbrack \) . Now \[ \left| {N\left( \alpha \right) }\right| = N\left( \left( \alpha \right) \right) = N\left( {I}^{ * }\right) N\left( J\right) \] Since the left-hand side is bounded by \( {C}_{d}N\left( {I}^{ * }\right) \), we can cancel \( N\left( {I}^{ * }\right) \) to obtain \( N\left( J\right) \leq {C}_{d} \) . Now the theorem follows easily: For any given integer \( k \geq 0 \), there are only finitely many ideals of norm \( k \) ; this is because any ideal must be a product of prime ideals of norm \( p \) or \( {p}^{2} \), where \( p \) runs through the prime factors of \( k \) . There are only finitely many such prime ideals and hence there are only finitely many ideals of norm \( k \) . Now apply this to the integers \( k \leq {C}_{d} \) to deduce that there are only finitely many ideals of norm bounded by \( {C}_{d} \) . Since each class contains an ideal whose norm is thus bounded, by the first part of the proof, it follows that there are only finitely many classes. Exercise 4.15. Investigate the relationship between quadratic forms and ideals in quadratic fields. In particular, show that Exercise 3.32 on p. 81 is equivalent to Theorem 4.14. (Hint: If \( I \) denotes an ideal with basis \( \{ \alpha ,\beta \} \), show that for \( x, y \in \mathbb{Z}, N\left( {{x\alpha } + {y\beta }}\right) /N\left( I\right) \) is a (binary) integral quadratic form. How does a change of basis for \( I \) relate to the form? What effect does multiplying \( I \) by a principal ideal have on the form?) ## 4.4.1 Prime Ideals To better understand prime ideals, we close with an exercise that links up the various trains of thought in this chapter and shows that ideal theory better explains the various phenomena encountered in Chapter 3. Exercise 4.16. Factorize the ideal (6) into prime ideals in \( \mathbb{Z}\left\lbrack \sqrt{-5}\right\rbrack \), expressing each prime factor in the form \( \left( {a, b + c\sqrt{-5}}\right) \) . Exercise 4.17. Let \( {O}_{\mathbb{K}} \) denote the ring of algebraic integers in the quadratic field \( \mathbb{K} = \mathbb{Q}\left( \sqrt{d}\right) \) for a square-free integer \( d \) . (a) If \( P \) is a prime ideal in \( {O}_{\mathbb{K}} \), show that \( P \mid \left( p\right) \) for some integer prime \( p \in \mathbb{Z} \) . (b) Show that there are only three possibilities for the factorization of the ideal \( \left( p\right) \) in \( {O}_{\mathbb{K}} \) : \( \left( p\right) = {P}_{1}{P}_{2} \) where \( {P}_{1} \) and \( {P}_{2} \) are prime ideals in \( {O}_{\mathbb{K}} \) ( \( p \) splits); \( \left( p\right) = P \), where \( P \) is a prime ideal in \( {O}_{\mathbb{K}} \) ( \( p \) is inert); \( \left( p\right) = {P}^{2} \), where \( P \) is a prime ideal in \( {O}_{\mathbb{K}} \) ( \( p \) is ramified). This should be compared with the possible primes in \( \mathbb{Z}\left\lbrack i\right\rbrack \) described in Theorem 2.8(3). The following exercise gives a complete description of splitting types in terms of the Legendre symbol. Exercise 4.18. Let \( {O}_{\mathbb{K}} \) denote the ring of algebraic integers in the quadratic field \( \mathbb{K} = \mathbb{Q}\left( \sqrt{d}\right) \) for a square-free integer \( d \) . Let \( D = d \) if \( d \equiv 1 \) modulo 4 and let \( D = {4d} \) otherwise. Show that an odd prime \( p \) is inert, ramified, or split as the Legendre symbol \( \left( \frac{D}{p}\right) \) is \( - 1,0 \), or +1, respectively. What are the possibilities when \( p = 2 \) ? We should say something about the terminology. Splitting and inertia are fairly obvious, the latter signifying that the prime \( p \) remains prime in this bigger ring, just as primes \( p \equiv 3 \) modulo 4 remain primes in \( \mathbb{Z}\left\lbrack i\right\rbrack \) . The term "ramify" means literally to branch, and we see here something of an overlap with the theory of functions. A function such as \( y = \sqrt{x} \) really consists of two possible branches. This notion was borrowed deliberately to name the phenomenon seen in number theory, where a prime in \( \mathbb{Z} \) becomes a power of a prime in a larger ring. We end this chapter with a definition because it is going to appear again in Chapter 11. Definition 4.17. Let \( \mathbb{K} = \mathbb{Q}\left( \sqrt{d}\right) \) denote a quadratic field, where \( d \) is a square-free integer. Define \( D \) by \[ D = \left\{ \begin{array}{ll} d & \text{ if }d \equiv 1{\;\operatorname{modulo}\;4}\text{ and } \\ {4d} & \text{ otherwise. } \end{array}\right. \] (4.3) Then \( D \) is called the discriminant of the quadratic field \( \mathbb{K} =

1139_(GTM44)Elementary Algebraic Geometry

Definition 3.5

Definition 3.5. If \( R \) is the coordinate ring of an irreducible variety \( V \subset {\mathbb{C}}^{n} \) , and if \( \mathfrak{p} = \mathrm{J}\left( W\right) \) is the prime ideal of an irreducible subvariety \( W \) of \( V \) , then the local ring \( {R}_{\mathrm{p}} \) is called the localization of \( V \) at \( W \) (or along \( W \) ), or the local ring of \( V \) at \( W \) ; in this case \( {R}_{\mathfrak{p}} \) is also denoted by \( \mathfrak{o}\left( {W;V}\right) \) . Definition 3.6. Let \( V \subset {\mathbb{P}}^{n}\left( \mathbb{C}\right) \) be irreducible, and let \( {K}_{V} \) be \( V \) ’s function field-that is, the set of quotients of equal-degree forms in \( {x}_{1},\ldots ,{x}_{n + 1} \), where \( \mathbb{C}\left\lbrack {{x}_{1},\ldots ,{x}_{n + 1}}\right\rbrack = \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n + 1}}\right\rbrack /\mathrm{J}\left( V\right) \) . If \( W \) is an irreducible subvariety of \( V \), then the set of all elements of \( {K}_{V} \) which can be written as \( p/q \) , where \( p \) and \( q \) are forms in \( {x}_{1},\ldots ,{x}_{n + 1} \) of the same degree, and where \( q \) is not identically zero on \( W \), forms a subring of \( {K}_{V} \) ; it is called the local ring of \( V \) at \( W \), and is denoted by \( \mathfrak{o}\left( {W;V}\right) \) . Remark 3.7. If \( W \subset V \) are irreducible varieties in \( {\mathbb{P}}^{n}\left( \mathbb{C}\right) \), and if \( R \) is the coordinate ring of any dehomogenization \( \mathrm{D}\left( V\right) \) of \( V \) (where \( W \) is not contained in the hyperplane at infinity), then \( \mathfrak{o}\left( {W;V}\right) \) is the localization \( {R}_{\mathfrak{p}} = \) \( \mathfrak{o}\left( {\mathrm{D}\left( W\right) ;\mathrm{D}\left( V\right) }\right) \) of \( R \) at \( \mathrm{D}\left( W\right) = \mathrm{V}\left( \mathfrak{p}\right) \) ; this follows from the fact that if we without loss of generality dehomogenize at \( {X}_{n + 1} \), then \[ \frac{p\left( {{x}_{1},\ldots ,{x}_{n + 1}}\right) }{q\left( {{x}_{1},\ldots ,{x}_{n + 1}}\right) } = \frac{p\left( {{x}_{1}/{x}_{n + 1},\ldots ,1}\right) }{q\left( {{x}_{1}/{x}_{n + 1},\ldots ,1}\right) }. \] The left-hand side is an element of \( \mathfrak{o}\left( {W;V}\right) \), while the right-hand side belongs to \( {R}_{\mathfrak{p}} \) . Many of the basic algebraic and geometric relations between \( R \) and \( {R}_{\mathfrak{p}} \) may be compactly expressed using a double sequence, as in Diagrams 2 and 3 of Chapter III. We explore this next. Again, for expository purposes we select a fixed variety \( V \subset {\mathbb{C}}_{{x}_{1},\ldots ,{x}_{n}} \) having \( R = \mathbb{C}\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) as coordinate ring, and we let \( W = \mathbf{V}\left( \mathfrak{p}\right) \) be an arbitrary, fixed irreducible subvariety of \( V \) . Our sequence is given in Diagram 1.  Diagram 1. In this diagram, \( \mathcal{I}\left( {R}_{\mathfrak{p}}\right) \) denotes the lattice \( \left( {\mathcal{I}\left( {R}_{\mathfrak{p}}\right) , \subset ,\cap , + }\right) \) of ideals of \( {R}_{\mathfrak{p}} \) and \( \mathcal{J}\left( {R}_{\mathfrak{p}}\right) \) denotes the lattice \( \left( {\mathcal{J}\left( {R}_{\mathfrak{p}}\right) , \subset ,\cap , + }\right) \) of closed ideals of \( {R}_{\mathfrak{p}} \) . Closure in \( \mathcal{I}\left( {R}_{\mathfrak{p}}\right) \) is with respect to the radical of Definition 1.1 of Chapter III; by Lemma 5.7 of Chapter III the radical of an ideal \( \mathfrak{a} \) in \( {R}_{\mathfrak{p}} \) will be seen to be the intersection of all prime ideals of \( {R}_{\mathfrak{p}} \) which contain \( \mathfrak{a} \), since \( {R}_{\mathfrak{p}} \) is Noetherian (Lemma 3.9). This radical is not in general the intersection of the \( \mathfrak{a} \) -containing maximal ideals of \( {R}_{\mathfrak{p}} \), since \( {R}_{\mathfrak{p}} \) has but one maximal ideal. Continuing the explanation of symbols in Diagram \( 1,\mathcal{G}\left( {R}_{\mathfrak{p}}\right) \) denotes the lattice \( \left( {\mathcal{G}\left( {R}_{\mathfrak{p}}\right) , \subset ,\cap , \cup }\right) \) of all \( {V}_{W} \) where \( V \in \mathcal{I} \) and \( W \) is fixed, with \( \subset , \cap \), and \( \cup \) as in Definition 3.3. The letter \( \mathcal{G} \) reminds us that these ordered pairs \( {V}_{W} \) are identified with germs (We remark that there exists an analogous sequence at the analytic level, where one uses germs instead of representatives, since there is not in general a canonical representative of each "analytic germ," as is the case with algebraic varieties, where there is a unique smallest algebraic variety representing a given "algebraic germ." One can even push certain aspects to the differential level.) It is easily seen that \( \mathcal{G}\left( {R}_{\mathfrak{p}}\right) \) actually is a lattice, using Definition 3.3 together with the fact that \( \varnothing \) and the subvarieties of \( V \) containing \( \mathbf{V}\left( \mathfrak{p}\right) \) form a lattice. As for the various maps, \( {\left( \;\right) }^{c} \) and \( {\left( \;\right) }^{e} \) are just contraction and extension of ideals. Since \( R \rightarrow {R}_{\mathfrak{p}} \) is an embedding, \( {\left( \;\right) }^{c} \) reduces to intersection with \( R \) . In contrast to extension in Section III,10, we shall see that \( {\left( \;\right) }^{e} \) maps closed ideals in \( \mathcal{I}\left( R\right) \) to closed ideals in \( \mathcal{I}\left( {R}_{\mathfrak{p}}\right) \) . The map \( {\left( \;\right) }_{W} \) sends \( V \) into \( {V}_{W} \) , and \( i \) assigns to each \( {V}_{W} \) the variety \( i\left( {V}_{W}\right) = {V}_{\left( W\right) } \) . (Thus \( i \) simply removes from \( {V}_{W} \) reference to the "center" \( W \) .) Finally, the bottom horizontal maps i* and \( \sqrt{} \) are the embedding and radical maps; \( {G}^{ * } \) and \( {J}^{ * } \) will be defined in terms of the other maps, and will turn out to be mutually inverse lattice-reversing isomorphisms. In establishing properties of these maps, extension and contraction between \( \mathcal{I}\left( R\right) \) and \( \mathcal{I}\left( {R}_{\mathfrak{p}}\right) \) play a basic part; we look at them first. \( {\left( \;\right) }^{e} : \mathcal{I}\left( R\right) \rightarrow \mathcal{I}\left( {R}_{\mathfrak{p}}\right) \) This map is onto \( \mathcal{I}\left( {R}_{\mathfrak{p}}\right) \) ; in particular, each ideal \( {\mathfrak{a}}^{ * } \subset {R}_{\mathfrak{p}} \) comes from the ideal \( {\mathfrak{a}}^{*c} \subset R \) -that is, For each \( {\mathfrak{a}}^{ * } \in {R}_{\mathfrak{p}} \) , \[ {\mathfrak{a}}^{ * } = {\mathfrak{a}}^{*{ce}} \] (8) Proof. That \( {\mathfrak{a}}^{*{ce}} \subset {\mathfrak{a}}^{ * } \) is obvious, since \( {a}^{ * } \in {\mathfrak{a}}^{*{ce}} \) implies that \( {a}^{ * } = a/m \) for some \( a \in {\mathfrak{a}}^{*c} \) and some \( m \in R \smallsetminus \mathfrak{p} \) . To show \( {\mathfrak{a}}^{ * } \subset {\mathfrak{a}}^{*{ce}} \), let \( {a}^{ * } \in {\mathfrak{a}}^{ * } \) . Then \( {a}^{ * } \in {R}_{\mathfrak{p}} \) , which implies \( {a}^{ * } = a/m \) for some \( a \in R \) and \( m \in R \smallsetminus \mathfrak{p} \) ; also \( a = m{a}^{ * } \), so \( a \in {a}^{ * } \) , which means \( a \in {\mathfrak{a}}^{ * } \cap R = {\mathfrak{a}}^{*c} \) . Hence \( {\mathfrak{a}}^{ * } = a/m \in {\mathfrak{a}}^{*{ce}} \) , Next note that \( {\left( \;\right) }^{e} \) is not necessarily \( 1 : 1 \), since \[ {\mathfrak{a}}^{e} = {R}_{\mathfrak{p}}\text{ for every ideal }\mathfrak{a} ⊄ \mathfrak{p}. \] (9) ( \( \mathfrak{a} \subset \mathfrak{p} \) implies that there is an \( m \in \mathfrak{a} \cap \left( {R \smallsetminus \mathfrak{p}}\right) \), hence \( m/m = 1 \in {\mathfrak{a}}^{e} \) .) However, (3.8) \( {\left( \;\right) }^{e} \) is \( 1 : 1 \) on the set of contracted ideals of \( \mathcal{I}\left( R\right) \) . For if \( \mathfrak{a} = {\mathfrak{a}}^{*c} \) and \( \mathfrak{b} = {\mathfrak{b}}^{*c} \), and if \( {\mathfrak{a}}^{e} = {\mathfrak{a}}^{*{ce}} = {\mathfrak{b}}^{e} = {\mathfrak{b}}^{*{ce}} \), then \( {\mathfrak{a}}^{ * } = {\mathfrak{b}}^{ * } \), so \( \mathfrak{a} = {\mathfrak{a}}^{*c} = \mathfrak{b} = {\mathfrak{b}}^{*c}. \) \( {\left( \;\right) }^{c} : \mathcal{I}\left( {R}_{\mathfrak{p}}\right) \rightarrow \mathcal{I}\left( R\right) \) This map is not necessarily onto, because \( {\mathfrak{a}}^{*c} \) is either \( R \) or is contained in \( \mathfrak{p} \) . (If \( {\mathfrak{a}}^{*c} \) is not contained in \( \mathfrak{p} \), then \( {\mathfrak{a}}^{ * } = {\mathfrak{a}}^{ce} = {R}_{\mathfrak{p}} \), whence \( {\mathfrak{a}}^{*c} = R \) .) Next note that \( {\left( \text{ }\text{ }\right) }^{c} \) is \( 1 : 1 \), for if \( {\mathfrak{a}}^{*c} = {\mathfrak{b}}^{*c} \), then \( {\mathfrak{a}}^{*{ce}} = {\mathfrak{b}}^{*{ce}} = {\mathfrak{a}}^{ * } = {\mathfrak{b}}^{ * } \) . In general \( \mathfrak{a} \neq {\mathfrak{a}}^{ec} \), but we always have \[ \mathfrak{a} \subset {\mathfrak{a}}^{ec}. \] (10) (Theorem 3.14 will supply geometric meaning to (10), and also to Theorem 3.10 below.) The following characterization of \( {\mathfrak{a}}^{ec} \) is useful: \[ {\mathfrak{a}}^{ec} = \{ a \in R \mid {am} \in \mathfrak{a}\text{, for some }m \in R \smallsetminus \mathfrak{p}\} . \] (11) Proof \( \subset \) : Each element of \( {\mathfrak{a}}^{e} \) is a sum of quotients of elements in \( \mathfrak{a} \) by elements in \( R \smallsetminus \mathfrak{p} \) ; obviously such a sum is itself such a quotient. Hence an element \( a \) is in \( {\mathfrak{a}}^{ec} \) iff it is in \( R \) and is of the form \( a = {a}^{\prime }/m \) where \( {a}^{\prime } \in \mathfrak{a} \) . Hence \( {am} = {a}^{\prime } \in \mathfrak{a} \) , proving the inclusion. \( \supset \) : Any \( a \) on the right-hand side of (11) can be w

18_Algebra Chapter 0

Definition 1.8

Definition 1.8. An element \( a \) in a ring \( R \) is a left-zero-divisor if there exist elements \( b \neq 0 \) in \( R \) for which \( {ab} = 0 \) . The reader will have no difficulty figuring out what a right-zero-divisor should be. The element 0 is a zero-divisor in all nonzero rings \( R \) ; the zero ring is the only ring without zero-divisors(!). Proposition 1.9. In a ring \( R, a \in R \) is not a left- (resp., right-) zero-divisor if and only if left (resp., right) multiplication by a is an injective function \( R \rightarrow R \) . In other words, \( a \) is not a left- (resp., right-) zero-divisor if and only if multiplicative left- (resp., right-) cancellation by the element \( a \) holds in \( R \) . Proof. Let's verify the 'left' statement (the 'right' statement is of course entirely analogous). Assume \( a \) is not a left-zero-divisor and \( {ab} = {ac} \) for \( b, c \in R \) . Then, by distributivity, \[ a\left( {b - c}\right) = {ab} - {ac} = 0, \] and this implies \( b - c = 0 \) since \( a \) is not a left-zero-divisor; that is, \( b = c \) . This proves that left-multiplication is injective in this case. Conversely, if \( a \) is a left-zero-divisor, then \( \exists b \neq 0 \) such that \( {ab} = 0 = a \cdot 0 \) ; this shows that left-multiplication is not injective in this case, concluding the proof. Rings such as \( \mathbb{Z},\mathbb{Q} \), etc., are commutative rings without (nonzero) zero-divisors. Such rings are very special, but very important, and they deserve their own terminology: Definition 1.10. An integral domain is a nonzero commutative ring \( R \) (with 1) such that \[ \left( {\forall a, b \in R}\right) : \;{ab} = 0 \Rightarrow a = 0\text{ or }b = 0. \] Chapter V will be entirely devoted to integral domains. An element which is not a zero-divisor is called a non-zero-divisor. Thus, integral domains are those nonzero commutative rings in which every nonzero element is a non-zero-divisor. By Proposition 1.9, multiplicative cancellation by nonzero elements holds in integral domains. The rings \( \mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C} \) are all integral domains. As we have seen, some \( \mathbb{Z}/n\mathbb{Z} \) are not integral domains. Here is one of those places where the reader can do him/herself a great favor by pausing a moment and figuring something out: answer the question, which \( \mathbb{Z}/n\mathbb{Z} \) are integral domains? This is entirely within reach, given what the reader knows already. Don't read ahead before figuring this out - this question will be answered within a few short paragraphs, spoiling all the fun. There are even subtler reasons why \( \mathbb{Z} \) is a very special ring: we will see in due time that it is a 'UFD' (unique factorization domain); in fact, it is a 'PID' (principal ideal domain); in fact, it is more special still!, as it is a 'Euclidean domain'. All of this will be discussed in Chapter V particularly V12 However, \( \mathbb{Q},\mathbb{R},\mathbb{C} \) are more special than all of that and then some, since they are fields. Definition 1.11. An element \( u \) of a ring \( R \) is a left-unit if \( \exists v \in R \) such that \( {uv} = 1 \) ; it is a right-unit if \( \exists v \in R \) such that \( {vu} = 1 \) . Units are two-sided units. Proposition 1.12. In a ring \( R \) : - \( u \) is a left- (resp., right-) unit if and only if left- (resp., right-) multiplication by \( u \) is a surjective functions \( R \rightarrow R \) ; - if \( u \) is a left- (resp., right-) unit, then right- (resp., left-) multiplication by \( u \) is injective; that is, \( u \) is not a right- (resp., left-) zero-divisor; - the inverse of a two-sided unit is unique; - two-sided units form a group under multiplication. Proof. These assertions are all straightforward. For example, denote by \( {\rho }_{u} : R \rightarrow \) \( R \) right-multiplication by \( u \), so that \( {\rho }_{u}\left( r\right) = {ru} \) . If \( u \) is a right-unit, let \( v \in R \) be such that \( {vu} = 1 \) ; then \( \forall r \in R \) \[ {\rho }_{u} \circ {\rho }_{v}\left( r\right) = {\rho }_{u}\left( {rv}\right) = \left( {rv}\right) u = r\left( {vu}\right) = r{1}_{R} = r. \] That is, \( {\rho }_{v} \) is a right-inverse to \( {\rho }_{u} \), and therefore \( {\rho }_{u} \) is surjective (Proposition 1.2.1). Conversely, if \( {\rho }_{u} \) is surjective, then there exists a \( v \) such that \( {1}_{R} = {\rho }_{\left( u\right) }\left( v\right) = {vu} \) , so that \( u \) is a right-unit. This checks the first statement, for right-units. For the second statement, denote by \( {\lambda }_{u} : R \rightarrow R \) left-multiplication by \( u \) : \( {\lambda }_{u}\left( r\right) = {ur} \) . Assume \( u \) is a right-unit, and let \( v \) be such that \( {vu} = {1}_{R} \) ; then \( \forall r \in R \) \[ {\lambda }_{v} \circ {\lambda }_{u}\left( r\right) = {\lambda }_{v}\left( {ur}\right) = v\left( {ur}\right) = \left( {vu}\right) r = {1}_{R}r = r. \] That is, \( {\lambda }_{v} \) is a left-inverse to \( {\lambda }_{u} \), so \( {\lambda }_{u} \) is injective (Proposition 112.1 again). The rest of the proof is left to the reader (Exercise 1.9). Since the inverse of a two-sided unit \( u \) is unique, we can give it a name; of course we denote it by \( {u}^{-1} \) . The reader should keep in mind that inverses of left- or right-units are not unique in general, so the 'inverse notation' is not appropriate for them. Definition 1.13. A division ring is a ring in which every nonzero element is a two-sided unit. We will mostly be concerned with the commutative case, which has its own name: Definition 1.14. A field is a nonzero commutative ring \( R \) (with 1) in which every nonzero element is a unit. The whole of Chapter VII will be devoted to studying fields. By Proposition 1.12 (second part), every field is an integral domain, but not conversely: indeed, \( \mathbb{Z} \) is an integral domain, but it is not a field. Remember: \[ \text{field} \Rightarrow \text{integral domain,} \] integral domain \( \Rightarrow \) field. There is a situation, however, in which the two notions coincide: Proposition 1.15. Assume \( R \) is a finite commutative ring; then \( R \) is an integral domain if and only if it is a field. Proof. One implication holds for all rings, as pointed out above; thus we only have to verify that if \( R \) is a finite integral domain, then it is a field. This amounts to verifying that if \( a \) is a non-zero-divisor in a finite (commutative) ring \( R \), then it is a unit in \( R \) . Now, if \( a \) is a non-zero-divisor, then multiplication by \( a \) in \( R \) is injective (Proposition 1.9); hence it is surjective, as the ring is finite, by the pigeon-hole principle; hence \( a \) is a unit, by Proposition 1.12, Remark 1.16. A little surprisingly, the hypothesis of commutativity in Proposition 1.15 is actually superfluous: a theorem known as Wedderburn's little theorem shows that finite division rings are necessarily commutative. The reader will prove this fact in a distant future (Exercise VII 5.14). Example 1.17. The group of units in the ring \( \mathbb{Z}/n\mathbb{Z} \) is precisely the group \( {\left( \mathbb{Z}/n\mathbb{Z}\right) }^{ * } \) introduced in [112.3] indeed, a class \( {\left\lbrack m\right\rbrack }_{n} \) is a unit if and only if (right-) multiplication by \( {\left\lbrack m\right\rbrack }_{n} \) is surjective (by Proposition 1.12), if and only if the map \( a \mapsto a{\left\lbrack m\right\rbrack }_{n} \) is surjective, if and only if \( {\left\lbrack m\right\rbrack }_{n} \) generates \( \mathbb{Z}/n\mathbb{Z} \), if and only if \( \gcd \left( {m, n}\right) = 1 \) (Corollary II 2.5), if and only if \( {\left\lbrack m\right\rbrack }_{n} \in {\left( \mathbb{Z}/n\mathbb{Z}\right) }^{ * } \) . In particular, those \( n \) for which all nonzero elements of \( \mathbb{Z}/n\mathbb{Z} \) are units (that is, for which \( \mathbb{Z}/n\mathbb{Z} \) is a field) are precisely those \( n \in \mathbb{Z} \) for which \( \gcd \left( {m, n}\right) = 1 \) for all \( m \) that are not multiples of \( n \) ; this is the case if and only if \( n \) is prime. Putting this together with Proposition 1.15, we get the pretty classification (for integers \( p \neq 0 \) ) \[ \mathbb{Z}/p\mathbb{Z}\text{ integral domain } \Leftrightarrow \mathbb{Z}/p\mathbb{Z}\text{ field } \Leftrightarrow p\text{ prime,} \] which the reader is well advised to remember firmly. Example 1.18. The rings \( \mathbb{Z}/p\mathbb{Z} \), with \( p \) prime, are not the only finite fields. In fact, for every prime \( p \) and every integer \( r > 0 \) there is a (unique, in a suitable sense) multiplication on the product group \[ \underset{r\text{ times }}{\underbrace{\mathbb{Z}/p\mathbb{Z} \times \cdots \times \mathbb{Z}/p\mathbb{Z}}} \] making it into a field. A discussion of these fields will have to wait until we have accumulated much more material (cf. (VII15.1), but the reader could already try to construct small examples 'by hand' (cf. Exercise 1.11). 1.3. Polynomial rings. We will study polynomial rings in some depth, especially over fields; they are another class of examples that is to some extent already familiar to our reader. I will capitalize on this familiarity and avoid a truly formal (and truly tedious) definition. Definition 1.19. Let \( R \) be a ring. A polynomial \( f\left( x\right) \) in the indeterminate \( x \) and with coefficients in \( R \) is a finite linear combination of nonnegative ’powers’ of \( x \) with coefficients in \( R \) : \[ f\left( x\right) = \mathop{\sum }\limits_{{i \geq 0}}{a}_{i}{x}^{i} = {a}_{0} + {a}_{1}x + {a}_{2}{x}^{2} + \cdots , \] where all \( {a}_{i} \) are elements of \( R \) (the coefficients) and we require \( {a}_{i} = 0 \) for \( i \gg 0 \) . Two polynomials are taken to be equal if all the coefficients are equal: \[ \mathop{\sum }\limits_{{i \geq 0}}{a}_{i}{x}^{i} = \mathop{\sum

108_The Joys of Haar Measure

Definition 7.1.7

Definition 7.1.7. Let \( E \) and \( {E}^{\prime } \) be two elliptic curves with identity elements \( \mathcal{O} \) and \( {\mathcal{O}}^{\prime } \) respectively. An isogeny \( \phi \) from \( E \) to \( {E}^{\prime } \) is a morphism of algebraic curves from \( E \) to \( {E}^{\prime } \) such that \( \phi \left( \mathcal{O}\right) = {\mathcal{O}}^{\prime } \) . A nonconstant isogeny is one such that there exists \( P \in E \) such that \( \phi \left( P\right) \neq {\mathcal{O}}^{\prime } \) . We say that \( E \) and \( {E}^{\prime } \) are isogenous if there exists a nonconstant isogeny from \( E \) to \( {E}^{\prime } \) . We will implicitly assume that our isogenies are nonconstant. By the above theorem, an isogeny \( \phi \) preserves the group law, in other words is such that \( \phi \left( {P + {P}^{\prime }}\right) = \phi \left( P\right) + \phi \left( {P}^{\prime }\right) \), where addition on the left is on the curve \( E \), and on the right is on the curve \( {E}^{\prime } \) . The following results summarize the main properties of isogenies; see [Sil1] for details and proofs. Theorem 7.1.8. Let \( \phi \) be a nonconstant isogeny from \( E \) to \( {E}^{\prime } \) defined over an algebraically closed field \( K \) . Then (1) The map \( \phi \) is surjective. (2) \( \phi \) is a finite map; in other words, the fiber over any point of \( {E}^{\prime } \) is constant and finite. From these properties it is easy to see that \( \phi \) induces an injective map from the function field of \( {E}^{\prime } \) to that of \( E \) over some algebraic closure of the base field. The degree of the corresponding field extension is finite and called the degree of \( \phi \) . If this field extension is separable, the degree of \( \phi \) is also equal to the cardinality of a fiber, in other words to \( \left| {\operatorname{Ker}\left( \phi \right) }\right| \), but this is not true in general. Thus, as algebraic curves, or equivalently, over an algebraically closed field extension of the base field, a nonconstant isogeny induces an isomorphism from \( E/\operatorname{Ker}\left( \phi \right) \) to \( {E}^{\prime } \), where \( E/\operatorname{Ker}\left( \phi \right) \) must be suitably defined as an elliptic curve. If there exists a nonconstant isogeny \( \phi \) from \( E \) to \( {E}^{\prime } \) of degree \( m \), we say that \( E \) and \( {E}^{\prime } \) are \( m \) -isogenous. Conversely, we have the following: Proposition 7.1.9. If \( G \) is a finite subgroup of \( E \) there exists a natural elliptic curve \( {E}^{\prime } \) and an isogeny \( \phi \) from \( E \) to \( {E}^{\prime } \) whose kernel (over some algebraic closure) is equal to \( G \) . The elliptic curve \( {E}^{\prime } \) is well defined up to isomorphism and is denoted by \( E/G \) . Note that the equation of \( {E}^{\prime } \) can be given explicitly by formulas due to Vélu [Vel]. Two isogenous elliptic curves are very similar, but are in general not isomorphic. For instance, Theorem 8.1.3 tells us that two elliptic curves defined over \( \mathbb{Q} \) that are isogenous over \( \mathbb{Q} \) have for instance the same rank and the same \( L \) -function. However, they do not necessarily have the same torsion subgroup: for instance, it follows from Proposition 8.4.3 that the elliptic curves \( {y}^{2} = {x}^{3} + 1 \) and \( {y}^{2} = {x}^{3} - {27} \) are 3-isogenous, but it is easily shown using for instance the Nagell-Lutz Theorem 8.1.10 that the torsion subgroup of the former has order 6 , while the torsion subgroup of the latter has order 2 . Proposition 7.1.10. Let \( \phi \) be a nonconstant isogeny from an elliptic curve \( E \) to \( {E}^{\prime } \) of degree \( m \) . There exists an isogeny \( \psi \) from \( {E}^{\prime } \) to \( E \), called the dual isogeny of \( \phi \), such that \[ \psi \circ \phi = {\left\lbrack m\right\rbrack }_{E}\;\text{ and }\;\phi \circ \psi = {\left\lbrack m\right\rbrack }_{{E}^{\prime }}, \] where \( \left\lbrack m\right\rbrack \) denotes the multiplication-by-m map on the corresponding curve. An isogeny of degree \( m \) will also be called an \( m \) -isogeny. We define the degree of the constant isogeny to be 0 . We will see several examples of isogenies in the next chapter, for instance in Section 8.2 on rational 2-descent, where the basic tools are 2-isogenies. ## 7.2 Transformations into Weierstrass Form ## 7.2.1 Statement of the Problem In this section, we explain how to transform the most commonly encountered equations of elliptic curves into Weierstrass form (simple or not, since it is trivial to transform into simple Weierstrass form by completing the square or the cube, if the characteristic permits). We will usually assume that the characteristic of the base field is different from 2 and 3 , although some of the transformations are valid in more general cases. Recall that a birational transformation is a rational map with rational inverse, outside of a finite number of poles. Although not entirely trivial, it can be shown that in the case of curves (but not of higher-dimensional varieties), two curves are isomorphic if and only if they are birationally equivalent, in other words if there exists a birational transformation from one to the other. It will be slightly simpler to work in projective coordinates instead of affine ones. Thus whenever projective coordinates \( \left( {x, y, z}\right) \) appear, it is always implicit that \( \left( {x, y, z}\right) \neq \left( {0,0,0}\right) \) . Apart from simple or generalized Weierstrass equations, an elliptic curve can be given in the following ways, among others: (1) \( f\left( {x, y, z}\right) = 0 \), where \( f \) is a homogeneous cubic polynomial whose three partial derivatives do not vanish simultaneously, together with a known rational point \( \left( {{x}_{0},{y}_{0},{z}_{0}}\right) \) . (2) \( {y}^{2}{z}^{2} = f\left( {x, z}\right) \), where \( f\left( {x, z}\right) \) is a homogeneous polynomial of degree 4 such that \( f\left( {1,0}\right) \neq 0 \) and without multiple roots, together with a known rational point \( \left( {{x}_{0},{y}_{0},{z}_{0}}\right) \) (this type of equation is called a hyperelliptic quartic). Note that in this case the point at infinity \( \left( {0,1,0}\right) \) is a singular point with distinct tangents, and if the given point is at infinity we ask that the slopes of the tangents be rational. This is equivalent to the fact that \( f\left( {x,1}\right) \) is a fourth-ndegree polynomial whose leading coefficient is a square. (3) \( {f}_{1}\left( {x, y, z, t}\right) = {f}_{2}\left( {x, y, z, t}\right) = 0 \), where \( {f}_{1} \) and \( {f}_{2} \) are two homogeneous quadratic polynomials together with a common projective rational solution \( \left( {{x}_{0},{y}_{0},{z}_{0},{t}_{0}}\right) \), and additional conditions to ensure that the corresponding curve is nonsingular and of genus 1 . We first explain how to transform each of the above equations into Weierstrass form. More precisely, we will show how (3) and (2) transform into (1), and explain how to transform (1) into Weierstrass form. In fact we will see that (2) can also be directly transformed into Weierstrass form. ## 7.2.2 Transformation of the Intersection of Two Quadrics Assume that we are given the homogeneous quadratic equations \( {f}_{1}\left( {x, y, z, t}\right) = \) \( {f}_{2}\left( {x, y, z, t}\right) = 0 \) with common projective rational solution \( \left( {{x}_{0},{y}_{0},{z}_{0},{t}_{0}}\right) \), and assume that the intersection of the corresponding quadrics is nonsingular and of genus 1 . For \( i = 1 \) and 2 write \[ {f}_{i}\left( {x, y, z, t}\right) = {A}_{i}{t}^{2} + {L}_{i}\left( {x, y, z}\right) t + {Q}_{i}\left( {x, y, z}\right) , \] where \( {A}_{i} \) is a constant, \( {L}_{i} \) is linear, and \( {Q}_{i} \) quadratic. By making a linear coordinate change, we may send the rational solution to the projective point \( \left( {0,0,0,1}\right) \), so that in the new coordinates we have \( {A}_{i} = 0 \) ; hence the equations take the form \( t{L}_{i}\left( {x, y, z}\right) + {Q}_{i}\left( {x, y, z}\right) = 0 \) . I claim that the linear forms \( {L}_{1} \) and \( {L}_{2} \) are linearly independent: indeed, otherwise we could replace one of the equations, \( {f}_{1} \) say, by a suitable linear combination of \( {f}_{1} \) and \( {f}_{2} \) to make the \( {L}_{1} \) term disappear, so that the equations would read \( {Q}_{1}\left( {x, y, z}\right) = 0 \) and \( t{L}_{2}\left( {x, y, z}\right) + {Q}_{2}\left( {x, y, z}\right) = 0 \) . This second equation expresses \( t \) rationally in terms of \( x, y \), and \( z \), and the first is a conic, which is of genus 0, a contradiction that proves my claim. Eliminating \( t \) between the two equations \( t{L}_{i}\left( {x, y, z}\right) + {Q}_{i}\left( {x, y, z}\right) = 0 \), we thus have a new equation \( C\left( {x, y, z}\right) = 0 \) with \( C = {L}_{1}{Q}_{2} - {L}_{2}{Q}_{1} \) . This is a homogeneous cubic equation with a projective rational point obtained by solving the homogeneous system of linear equations \( {L}_{1}\left( {x, y, z}\right) = {L}_{2}\left( {x, y, z}\right) = 0 \) , which has a unique projective solution since the \( {L}_{i} \) are independent. This shows how (3) can be transformed into (1). ## 7.2.3 Transformation of a Hyperelliptic Quartic Assume now that we are given the equation \( {y}^{2}{z}^{2} = f\left( {x, z}\right) \) with \( f\left( {x, z}\right) \) a homogeneous polynomial of degree 4, and a rational point \( \left( {{x}_{0},{y}_{0},{z}_{0}}\right) \), assumed to have rational tangents if \( {z}_{0} = 0 \) . If \( {z}_{0} = 0 \), we do nothing. Otherwise, by a translation \( x \mapsto x + {kz} \) for a suitable \( k \in K \), we may assume that \( {x}_{0} = 0 \) , so that the equation is \( {y}^{2}{z}^{2} = f\left( {x, z}\right) \) with \( f\left( {0, z}\right) = \left( {{y}_{0}^{2}/{z}_{0}^{2}}\right) {z}^{

1359_[陈省身] Lectures on Differential Geometry

Definition 2.1

Definition 2.1. Suppose \( C : {u}^{i} = {u}^{i}\left( t\right) \) is a parametrized curve on \( M \), and \( X\left( t\right) \) is a tangent vector field defined on \( C \) given by \[ X\left( t\right) = {x}^{i}\left( t\right) {\left( \frac{\partial }{\partial {u}^{i}}\right) }_{C\left( t\right) }. \] (2.17) We say that \( X\left( t\right) \) is parallel along \( C \) if its absolute differential along \( C \) is zero, i.e., if \[ \frac{DX}{dt} = 0 \] (2.18) If the tangent vectors of a curve \( C \) are parallel along \( C \), then we call \( C \) a self-parallel curve, or a geodesic. Equation (2.18) is equivalent to \[ \frac{d{x}^{i}}{dt} + {x}^{j}{\Gamma }_{jk}^{i}\frac{d{u}^{k}}{dt} = 0 \] \( \left( {2.19}\right) \) This is a system of first-order ordinary differential equations. Thus a given tangent vector \( X \) at any point on \( C \) gives rise to a parallel tangent vector field, called the parallel displacement of \( X \) along the curve \( C \) . By the general discussion in \( §4 - 1 \), we see that a parallel displacement along \( C \) establishes an isomorphism between the tangent spaces at any two points on \( C \) . If \( C \) is a geodesic, then its tangent vector \[ X\left( t\right) = \frac{d{u}^{i}\left( t\right) }{dt}{\left( \frac{\partial }{\partial {u}^{i}}\right) }_{C\left( t\right) } \] is parallel along \( C \) . Therefore a geodesic curve \( C \) should satisfy: \[ \frac{{d}^{2}{u}^{i}}{d{t}^{2}} + {\Gamma }_{jk}^{i}\frac{d{u}^{j}}{dt}\frac{d{u}^{k}}{dt} = 0 \] \( \left( {2.20}\right) \) This is a system of second-order ordinary differential equations. Thus there exists a unique geodesic through a given point of \( M \) which is tangent to a given tangent vector at that point. We now discuss the curvature matrix \( \Omega \) of an affine connection. Since \[ {\omega }_{i}^{j} = {\Gamma }_{ik}^{j}d{u}^{k} \] (2.21) we have \[ d{\omega }_{i}^{j} - {\omega }_{i}^{h} \land {\omega }_{h}^{j} = \frac{\partial {\Gamma }_{ik}^{j}}{\partial {u}^{l}}d{u}^{l} \land d{u}^{k} - {\Gamma }_{il}^{h}{\Gamma }_{hk}^{j}d{u}^{l} \land d{u}^{k} \] \[ = \frac{1}{2}\left( {\frac{\partial {\Gamma }_{il}^{j}}{\partial {u}^{k}} - \frac{\partial {\Gamma }_{ik}^{j}}{\partial {u}^{l}} + {\Gamma }_{il}^{h}{\Gamma }_{hk}^{j} - {\Gamma }_{ik}^{h}{\Gamma }_{hl}^{j}}\right) d{u}^{k} \land d{u}^{l}. \] Therefore \[ {\Omega }_{i}^{j} = \frac{1}{2}{R}_{ikl}^{j}d{u}^{k} \land d{u}^{l} \] (2.22) where \[ {R}_{ikl}^{j} = \frac{\partial {\Gamma }_{il}^{j}}{\partial {u}^{k}} - \frac{\partial {\Gamma }_{ik}^{j}}{\partial {u}^{l}} + {\Gamma }_{il}^{h}{\Gamma }_{hk}^{j} - {\Gamma }_{ik}^{h}{\Gamma }_{hl}^{j}. \] (2.23) If \( \left( {W;{w}^{i}}\right) \) is another coordinate system of \( M \), then the local frame field on \( W,{S}^{\prime } = {}^{t}\left( {\frac{\partial }{\partial {w}^{1}},\ldots ,\frac{\partial }{\partial {w}^{m}}}\right) \), is related to \( S \) on \( U \cap W \) by (2.2). By (1.29) we have \[ {\Omega }^{\prime } = {J}_{WU} \cdot \Omega \cdot {J}_{WU}^{-1} \] (2.24) where \( {\Omega }^{\prime } \) is the curvature matrix of the connection \( D \) under the coordinate system \( \left( {W;{w}^{i}}\right) \) . Componentwise the above equation can be written \[ {\Omega }^{\prime j}{}_{i} = {\Omega }_{p}^{q}\frac{\partial {u}^{p}}{\partial {w}^{i}}\frac{\partial {w}^{j}}{\partial {u}^{q}}. \] Thus \[ {R}^{\prime }{}_{ikl}^{j} = {R}_{prs}^{q}\frac{\partial {w}^{j}}{\partial {u}^{q}}\frac{\partial {u}^{p}}{\partial {w}^{i}}\frac{\partial {u}^{r}}{\partial {w}^{k}}\frac{\partial {u}^{s}}{\partial {w}^{l}}, \] \( \left( {2.25}\right) \) where \( {R}^{\prime j}{}_{ikl} \) is determined by \[ {\Omega }_{i}^{\prime j} = \frac{1}{2}{R}_{ikl}^{\prime j}d{w}^{k} \land d{w}^{l} \] Comparing (2.25) with (2.9) of Chapter 2 we observe that \( {R}_{ikl}^{j} \) satisfies the transformation rule for the components of type- \( \left( {1,3}\right) \) tensors. Therefore \[ R = {R}_{ikl}^{j}\frac{\partial }{\partial {u}^{j}} \otimes d{u}^{i} \otimes d{u}^{k} \otimes d{u}^{l} \] (2.26) is independent of the choice of local coordinates, and is called the curvature tensor of the affine connection \( D \) . For any two smooth tangent vector fields \( X, Y \) on \( M \) we have the curvature operator \( R\left( {X, Y}\right) \) [see (1.30)] which maps a tangent vector field on \( M \) to another tangent vector field. By Theorem 1.3, \( R\left( {X, Y}\right) \) can be written \[ R\left( {X, Y}\right) = {D}_{X}{D}_{Y} - {D}_{Y}{D}_{X} - {D}_{\left\lbrack X, Y\right\rbrack }. \] \( \left( {2.27}\right) \) Now we can express \( R\left( {X, Y}\right) \) in terms of the curvature tensor. Suppose \( X, Y \) , \( Z \) are tangent vector fields with local expressions \[ X = {X}^{i}\frac{\partial }{\partial {u}^{i}},\;Y = {Y}^{i}\frac{\partial }{\partial {u}^{i}},\;Z = {Z}^{i}\frac{\partial }{\partial {u}^{i}}. \] (2.28) Then \[ R\left( {X, Y}\right) Z = {Z}^{i}\left\langle {X \land Y,{\Omega }_{i}^{j}}\right\rangle \frac{\partial }{\partial {u}^{j}} \] (2.29) \[ = {{R}^{j}}_{ikl}{Z}^{i}{X}^{k}{Y}^{l}\frac{\partial }{\partial {u}^{j}}. \] Thus \[ {R}_{ikl}^{j} = \left\langle {R\left( {\frac{\partial }{\partial {u}^{k}},\frac{\partial }{\partial {u}^{l}}}\right) \frac{\partial }{\partial {u}^{i}}, d{u}^{j}}\right\rangle . \] \( \left( {2.30}\right) \) We know that the connection coefficients \( {\Gamma }_{ik}^{j} \) do not satisfy the transformation rule for tensors. But if we define \[ {T}_{ik}^{j} = {\Gamma }_{ki}^{j} - {\Gamma }_{ik}^{j} \] (2.31) then (2.5) implies \[ {T}^{\prime }{}_{ik}^{j} = {T}_{pr}^{q}\frac{\partial {w}^{j}}{\partial {u}^{q}}\frac{\partial {u}^{p}}{\partial {w}^{i}}\frac{\partial {u}^{r}}{\partial {w}^{k}}. \] (2.32) Hence \( {T}_{ik}^{j} \) satisfies the transformation rule for the components of \( \left( {1,2}\right) \) -type tensors. Thus \[ T = {T}_{ik}^{j}\frac{\partial }{\partial {u}^{j}} \otimes d{u}^{i} \otimes d{u}^{k} \] (2.33) is a \( \left( {1,2}\right) \) -type tensor, called the torsion tensor of the affine connection \( D \) . By (2.31) the components of the torsion tensor \( T \) are skew-symmetric with respect to the lower indices, that is, \[ {T}_{ik}^{j} = - {T}_{ki}^{j} \] \( \left( {2.34}\right) \) Being a \( \left( {1,2}\right) \) -type tensor, \( T \) can be viewed as a map from \( \Gamma \left( {T\left( M\right) }\right) \times \Gamma \left( {T\left( M\right) }\right) \) to \( \Gamma \left( {T\left( M\right) }\right) \) . Suppose \( X, Y \) are any two tangent vector fields on \( M \) . Then \( T\left( {X, Y}\right) \) is a tangent vector field on \( M \) with local expression \[ T\left( {X, Y}\right) = {T}_{ij}^{k}{X}^{i}{Y}^{j}\frac{\partial }{\partial {u}^{k}}. \] \( \left( {2.35}\right) \) The reader should verify that \[ T\left( {X, Y}\right) = {D}_{X}Y - {D}_{Y}X - \left\lbrack {X, Y}\right\rbrack . \] (2.36) Definition 2.2. If the torsion tensor of an affine connection \( D \) is zero, then the connection is said to be torsion-free. A torsion-free affine connection always exists. In fact, if the coefficients of a connection \( D \) are \( {\Gamma }_{ik}^{j} \), then set \[ {\widetilde{\Gamma }}_{ik}^{j} = \frac{1}{2}\left( {{\Gamma }_{ik}^{j} + {\Gamma }_{ki}^{j}}\right) \] (2.37) Obviously, \( {\widetilde{\Gamma }}_{ik}^{j} \) is symmetric with respect to the lower indices and satisfies (2.5) under a local change of coordinates. Therefore the \( {\widetilde{\Gamma }}_{ik}^{j} \) are the coefficients of some connection \( \widetilde{D} \), and \( \widetilde{D} \) is torsion-free. Any connection can be decomposed into a sum of a multiple of its torsion tensor and a torsion-free connection. In fact, (2.31) and (2.37) give \[ {\Gamma }_{ik}^{j} = - \frac{1}{2}{T}_{ik}^{j} + {\widetilde{\Gamma }}_{ik}^{j} \] (2.38) that is, \[ {D}_{X}Z = \frac{1}{2}T\left( {X, Z}\right) + {\widetilde{D}}_{X}Z \] (2.39) The geodesic equation (2.20) is equivalent to \[ \frac{{d}^{2}{u}^{i}}{d{t}^{2}} + {\widetilde{\Gamma }}_{jk}^{i}\frac{d{u}^{j}}{dt}\frac{d{u}^{k}}{dt} = 0 \] \( \left( {2.40}\right) \) Thus a connection \( D \) and the corresponding torsion-free connection \( \widetilde{D} \) have the same geodesics. The following two theorems indicate that torsion-free affine connections have relatively desirable properties. Theorem 2.1. Suppose \( D \) is a torsion-free affine connection on \( M \) . Then for any point \( p \in M \) there exists a local coordinate system \( {u}^{i} \) such that the corresponding connection coefficients \( {\Gamma }_{ik}^{j} \) vanish at \( p \) . Proof. Suppose \( \left( {W;{w}^{i}}\right) \) is a local coordinate system at \( p \) with connection coefficients \( {\Gamma }^{\prime j}{}_{ik} \) . Let \[ {u}^{i} = {w}^{i} + \frac{1}{2}{\Gamma }^{\prime }{}_{jk}^{i}\left( p\right) \left( {{w}^{j} - {w}^{j}\left( p\right) }\right) \left( {{w}^{k} - {w}^{k}\left( p\right) }\right) . \] (2.41) Then \[ {\left. \frac{\partial {u}^{i}}{\partial {w}^{j}}\right| }_{p} = {\delta }_{j}^{i},{\left. \;\frac{{\partial }^{2}{u}^{i}}{\partial {w}^{j}\partial {w}^{k}}\right| }_{p} = {\Gamma }^{\prime }{}_{jk}^{i}\left( p\right) . \] \( \left( {2.42}\right) \) Thus the matrix \( \left( \frac{\partial {u}^{i}}{\partial {w}^{j}}\right) \) is nondegenerate near \( p \), and (2.41) provides for a change of local coordinates in a neighborhood of \( p \) . From (2.5) we see that the connection coefficients \( {\Gamma }_{ik}^{j} \) in the new coordinate system \( {u}^{i} \) satisfy \[ {\Gamma }_{ik}^{j}\left( p\right) = 0,\;1 \leq i, j, k \leq m. \] Theorem 2.2. Suppose \( D \) is a torsion-free affine connection on \( M \) . Then we have the Bianchi identity: \[ {R}_{{ikl}, h}^{j} + {R}_{{ilh}, k}^{j} + {R}_{{ihk}, l}^{j} = 0. \] (2.43) Proof. From Theorem 1.4 we have \[ d{\Omega }_{i}^{j} = {\omega }_{i}^{k} \land {\Omega }_{k}^{j} - {\Omega }_{i}^{k} \land {\omega }_{k}^{j} \] that is, \[ \fra

1139_(GTM44)Elementary Algebraic Geometry

Definition 8.9

Definition 8.9. Let \( \left( {A, M,\pi }\right) \) be a near cover (Definition 5.2). A connected open set \( \mathcal{O} \subset M \) is said to be liftable to \( A \) if there is an open set \( \mathcal{Q} \subset A \) such that \( \pi \mid \mathcal{Q} \) is a homeomorphism from \( \mathcal{Q} \) to \( \mathcal{O};\mathcal{Q} \) is then a lifting of \( \mathcal{O} \) . If \( P \in \mathbb{Q} \) we say \( \mathbb{Q} \) is a lifting through \( P \) , and that \( \mathbb{Q} \) lifts \( \mathbb{O} \) through \( P \) . A chain \( \left( {{\mathcal{O}}_{1},\ldots ,{\mathcal{O}}_{m}}\right) \) in \( M \) is liftable to \( A \) if there is a chain \( \left( {{\mathcal{Q}}_{1},\ldots ,{\mathcal{Q}}_{m}}\right) \) in \( A \) such that each \( {Q}_{i} \) is a lifting of \( {O}_{i} \) . Then \( \left( {{Q}_{1},\ldots ,{Q}_{m}}\right) \) is called a lifting of \( \left( {{\mathcal{O}}_{1},\ldots ,{\mathcal{O}}_{m}}\right) \), and a lifting through \( P \) if \( P \in {\mathcal{Q}}_{1} \cup \ldots \cup {\mathcal{Q}}_{m} \) . Definition 8.10. Let \( \mathcal{O} \) be a connected open subset of \( \mathbb{C} = {\mathbb{C}}_{X} \) . The graph in \( \mathcal{O} \times {\mathbb{C}}_{Y} \) of a function single-valued and complex-analytic on \( \mathcal{O} \), is called an analytic function element. Note that an analytic function element describes in a natural way a lifting of \( \mathcal{O} \) ; we therefore write \( \mathcal{Q} \) to denote such a function element. If \( P \in Q \), then \( Q \) is an analytic function element through \( P \) . A chain \( \left( {{\mathcal{Q}}_{1},\ldots ,{\mathcal{Q}}_{m}}\right) \) of analytic function elements lifting a chain \( \left( {{\mathcal{O}}_{1},\ldots ,{\mathcal{O}}_{m}}\right) \) of \( \mathbb{C} \) is called the analytic continuation from \( {\mathcal{Q}}_{1} \) to \( {\mathcal{Q}}_{m} \) along \( \left( {{\mathcal{O}}_{1},\ldots ,{\mathcal{O}}_{m}}\right) \) , or the analytic continuation of \( {\mathcal{Q}}_{1} \) along \( \left( {{\mathcal{O}}_{1},\ldots ,{\mathcal{O}}_{m}}\right) \) ; if \( P \in {\mathcal{Q}}_{1} \) and \( {P}^{\prime } \in {\mathcal{Q}}_{m} \) , \( \left( {{\mathcal{Q}}_{1},\ldots ,{\mathcal{Q}}_{m}}\right) \) is the analytic continuation from \( P \) to \( {P}^{\prime } \) along \( \left( {{\mathcal{O}}_{1},\ldots ,{\mathcal{O}}_{m}}\right) \) . Relative to the cover of special interest to us, namely \( \left( {\mathrm{V}\left( p\right) \smallsetminus {\pi }_{Y}{}^{-1}\left( \mathcal{D}\right) }\right. \) , \( \mathbb{C} \smallsetminus \mathcal{D},{\pi }_{Y} \) ), there is about each point of \( \mathbb{C} \smallsetminus \mathcal{D} \) a connected open neighborhood \( \mathcal{O} \) which has a lifting \( \mathcal{Q} \) . Any such lifting is the graph of a function analytic on \( \mathcal{C} \) (from Theorem 3.6)-that is, any such \( \mathcal{Q} \) is an analytic function element. When considering chains in our proof of Theorem 8.5, it will be of technical convenience to restrict our attention to connected open sets \( {\mathcal{O}}_{i} \) of \( \mathbb{C} \smallsetminus \mathcal{D} \) which are liftable through each point of \( {\pi }_{Y}{}^{-1}\left( {\mathcal{O}}_{i}\right) \) (which means that \( {\pi }_{Y}{}^{-1}\left( {\mathcal{O}}_{i}\right) \) consists of \( n\left( { = {\deg }_{Y}p}\right) \) functional elements. Note that there is such an \( \mathcal{O} \) about each point of \( \mathbb{C} \smallsetminus \mathcal{D} \) . Definition 8.11. Relative to \( \left( {\mathrm{V}\left( p\right) \smallsetminus {\pi }_{Y}{}^{-1}\left( \mathcal{D}\right) ,\mathbb{C} \smallsetminus \mathcal{D},{\pi }_{Y}}\right) \), any connected open set \( \mathcal{O} \) of \( \mathbb{C} \smallsetminus \mathcal{D} \) which lifts through each point of \( {\pi }_{Y}{}^{-1}\left( \mathcal{O}\right) \) is an allowable set. Any chain of allowable open sets is an allowable chain. Lemma 8.13 below is used in our proof of Theorem 8.5 and gives an important class of allowable open sets. Definition 8.12. An open set \( \Omega \subset \mathbb{C} \) is simply connected if it is homeomorphic to an open disk. Examples are: \( \mathbb{C} \) itself; \( \mathbb{C} \smallsetminus \) (nonnegative real axis); \( \mathbb{C} \smallsetminus \Phi \), where \( \Phi \) is any closed, non-self-intersecting polygonal path that goes out to the infinite point of \( {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) (see Figure 20).  Figure 20 Lemma 8.13. Relative to \( \left( {\mathrm{V}\left( p\right) \smallsetminus {\pi }_{Y}{}^{-1}\left( \mathcal{D}\right) ,\mathbb{C} \smallsetminus \mathcal{D},{\pi }_{Y}}\right) \), any simply connected open subset of \( \mathbb{C} \smallsetminus \mathcal{D} \) is allowable. This is an immediate consequence of the familiar "monodromy theorem" (proved in most standard texts on elementary complex analysis). To state it, we use the following ideas: First, let \( U \) be a nonempty open subset of \( \mathbb{C} \) . A polygonal path in \( U \) is the union of closed line segments \( {\overline{{P}_{i}, P}}_{i + 1} \subset U(i = 0,\ldots \) , \( r - 1) \) connecting finitely many ordered points \( \left( {{P}_{0},\ldots ,{P}_{r}}\right) \left( {{P}_{i} \neq {P}_{i + 1}}\right) \) in \( U \) . Now suppose \( \mathcal{Q} \) is an analytic function element which is a lifting of a connected open set \( \mathcal{O} \subset U \) . We say \( \mathcal{Q} \) can be continued along a polygonal path \( \overline{{P}_{0},{P}_{1}} \cup \ldots \cup \overline{{P}_{r - 1},{P}_{r}} \) in \( U \) if \( {P}_{0} \in \mathcal{O} \) and if there is a chain \( \mathcal{O},{\mathcal{O}}_{1},\ldots ,{\mathcal{O}}_{r} \) in \( U \) such that \( {\bar{P}}_{i},{\bar{P}}_{i + 1} \subseteq {\mathcal{O}}_{i + 1}\left( {i = 0,\ldots, r - 1}\right) \), and such that there is an analytic continuation of \( Q \) along \( O,\ldots ,{O}_{r} \) . Theorem 8.14 (Monodromy theorem). Let \( \Omega \) be a simply connected open set in \( \mathbb{C} \), and suppose an analytic function element \( \mathcal{Q} \) is a lifting of a connected open set \( \mathcal{O} \subset \Omega \) . If \( \mathcal{Q} \) can be analytically continued along any polygonal path in \( \Omega \) , then \( \mathcal{Q} \) has a unique extension to a (single-valued) function which is analytic at each point of \( \Omega \) . For a proof of Theorem 8.14, see, e.g., [Ahlfors, Chapter VI, Theorem 2]. To prove Lemma 8.13, we need only verify that in our case, the hypothesis of Theorem 8.14 is satisfied, i.e., that for any simply connected open subset \( \Omega \) of \( \mathbb{C} \smallsetminus \mathcal{D} \), we can analytically continue any analytic function element along any polygonal path in \( \Omega \) . The argument is easy, and may be left to the exercises (Exercise 8.1). We now prove that \( \mathrm{V}\left( p\right) \smallsetminus {\pi }_{Y}{}^{-1}\left( \mathcal{D}\right) \) is chainwise connected by contradiction. Suppose \( P \) and \( Q \) are two points of \( \mathrm{V}\left( p\right) \smallsetminus {\pi }_{Y}{}^{-1}\left( \mathcal{D}\right) \) such that there is no analytic continuation from \( P \) to \( Q \) along any allowable chain in \( \mathbb{C} \smallsetminus \mathcal{D} \) . Choose a non-self-intersecting polygonal path \( \Phi \) in \( \mathbb{C} \) connecting the finitely many points of \( \mathcal{D} \), and the infinite point of \( {\mathbb{P}}^{1}\left( \mathbb{C}\right) = {\mathbb{C}}_{X} \cup \{ \infty \} \), as suggested by Figure 20. We can obviously choose \( \Phi \) so it does not go through \( {\pi }_{Y}\left( P\right) \) or \( {\pi }_{Y}\left( Q\right) \) . The "slit sphere" \( {\mathbb{P}}^{1}\left( \mathbb{C}\right) \smallsetminus \Phi \) is then topologically an open disk of \( \mathbb{C} \), and is therefore simply connected. Now each point of \( C \) above any point of \( \mathbb{C} \smallsetminus \Phi \) is contained in an analytic function element, and by Lemma 8.13 each such function element extends to an analytic function on \( \mathbb{C} \smallsetminus \Phi \) . Since there are \( n \) points of \( C \) above each point of \( \mathbb{C} \smallsetminus \Phi \), there are just \( n \) such functions \( {f}_{i} \) on \( \mathbb{C} \smallsetminus \Phi \) . Call their graphs \( {F}_{1},\ldots ,{F}_{n} \) . Suppose, to be specific, that \( P \in {F}_{1} \) and \( Q \in {F}_{n} \) . Now let \( P \) and \( {P}^{\prime } \) be two points of \( C \) lying over \( \mathbb{C} \smallsetminus \Phi \), and suppose that we can analytically continue from \( P \) to \( {P}^{\prime } \) along some allowable chain \( \left( {{\mathcal{O}}_{1},\ldots ,{\mathcal{O}}_{r}}\right) \) in \( C \smallsetminus \mathcal{D} \) . Choose open sets such that \( {\mathcal{O}}_{0},{\mathcal{O}}_{n + 1} \subset \mathbb{C} \smallsetminus \Phi, P \in {\mathcal{O}}_{0} \subset {\mathcal{O}}_{1} \) and \( {P}^{\prime } \in {\mathcal{O}}_{r + 1} \supset {\mathcal{O}}_{r} \) . Since \( \mathbb{C} \smallsetminus \Phi \) is simply connected, it is allowable by Lemma 8.13. Hence its subsets \( {\mathcal{O}}_{0},{\mathcal{O}}_{r + 1} \) are also allowable, and therefore \( (\mathbb{C} \smallsetminus \Phi ,{\mathcal{O}}_{0} \) , \( {\mathcal{O}}_{1},\ldots ,{\mathcal{O}}_{r},{\mathcal{O}}_{r + 1},\mathbb{C} \smallsetminus \Phi \) ) is an allowable chain. Thus we may assume without loss of generality that any such analytic continuation in \( \mathbb{C} \smallsetminus \mathcal{D} \) from a point \( P \in C \) to any other point \( {P}^{\prime } \in C \), where \( {\pi }_{Y}\left( P\right) \) and \( {\pi }_{Y}\left( {P}^{\prime }\right) \in \mathbb{C} \smallsetminus \Phi \), is the lifting of some allowable chain from \( \mathbb{C} \smallsetminus \Phi \) to \( \mathbb{C} \smallsetminus \Phi \) . If \( P \in {F}_{i} \) and \( {P}^{\prime } \in {F}_{j} \), then this same chain also defines a continuation from any point in \( {F}_{i}

18_Algebra Chapter 0

Definition 2.7

Definition 2.7. A Euclidean valuation on an integral domain \( R \) is a valuation satisfying the following property 8 : for all \( a \in R \) and all nonzero \( b \in R \) there exist \( q, r \in R \) such that \[ a = {qb} + r \] with either \( r = 0 \) or \( v\left( r\right) < v\left( b\right) \) . An integral domain \( R \) is a Euclidean domain if it admits a Euclidean valuation. \( {}^{6} \) This fact is known as the fundamental theorem of arithmetic. \( {}^{7} \) Entire libraries have been written on the subject of valuations, studying a more precise notion than what is needed here. \( {}^{8} \) It is not uncommon to also require that \( v\left( {ab}\right) \geq v\left( b\right) \) for all nonzero \( a, b \in R \) ; but this is not needed in the considerations that follow, and cf. Exercise 2.15 We say that \( q \) is the quotient of the division and \( r \) is the remainder. Division with remainder in \( \mathbb{Z} \) and in \( k\left\lbrack x\right\rbrack \) (where \( k \) is a field) provide examples, so that \( \mathbb{Z} \) and \( k\left\lbrack x\right\rbrack \) are Euclidean domains. Proposition 2.8. Let \( R \) be a Euclidean domain. Then \( R \) is a PID. The proof is modeled after the instances encountered for \( \mathbb{Z} \) (Proposition 1114.4) and \( k\left\lbrack x\right\rbrack \) (which the reader has hopefully worked out in Exercise 11114.4). Proof. Let \( I \) be an ideal of \( R \) ; we have to prove that \( I \) is principal. If \( I = \{ 0\} \) , there is nothing to show; therefore, assume \( I \neq \{ 0\} \) . The valuation maps the nonzero elements of \( I \) to a subset of \( {\mathbb{Z}}^{ \geq 0} \) ; let \( b \in I \) be an element with the smallest valuation. Then I claim that \( I = \left( b\right) \) ; therefore \( I \) is principal, as needed. Since clearly \( \left( b\right) \subseteq I \), we only need to verify that \( I \subseteq \left( b\right) \) . For this, let \( a \in I \) and apply division with remainder: we have \[ a = {qb} + r \] for some \( q, r \) in \( R \), with \( r = 0 \) or \( v\left( r\right) < v\left( b\right) \) . But \[ r = a - {qb} \in I : \] by the minimality of \( v\left( b\right) \) among nonzero elements of \( I \), we cannot have \( v\left( r\right) < v\left( b\right) \) . Therefore \( r = 0 \), showing that \( a = {qb} \in \left( b\right) \), as needed. Proposition 2.8 justifies one more feature of the picture at the beginning of the chapter: the class of Euclidean domains is contained in the class of principal ideal domains. This inclusion is proper, as suggested in the picture. Producing an explicit example of a PID which is not a Euclidean domain is not so easy, but the gap between PIDs and Euclidean domains can in fact be described very sharply: PID may be characterized as domains satisfying a weaker requirement than 'division with remainder'. More precisely, a ’Dedekind-Hasse valuation’ is a valuation \( v \) such that \( \forall a, b \) , either \( \left( {a, b}\right) = \left( b\right) \) (that is, \( b \) divides \( a \) ) or there exists \( r \in \left( {a, b}\right) \) such that \( v\left( r\right) < v\left( b\right) \) . This latter condition amounts to requiring that there exist \( q, s \in R \) such that \( {as} = {bq} + r \) with \( v\left( r\right) < v\left( b\right) \) ; hence a Euclidean valuation (for which we may in fact choose \( s = 1 \) ) is a Dedekind-Hasse valuation. It is not hard to show that an integral domain is a PID if and only if it admits a Dedekind-Hasse valuation (Exercise 2.21). For example, this can be used to show that the ring \( \mathbb{Z}\left\lbrack {\left( {1 + \sqrt{-{19}}}\right) /2}\right\rbrack \) is a PID: the norm considered in Exercise 2.18 in order to prove that this ring is not a Euclidean domain turns out to be a Dedekind-Hasse valuation . Thus, this ring gives an example of a PID that is not a Euclidean domain. One excellent feature of Euclidean domains, and the one giving them their names, is the presence of an effective algorithm computing greatest common divisors: the Euclidean algorithm. As Euclidean domains are PIDs, and hence UFDs, we know that they do have greatest common divisors. However, the 'algorithm' \( {}^{9} \) This boils down to a case-by-case analysis, which I am happily leaving to my most patient readers. obtained by distilling the proof of Lemma 2.3 is highly impractical: if we had to factor two integers \( a, b \) in order to compute their gcd, this would make it essentially impossible (with current technologies and factorization algorithms) for integers of a few hundred digits. The Euclidean algorithm bypasses the necessity of factorization: greatest common divisors of thousand-digit integers may be computed in a fraction of a second. The key lemma on which the algorithm is based is the following trivial general fact: Lemma 2.9. Let \( a = {bq} + r \) in \( a \) ring \( R \) . Then \( \left( {a, b}\right) = \left( {b, r}\right) \) . Proof. Indeed, \( r = a - {bq} \in \left( {a, b}\right) \), proving \( \left( {b, r}\right) \subseteq \left( {a, b}\right) \) ; and \( a = {bq} + r \in \left( {b, r}\right) \) , proving \( \left( {a, b}\right) \subseteq \left( {b, r}\right) \) . In particular, \[ \left( {\forall c \in R}\right) ,\;\left( {a, b}\right) \subseteq \left( c\right) \Leftrightarrow \left( {b, r}\right) \subseteq \left( c\right) ; \] that is, the set of common divisors of \( a, b \) and the set of common divisors of \( b, r \) coincide. Therefore, Corollary 2.10. Assume \( a = {bq} + r \) . Then \( a, b \) have a gcd if and only if \( b, r \) have \( {agcd} \), and in this case \( \gcd \left( {a, b}\right) = \gcd \left( {b, r}\right) \) . Of course ’ \( \gcd \left( {a, b}\right) = \gcd \left( {b, r}\right) \) ’ means that the two classes of associate elements coincide. These considerations hold over any integral domain; assume now that \( R \) is a Euclidean domain. Then we can use division with remainder to gain some control over the remainders \( r \) . Given two elements \( a, b \) in \( R \), with \( b \neq 0 \), we can apply division with remainder repeatedly: \[ a = b{q}_{1} + {r}_{1} \] \[ b = {r}_{1}{q}_{2} + {r}_{2} \] \[ {r}_{1} = {r}_{2}{q}_{3} + {r}_{3} \] \( \ldots \) as long as the remainder \( {r}_{i} \) is nonzero. Claim 2.11. This process terminates: that is, \( {r}_{N} = 0 \) for some \( N \) . Proof. Each line in the table is a division with remainder. If no \( {r}_{i} \) were zero, we would have an infinite decreasing sequence \[ v\left( b\right) > v\left( {r}_{1}\right) > v\left( {r}_{2}\right) > v\left( {r}_{3}\right) > \cdots \] of nonnegative integers, which is nonsense. Thus the table of divisions with remainders must be as follows: letting \( {r}_{0} = b \) , \[ a = {r}_{0}{q}_{1} + {r}_{1} \] \[ b = {r}_{1}{q}_{2} + {r}_{2} \] \[ {r}_{1} = {r}_{2}{q}_{3} + {r}_{3} \] \[ \text{...} \] \[ {r}_{N - 3} = {r}_{N - 2}{q}_{N - 1} + {r}_{N - 1} \] \[ {r}_{N - 2} = {r}_{N - 1}{q}_{N} \] with \( {r}_{N - 1} \neq 0 \) . Proposition 2.12. With notation as above, \( {r}_{N - 1} \) is a gcd of \( a, b \) . Proof. By Corollary 2.10, \[ \gcd \left( {a, b}\right) = \gcd \left( {b,{r}_{1}}\right) = \gcd \left( {{r}_{1},{r}_{2}}\right) = \cdots = \gcd \left( {{r}_{N - 2},{r}_{N - 1}}\right) . \] But \( {r}_{N - 2} = {r}_{N - 1}{q}_{N - 1} \) gives \( {r}_{N - 2} \in \left( {r}_{N - 1}\right) \) ; hence \( \left( {{r}_{N - 2},{r}_{N - 1}}\right) = \left( {r}_{N - 1}\right) \) . Therefore \( {r}_{N - 1} \) is a gcd for \( {r}_{N - 2} \) and \( {r}_{N - 1} \), hence for \( a \) and \( b \), as needed. The ring of integers and the polynomial ring over a field are both Euclidean domains. Fields are Euclidean domains (as represented in the picture at the beginning of the chapter), but not for a very interesting reason: the remainder of the division by a nonzero element in a field is always zero, so every function qualifies as a 'Euclidean valuation' for trivial reasons. We will study another interesting Euclidean domain later in this chapter (§6.2). ## Exercises ## 2.1. \( \vartriangleright \) Prove Lemma 2.1 [92.1 2.2. Let \( R \) be a UFD, and let \( a, b, c \) be elements of \( R \) such that \( a \mid {bc} \) and \( \gcd \left( {a, b}\right) = \) 1. Prove that \( a \) divides \( c \) . 2.3. Let \( n \) be a positive integer. Prove that there is a one-to-one correspondence preserving multiplicities between the irreducible factors of \( n \) (as an integer) and the composition factors of \( \mathbb{Z}/n\mathbb{Z} \) (as a group). (In fact, the Jordan-Hölder theorem may be used to prove that \( \mathbb{Z} \) is a UFD.) 2.4. \( \vartriangleright \) Consider the elements \( x, y \) in \( \mathbb{Z}\left\lbrack {x, y}\right\rbrack \) . Prove that 1 is a gcd of \( x \) and \( y \), and yet 1 is not a linear combination of \( x \) and \( y \) . (Cf. Exercise 112.13) [42.1,32.3] 2.5. \( \vartriangleright \) Let \( R \) be the subring of \( \mathbb{Z}\left\lbrack t\right\rbrack \) consisting of polynomials with no term of degree 1: \( {a}_{0} + {a}_{2}{t}^{2} + \cdots + {a}_{d}{t}^{d} \) . - Prove that \( R \) is indeed a subring of \( \mathbb{Z}\left\lbrack t\right\rbrack \), and conclude that \( R \) is an integral domain. - List all common divisors of \( {t}^{5} \) and \( {t}^{6} \) in \( R \) . - Prove that \( {t}^{5} \) and \( {t}^{6} \) have no gcd in \( R \) . ## [82.1] 2.6. Let \( R \) be a domain with the property that the intersection of any family of principal ideals in \( R \) is necessarily a principal ideal. - Show that greatest common divisors exist in \( R \) . - Show that UFDs satisfy this property. 2.7. \( \vartriangleright \) Let \( R \) be a Noetherian domain, and assume that for all nonzero \( a, b \) in \( R \) , the greatest common divisors of \( a \) and \( b \) are linear combinations of \( a \) and \( b \) . Prove that \( R \) is a PID. [2.3] 2.8. Let \( R \) be a UFD, and let \

1009_(GTM175)An Introduction to Knot Theory

Definition 3.4

Definition 3.4. The writhe \( w\left( D\right) \) of a diagram \( D \) of an oriented link is the sum of the signs of the crossings of \( D \), where each crossing has sign +1 or -1 as defined (by convention) in Figure 1.11. Note that this definition of \( w\left( D\right) \) uses the orientation of the plane and that of the link. Note, too, that \( w\left( D\right) \) does not change if \( D \) is changed under a Type II or Type III Reidemeister move. However, \( w\left( D\right) \) does change by +1 or -1 if \( D \) is changed by a Type I Reidemeister move. It is thought that nineteenth-century knot tabulators believed that the writhe of a diagram was a knot invariant, at least when no reduction in the number of crossings by a Type I move was possible in a diagram. That lead to the famous error of the inclusion, in the early knot tables, of both a knot and its reflection, listed as \( {10}_{161} \) and \( {10}_{162} \) (an error detected by \( \mathrm{K} \) . Perko in the 1970's). See Figure 3.1. The writhes of the diagrams are -8 and 10, respectively; yet, modulo reflection, these diagrams represent the same knot.  Figure 3.1 The writhe of an oriented link diagram and the bracket polynomial of the diagram with orientation neglected are, then, both invariant under Reidemeister moves of Types II and III, and both behave in a predictable way under Type I moves. This leads to the following result, which is essentially a statement of the existence of the Jones invariant. Theorem 3.5. Let \( D \) be a diagram of an oriented link \( L \) . Then the expression \[ {\left( -A\right) }^{-{3w}\left( D\right) }\langle D\rangle \] is an invariant of the oriented link \( L \) . Proof. It follows from Lemma 3.3 that the given expression is unchanged by Reidemeister moves of Types II and III; Lemma 3.2 and the above remarks on \( w\left( D\right) \) show it is unchanged by a Type I move. As any two diagrams of two equivalent links are related by a sequence of such moves, the result follows at once. Definition 3.6. The Jones polynomial \( V\left( L\right) \) of an oriented link \( L \) is the Laurent polynomial in \( {t}^{1/2} \), with integer coefficients, defined by \[ V\left( L\right) = {\left( {\left( -A\right) }^{-{3w}\left( D\right) }\langle D\rangle \right) }_{{t}^{1/2} = {A}^{-2}} \in \mathbb{Z}\left\lbrack {{t}^{-1/2},{t}^{1/2}}\right\rbrack , \] where \( D \) is any oriented diagram for \( L \) . Here \( {t}^{1/2} \) is just an indeterminate the square of which is \( t \) . In fact, links with an odd number of components, including knots, have polynomials consisting of only integer powers of \( t \) . It is easy to show, by induction on the number of crossings in a diagram, that the given expression does indeed belong to \( \mathbb{Z}\left\lbrack {{t}^{-1/2},{t}^{1/2}}\right\rbrack \) . Note that by Theorem 3.5, the Jones polynomial invariant is well defined and that \( V \) (unknot) \( = 1 \) . At the time of writing, it is unknown whether there is a nontrivial knot \( K \) with \( V\left( K\right) = 1 \) and finding such a \( K \), or proving none exists, is thought to be an important problem. The following table gives the Jones polynomial of knots with diagrams of at most eight crossings. It does not take very long to calculate such a table directly from the definition. It is clear that if the orientation of every component of a link is changed, then the sign of each crossing does not change. Thus the Jones polynomial of a knot does not depend upon the orientation chosen for the knot. It is easy to check that if the oriented link \( {L}^{ * } \) is obtained from the oriented link \( L \) by reversing the orientation of one component \( K \), then \( V\left( {L}^{ * }\right) = {t}^{-3\operatorname{lk}\left( {K, L - K}\right) }V\left( L\right) \) . Thus the Jones polynomial depends on orientations in a very elementary way. Displayed in Table 3.1 are the coefficients of the Jones polynomials of the knots shown in Chapter 1. A bold entry in the table is a coefficient of \( {t}^{0} \) . For example, \[ V\left( {6}_{1}\right) = {t}^{-4} - {t}^{-3} + {t}^{-2} - 2{t}^{-1} + 2 - t + {t}^{2}. \] The bracket polynomial of a diagram can be regarded as an invariant of framed unoriented links. For the moment, regard a framed link as a link \( L \) with an integer TABLE 3.1. Jones Polynomial Table <table><tr><td>1.40.86.88.9</td><td>49.8</td><td>SS</td><td>物8.8</td><td>231.000.988.008%s88So8L2之乙L乙L</td><td>a99yS少S</td><td></td></tr><tr><td>I上0</td><td>1</td><td></td><td></td><td>上1II1\( - \)\( - \)\( - \)上L10011</td><td>上【I11【L</td><td>iADLC J. I.</td></tr><tr><td>乙I0r</td><td>山s</td><td>出</td><td>山乙</td><td>心乙乙乙乙乙乙1乙上下1S乙乙I01I</td><td>乙心上【I-I</td><td></td></tr><tr><td>乙上09</td><td>Ss</td><td>力</td><td>力山</td><td>力s山S山下SSS乙乙I山山山下I上上</td><td>乙乙1上上I0</td><td>JOHCOTOTYING</td></tr><tr><td>出乙【上</td><td>外9</td><td>少</td><td>sS</td><td>ssSrS力t山山山出乙力力SSL乙I</td><td>s下下乙【上1</td><td></td></tr><tr><td>sL06</td><td>L少</td><td>9</td><td>9s</td><td>SStStttSSSS乙t山山乙乙乙上</td><td>乙乙乙10I0</td><td></td></tr><tr><td>乙乙1山</td><td>少9</td><td>s</td><td>sUn</td><td>stStttrt山山下下SSSS下乙T</td><td>乙上1I</td><td></td></tr><tr><td>乙上09</td><td>Sr</td><td>S</td><td>力t</td><td>t力tS出下Ss乙乙乙乙乙下上下s10</td><td>上II0O</td><td>1.001C</td></tr><tr><td>00扩</td><td>山w</td><td>乙</td><td>下w</td><td>乙了乙乙乙乙上乙L上上1II乙II</td><td>0</td><td></td></tr><tr><td>1【</td><td>I1</td><td>【</td><td>I上</td><td>I1I111I10上【00</td><td></td><td></td></tr><tr><td></td><td></td><td>0</td><td></td><td>0上0</td><td></td><td></td></tr><tr><td></td><td></td><td>0</td><td></td><td>0</td><td></td><td></td></tr></table> assigned to each component. Let \( D \) be a diagram for \( L \) with the property that for each component \( K \) of \( L \), the part of \( D \) corresponding to \( K \) has as its writhe the integer assigned to \( K \) . Then \( \langle D\rangle \) is an invariant of the framed link. Note that any diagram for \( L \) can be adjusted by moves of Type I (or its reflection) to achieve any given framing. The Jones polynomial is characterised by the following proposition, which follows easily from the above definition (though historically it preceded that definition). Proposition 3.7. The Jones polynomial invariant is a function \[ V : \left\{ {\text{ Oriented links in }{S}^{3}}\right\} \rightarrow \mathbb{Z}\left\lbrack {{t}^{-1/2},{t}^{1/2}}\right\rbrack \] such that (i) \( V \) (unknot) \( = 1 \) , (ii) whenever three oriented links \( {L}_{ + },{L}_{ - } \) and \( {L}_{0} \) are the same, except in the neighbourhood of a point where they are as shown in Figure 3.2, then \[ {t}^{-1}V\left( {L}_{ + }\right) - {tV}\left( {L}_{ - }\right) + \left( {{t}^{-1/2} - {t}^{1/2}}\right) V\left( {L}_{0}\right) = 0. \]  Figure 3.2 Proof. \[ \langle X\rangle = A\langle X\rangle + {A}^{-1}\langle X\rangle \] \[ \langle X\rangle = {A}^{-1}\langle X\rangle + A\langle X\rangle . \] Multiplying the first equation by \( A \), the second by \( {A}^{-1} \), and subtracting gives \[ A\langle > < \rangle - {A}^{-1}\langle > < \rangle = \left( {{A}^{2} - {A}^{-2}}\right) \langle \rangle (\rangle . \] Thus, for the oriented links with diagrams as shown, using the fact that in those diagrams \( w\left( {L}_{ + }\right) - 1 = w\left( {L}_{0}\right) = w\left( {L}_{ - }\right) + 1 \), it follows that \[ - {A}^{4}V\left( {L}_{ + }\right) + {A}^{-4}V\left( {L}_{ - }\right) = \left( {{A}^{2} - {A}^{-2}}\right) V\left( {L}_{0}\right) . \] The substitution \( {t}^{1/2} = {A}^{-2} \) gives the required answer. Working from Proposition 3.7, a straightforward exercise shows that if \( {L}^{\prime } \) is \( L \) together with an additional trivial (unknotted, unlinking) component, then its Jones polynomial is given by \( V\left( {L}^{\prime }\right) = \left( {-{t}^{-1/2} - {t}^{1/2}}\right) V\left( L\right) \) . Proposition 3.7 characterises the invariant in that using it allows the Jones polynomial of any oriented link to be calculated. This follows from the fact that any link can be changed to an unlink of \( c \) unknots (for which the Jones polynomial is \( \left( {-{t}^{-1/2} - }\right. \) \( {t}^{1/2}{)}^{c - 1} \) ) by changing crossings in some diagram; formula (ii) of Proposition 3.7 relates the polynomials before and after such a change with the that of a link diagram with fewer crossings (which has a known polynomial by induction). The Jones polynomial of the sum of two knots is just the product of their Jones polynomials, that is, \[ V\left( {{K}_{1} + {K}_{2}}\right) = V\left( {K}_{1}\right) V\left( {K}_{2}\right) . \] This follows at once by considering a calculation of the polynomial of \( {K}_{1} + {K}_{2} \) and operating firstly on the crossings of just one summand. The same formula is true for links, but the sum of two links is not well defined; the result depends on which two components are fused together in the summing operation. That fact can easily be used, in a straightforward exercise, to produce two distinct links with the same Jones polynomial. If an oriented link has a diagram \( D \), its reflection has \( \bar{D} \) as a diagram; of course, \( w\left( D\right) = - w\left( \bar{D}\right) \) . As \( \langle \bar{D}\rangle = \overline{\langle D\rangle } \), this means that if \( \bar{L} \) is the reflection of the oriented link \( L \), then \( V\left( \bar{L}\right) \) is obtained from \( L \) by interchanging \( {t}^{-1/2} \) and \( {t}^{1/2} \) . The bracket polynomial of a diagram, of writhe equal to 3 , for the right-handed trefoil knot \( {3}_{1} \) has already been calculated, and that at once determines that \( - {t}^{4} + {t}^{3} + t \) is the Jones polynomial of the right-hand trefoil knot. Thus its reflection, the left-hand trefoil knot, has Jon

1116_(GTM270)Fundamentals of Algebraic Topology

Definition 6.2.25

Definition 6.2.25. Let \( \left\{ {\bar{\varphi }}_{y}\right\} \) be an orientation of int \( \left( M\right) \), i.e., a compatible system of local orientations. The induced orientation of \( \partial M \) is the compatible system of local orientations \( \left\{ {\bar{\varphi }}_{x}\right\} \) obtained as follows: \( {\bar{\varphi }}_{x} \) is the composition of isomorphisms and their inverses \[ \mathbb{Z}\xrightarrow[]{{\bar{\varphi }}_{y}}{H}_{n}\left( {M, M - y}\right) \prec \;{H}_{n}\left( {{\varphi }_{\alpha }\left( B\right) ,{\varphi }_{\alpha }\left( B\right) - y}\right) \overset{{\left( {\varphi }_{\alpha }\right) }_{ * }}{ \prec }{H}_{n}\left( {B, B - q}\right) \] \[ {H}_{n - 1}\left( {\partial M,\partial M - x}\right) < \rightharpoonup {H}_{n - 1}\left( {{\varphi }_{\alpha }\left( C\right) ,{\varphi }_{\alpha }\left( C\right) - x}\right) \overset{{\left( {\varphi }_{\alpha }\right) }_{ * }}{ < }{H}_{n - 1}\left( {C, C - p}\right) \] Here \( {\varphi }_{\alpha } : {\mathbb{R}}_{ + }^{n} \rightarrow {U}_{\alpha } \subseteq M \) is a coordinate patch with \( {\varphi }_{\alpha }\left( p\right) = x \) and \( {\varphi }_{\alpha }\left( q\right) = y \) . The maps labelled \( {\left( {\varphi }_{\alpha }\right) }_{ * } \) are both restrictions of \( {\varphi }_{\alpha } \) to the respective domains, and the unlabelled maps are excision isomorphisms. Theorem 6.2.26. Let \( M \) be an oriented manifold with boundary. Then \( \partial M \) has a well-defined induced orientation given by the construction in Definition 6.2.25. In particular, \( \partial M \) is orientable. Proof. This is simply a matter of checking that the local orientations \( \left\{ {\bar{\varphi }}_{x}\right\} \) are indeed compatible, and that they are independent of the choice of coordinate patches \( \left( {{U}_{\alpha },{\varphi }_{\alpha }}\right) \) used in the construction. Remark 6.2.27. If \( M \) is not orientable, then \( \partial M \) may or may not be orientable (i.e., both possibilities may arise). In practice, we often also want to consider (co)homology with coefficients in a field. In this regard we have the following result. Lemma 6.2.28. Let \( G = \mathbb{F} \) be a field of characteristic 0 or odd characteristic. Then a manifold \( M \) is \( G \) -orientable if and only if it is orientable. If \( G = \mathbb{F} \) is a field of characteristic 2, then every manifold \( M \) is \( G \) -orientable. Proof. We do the more interesting case of a field \( \mathbb{F} \) of characteristic \( \neq 2 \) . Consider the diagram in Definition 6.2.3. If we let \( V = {\varphi }_{x}\left( D\right) \) and replace \( G \) in that diagram by \( {H}_{n}\left( {M, M - V;G}\right) \) and the two vertical maps by the isomorphisms induced by inclusions, we obtain a commutative diagram. This is true whether we use \( \mathbb{F} \) coefficients or \( \mathbb{Z} \) coefficients. But we also have the commutative diagram with the horizontal maps induced by the map \( \mathbb{Z} \rightarrow \mathbb{F} \) of coefficients  Now if \( M \) is \( \mathbb{Z} \) -orientable it has a compatible collection of local \( \mathbb{Z} \) -orientations \( \left\{ {\bar{\varphi }}_{x}\right\} \) and then \( \left\{ {{\bar{\varphi }}_{x} \otimes 1}\right\} \) is a compatible collection of local \( \mathbb{F} \) -orientations. On the other hand, suppose we have a compatible collection \( \left\{ {\bar{\psi }}_{x}\right\} \) of local \( \mathbb{F} \) - orientations. Fix a point \( x \in M \) . Then \( {\bar{\psi }}_{x}\left( 1\right) \in {H}_{n}\left( {M, M - x;\mathbb{F}}\right) \) is a generator, i.e., a nonzero element. This element may not be in the image of \( {H}_{n}\left( {M, M - x;\mathbb{Z}}\right) \) . But there is a nonzero element \( f \) of \( \mathbb{F} \) (in fact, exactly two such) such that \( f{\bar{\psi }}_{x}\left( 1\right) \) is the image in \( {H}_{n}\left( {M, M - x;\mathbb{F}}\right) \) of a generator of \( {H}_{n}\left( {M, M - x;\mathbb{Z}}\right) \) . By the commutativity of the above diagram that implies the same is true for \( f{\bar{\psi }}_{y}\left( 1\right) \) for every \( y \in M \) . Hence \( \left\{ {f{\bar{\psi }}_{x}}\right\} \) is a compatible system of local \( \mathbb{Z} \) -orientations of \( M \), where \( x \) varies over \( M \) . The proof of this lemma also shows how to obtain \( \mathbb{F} \) -orientations. Definition 6.2.29. Let \( M \) be orientable and let \( \mathbb{F} \) be an arbitrary field. An \( \mathbb{F} \) -orientation of \( M \) is a compatible system of local \( \mathbb{F} \) -orientations of the form \( \left\{ {{\bar{\varphi }}_{x} \otimes 1}\right\} \) where \( \left\{ {\bar{\varphi }}_{x}\right\} \) is a compatible system of local \( \mathbb{Z} \) -orientations of \( M \) . Let \( M \) be arbitrary and let \( \mathbb{F} \) be a field of characteristic 2. An \( \mathbb{F} \) -orientation of \( M \) is a compatible system of local \( \mathbb{F} \) -orientations of the form \( \left\{ {{\bar{\varphi }}_{x} \otimes 1}\right\} \) where \( \left\{ {\bar{\varphi }}_{x}\right\} \) is a compatible system of local \( \mathbb{Z}/2\mathbb{Z} \) -orientations of \( M \) . (In the characteristic 2 case compatibility is automatic.) It is easy to check that the two parts of this definition agree when \( M \) is orientable and \( \mathbb{F} \) has characteristic 2. Orientability has very important homological implications, given by the following theorem. We state this theorem for manifolds with boundary, which includes the case of manifolds by taking \( \partial M = \varnothing \) . The hypothesis that \( M \) be connected is not essentially restrictive, as otherwise we could consider each component of \( M \) separately. Theorem 6.2.30. Let \( M \) be a compact connected \( n \) -manifold with boundary. Let \( G = \mathbb{Z}/2\mathbb{Z} \) or \( \mathbb{Z} \) . If \( M \) is G-oriented, suppose that \( \left\{ {\bar{\varphi }}_{x}\right\} \) is a compatible system of local \( G \) -orientations giving the \( G \) -orientation of \( M \) . In this case, there is a unique homology class \( \left\lbrack {M,\partial M}\right\rbrack \in {H}_{n}\left( {M,\partial M;G}\right) \) with \( {i}_{ * }\left( \left\lbrack {M,\partial M}\right\rbrack \right) = {\varphi }_{x}\left( 1\right) \in {H}_{n}(M, M - \) \( x;G) \) for every \( x \in M \), where \( i : \left( {M,\partial M}\right) \rightarrow \left( {M, M - x}\right) \) is the inclusion of pairs. Furthermore, \( \left\lbrack {M,\partial M}\right\rbrack \) is a generator of \( {H}_{n}\left( {M,\partial M;G}\right) \) . If \( M \) is not \( G \) -orientable, then \( {H}_{n}\left( {M,\partial M;G}\right) = 0 \) . Since this theorem is so important, we will explicitly state one of its immediate consequences. Corollary 6.2.31. Let \( M \) be a compact connected \( n \) -manifold with boundary. (1) For any such \( M,{H}_{n}\left( {M,\partial M;\mathbb{Z}/2\mathbb{Z}}\right) \cong \mathbb{Z}/2\mathbb{Z} \) and \( {H}^{n}\left( {M,\partial M;\mathbb{Z}/2\mathbb{Z}}\right) \cong \mathbb{Z}/2\mathbb{Z} \) . (2) If \( M \) is orientable, then \( {H}_{n}\left( {M,\partial M;\mathbb{Z}}\right) \cong \mathbb{Z} \) and \( {H}^{n}\left( {M,\partial M;\mathbb{Z}}\right) \cong \mathbb{Z} \) . If \( M \) is not orientable, then \( {H}_{n}\left( {M,\partial M;\mathbb{Z}}\right) = 0 \) and \( {H}^{n}\left( {M,\partial M;\mathbb{Z}}\right) = 0 \) . Proof. The statements on homology are a direct consequence of Theorems 6.2.6 and 6.2.30. The statements for cohomology then follow from the universal coefficient theorem and Theorem 6.1.12. Example 6.2.32. We computed the homology of \( \mathbb{R}{P}^{n} \) in Theorem 4.3.4. Combining that result with Corollary 6.2.31, we see that \( \mathbb{R}{P}^{n} \) is orientable for \( n \) odd and nonorientable for \( n \) even. Definition 6.2.33. Let \( M \) be a compact connected \( G \) -oriented \( n \) -manifold, \( G = \mathbb{Z}/2\mathbb{Z} \) or \( \mathbb{Z} \) . The homology class \( \left\lbrack {M,\partial M}\right\rbrack \in {H}_{n}\left( {M,\partial M;G}\right) \) as in Theorem 6.2.30 is called the fundamental homology class (or simply fundamental class) of \( \left( {M,\partial M}\right) \) . Its dual \( \{ M,\partial M\} \) in \( {H}^{n}\left( {M,\partial M;G}\right) \), i.e., the cohomology class with \( e\left( {\{ M,\partial M\} ,\left\lbrack {M,\partial M}\right\rbrack }\right) = 1 \), is the fundamental cohomology class of \( \left( {M,\partial M}\right) \) . If \( M \) is oriented and \( G \) is any coefficient group, the image of \( \left\lbrack {M,\partial M}\right\rbrack \) in \( {H}_{n}\left( {M,\partial M;G}\right) \) under the coefficient map \( \mathbb{Z} \rightarrow G \) is also called a fundamental homology class, and similarly the image of \( \{ M,\partial M\} \) on \( {H}^{n}\left( {M,\partial M;G}\right) \) under the same coefficient map is also called a fundamental cohomology class. Remark 6.2.34. We need to be careful in Definition 6.2.33 when we referred to the dual of a homology class. In general, if \( V \) is a free abelian group (or a vector space) if does not make sense to speak of the dual of an element \( v \) of \( V \) . But it does make sense here. Suppose that \( M \) is connected. Then \( {H}_{n}\left( {M;G}\right) \) is free of rank 1 (in case \( M \) is orientable and \( G = \mathbb{Z} \) ) or is a 1-dimensional vector space over \( \mathbb{Z}/2\mathbb{Z} \) (for \( M \) arbitrary and \( G = \mathbb{Z}/2\mathbb{Z} \) ) and we have the pairing \( e : {H}^{n}\left( {M;G}\right) \otimes {H}_{n}\left( {M;G}\right) \rightarrow G \) . In this situation, given a generator \( v \) of \( {H}_{n}\left( {M;G}\right) \), there is a unique element (also a generator) \( {v}^{ * } \) in \( {H}^{n}\left( {M;G}\right) \) with \( e\left( {{v}^{ * }, v}\right) = 1 \), and \( {v}^{ * } \) is w

1088_(GTM245)Complex Analysis

Definition 9.10

Definition 9.10. A harmonic conjugate of a real-valued harmonic function \( u \) is any real-valued function \( v \) such that \( u + {uv} \) is holomorphic. Harmonic conjugates always exist locally, and globally on simply connected domains. They are unique up to additive real constants. In fact, it is easy to see that they are given locally as follows. Proposition 9.11. If \( g \) is harmonic and real-valued in \( \left| z\right| < \rho \) for some \( \rho > 0 \), then the harmonic conjugate of \( g \) vanishing at the origin is given by \[ \frac{1}{{2\pi }\imath }{\int }_{0}^{2\pi }g\left( {r{\mathrm{e}}^{\iota \theta }}\right) \cdot \frac{r{\mathrm{e}}^{-{\iota \theta }}z - r{\mathrm{e}}^{\iota \theta }\bar{z}}{{\left| r{\mathrm{e}}^{\iota \theta } - z\right| }^{2}}\mathrm{\;d}\theta ,\;\text{ for }\left| z\right| < r < \rho . \] The following result is interesting and useful. Theorem 9.12 (Harnack’s Inequalities). If \( g \) is a positive harmonic function on \( \left| z\right| < r \) that is continuous on \( \left| z\right| \leq r \), then \[ \frac{r - \left| z\right| }{r + \left| z\right| } \cdot g\left( 0\right) \leq g\left( z\right) \leq \frac{r + \left| z\right| }{r - \left| z\right| } \cdot g\left( 0\right) ,\text{ for all }\left| z\right| < r. \] Proof. Our starting point is (9.3). We use elementary estimates for the Poisson kernel: \[ \frac{r - \left| z\right| }{r + \left| z\right| } = \frac{{r}^{2} - {\left| z\right| }^{2}}{{\left( r + \left| z\right| \right) }^{2}} \leq \frac{{r}^{2} - {\left| z\right| }^{2}}{{\left| r{\mathrm{e}}^{t\theta } - z\right| }^{2}} \leq \frac{{r}^{2} - {\left| z\right| }^{2}}{{\left( r - \left| z\right| \right) }^{2}} = \frac{r + \left| z\right| }{r - \left| z\right| }. \] Multiplying these inequalities by the positive number \( g\left( w\right) = g\left( {r{\mathrm{e}}^{t\theta }}\right) \) and then averaging the resulting function over the circle \( \left| w\right| = r \), we obtain \[ \frac{r - \left| z\right| }{r + \left| z\right| } \cdot \frac{1}{2\pi }{\int }_{0}^{2\pi }g\left( {r{\mathrm{e}}^{t\theta }}\right) \mathrm{d}\theta \leq \frac{1}{2\pi }{\int }_{0}^{2\pi }g\left( {r{\mathrm{e}}^{t\theta }}\right) \cdot \frac{{r}^{2} - {\left| z\right| }^{2}}{{\left| r{\mathrm{e}}^{t\theta } - z\right| }^{2}}\mathrm{\;d}\theta \] \[ \leq \frac{r + \left| z\right| }{r - \left| z\right| } \cdot \frac{1}{2\pi }{\int }_{0}^{2\pi }g\left( {r{\mathrm{e}}^{\iota \theta }}\right) \mathrm{d}\theta . \] The middle term in the above inequalities is \( g\left( z\right) \) as a consequence of (9.3), while the extreme averages are equal to \( g\left( 0\right) \) by the MVP. Remark 9.13. Exercise 9.6 gives a remarkable consequence of Harnack's inequalities that we use in establishing our next result. Theorem 9.14 (Harnack's Convergence Theorem). Let \( D \) be a domain and let \( \left\{ {u}_{j}\right\} \) be a nondecreasing sequence of real-valued harmonic functions on \( D \) . Then (a) Either \( \mathop{\lim }\limits_{{j \rightarrow \infty }}{u}_{j}\left( z\right) = + \infty \) for all \( z \in D \) (b) The function on \( D \) defined by \( U\left( z\right) = \mathop{\lim }\limits_{{j \rightarrow \infty }}{u}_{j}\left( z\right) \) is harmonic in \( D \) . Proof. Since a nondecreasing sequence of real numbers converges if and only if it is bounded, the assumption that \( \mathop{\lim }\limits_{{j \rightarrow \infty }}{u}_{j}\left( z\right) \) is not \( + \infty \) for all \( z \in D \) allows us to conclude that there exist \( {z}_{0} \) in \( D \) and a real number \( M \) such that \( {u}_{j}\left( {z}_{0}\right) < M \) for all \( j \) . Then \( \mathop{\lim }\limits_{{j \rightarrow \infty }}{u}_{j}\left( {z}_{0}\right) \) exists, and it equals the value of the series \[ {u}_{1}\left( {z}_{0}\right) + \mathop{\sum }\limits_{{n = 1}}^{\infty }\left\lbrack {{u}_{n + 1}\left( {z}_{0}\right) - {u}_{n}\left( {z}_{0}\right) }\right\rbrack \] which is therefore convergent. Let \( K \) denote a compact subset of \( D \) . By enlarging \( K \) if necessary, we may assume that \( {z}_{0} \in K \) . It follows from Harnack’s inequalities (see Exercise 9.6) that there exists a real constant \( c \) such that \[ 0 \leq {u}_{n + 1}\left( z\right) - {u}_{n}\left( z\right) \leq c\left\lbrack {{u}_{n + 1}\left( {z}_{0}\right) - {u}_{n}\left( {z}_{0}\right) }\right\rbrack \] for all \( z \) in \( K \) and all \( n \) in \( \mathbb{N} \) . It follows immediately that the series \( {u}_{1}\left( z\right) + \mathop{\sum }\limits_{{n = 1}}^{\infty }\left\lbrack {{u}_{n + 1}\left( z\right) - {u}_{n}\left( z\right) }\right\rbrack \) converges uniformly on \( K \) ; that is, \( {u}_{j} \) converges uniformly to a function \( U \) on compact subsets of \( D \) . It is now easy to show that \( U \) is harmonic in \( D \) . ## 9.3 The Dirichlet Problem Let \( D \) be a bounded region in \( \mathbb{C} \) and let \( f \in \mathbf{C}\left( {\partial D}\right) \) . The Dirichlet problem is to find a continuous function \( u \) defined on the closure of \( D \) that agrees with \( f \) on the boundary of \( D \) and whose restriction to \( D \) is harmonic. We will consider, for the moment, only the special case where \( D \) is a disc; without loss of generality we may assume that the disc has radius one and center at zero. For a piecewise continuous function \( u \) on \( {S}^{1} \) and \( z \in \mathbb{C} \) with \( \left| z\right| < 1 \), we define (compare with (9.3)) \[ P\left\lbrack u\right\rbrack \left( z\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }u\left( {\mathrm{e}}^{\iota \theta }\right) \cdot \Re \left( \frac{{\mathrm{e}}^{\iota \theta } + z}{{\mathrm{e}}^{\iota \theta } - z}\right) \mathrm{d}\theta \] (9.7) or, equivalently, \[ P\left\lbrack u\right\rbrack \left( z\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }u\left( {\mathrm{e}}^{\iota \theta }\right) \cdot \frac{1 - {\left| z\right| }^{2}}{{\left| {\mathrm{e}}^{\iota \theta } - z\right| }^{2}}\mathrm{\;d}\theta . \] (9.8)  Fig. \( {9.1}{u}_{1} \) and \( {u}_{2} \) . (a) The function \( {u}_{1} \) . (b) The function \( {u}_{2} \) . ## The following properties of the operator \( P \) are easily established: 1. \( P\left\lbrack u\right\rbrack \) is a well-defined function on the open unit disc. Hence we view \( P \) as an operator that assigns the function \( P\left\lbrack u\right\rbrack \), on the open unit disc to each piecewise continuous function \( u \) on the unit circle. 2. \( P\left\lbrack {u + v}\right\rbrack = P\left\lbrack u\right\rbrack + P\left\lbrack v\right\rbrack \) and \( P\left\lbrack {cu}\right\rbrack = c \cdot {Pu} \) for all piecewise continuous functions \( u \) and \( v \) on \( {S}^{1} \) and every constant \( c \) (thus \( P \) is a linear operator). 3. If \( u \) is a real nonnegative piecewise continuous function on \( {S}^{1} \), then \( P\left\lbrack u\right\rbrack \) is a real-valued nonnegative function on the open unit disc. 4. \( P\left\lbrack u\right\rbrack \) is harmonic in the open unit disc. To establish this claim we may assume (by linearity of the operator \( P \) ) that \( u \) is real-valued. In this case, \( P\left\lbrack u\right\rbrack \) is obviously the real part of an analytic function on the disc. 5. For all constants \( c \) , \[ P\left\lbrack c\right\rbrack = c, \] as follows from (9.6) (or directly because constant functions are harmonic). 6. Properties 5 and 3 imply that any bound on \( u \) yields the same bound on \( {Pu} \) . For example, for real-valued function \( u \) satisfying \( m \leq u \leq M \) for some real constants \( m \) and \( M \), we have \( m \leq P\left\lbrack u\right\rbrack \leq M \) . We now establish the solvability of the Dirichlet problem for discs. Theorem 9.15 (H. A. Schwarz). If \( u \) is a piecewise continuous function on the unit circle \( {S}^{1} \), then the function \( P\left\lbrack u\right\rbrack \) is harmonic on \( \{ \left| z\right| < 1\} \) ; furthermore, for \( {\theta }_{0} \in \mathbb{R} \), its limit as \( z \) approaches \( {\mathrm{e}}^{\iota {\theta }_{0}} \) is \( u\left( {\mathrm{e}}^{\iota {\theta }_{0}}\right) \) provided \( u \) is continuous at \( {\mathrm{e}}^{\iota {\theta }_{0}} \) . In particular, the Dirichlet problem is solvable for discs. Proof. We only have to study the boundary values for \( P\left\lbrack u\right\rbrack \) . Let \( {C}_{1} \) and \( {C}_{2} \) be complementary arcs on the unit circle. Let \( {u}_{1} \) be the function which coincides with \( u \) on \( {C}_{1} \) and vanishes on \( {C}_{2} \) ; let \( {u}_{2} \) be the corresponding function for \( {C}_{2} \) (see Fig. 9.1). Clearly \( P\left\lbrack u\right\rbrack = P\left\lbrack {u}_{1}\right\rbrack + P\left\lbrack {u}_{2}\right\rbrack \) . The function \( P\left\lbrack {u}_{1}\right\rbrack \) can be regarded as an integral over the arc \( {C}_{1} \) ; hence it is harmonic on \( \mathbb{C} - {C}_{1} \) . The expression \[ \Re \left( \frac{{\mathrm{e}}^{i\theta } + z}{{\mathrm{e}}^{i\theta } - z}\right) = \frac{1 - {\left| z\right| }^{2}}{{\left| {\mathrm{e}}^{i\theta } - z\right| }^{2}} \] vanishes on \( \left| z\right| = 1 \) for \( z \neq {\mathrm{e}}^{\iota \theta } \) . It follows that \( P\left\lbrack {u}_{1}\right\rbrack \) is zero on the one-dimensional interior of the arc \( {C}_{2} \) . By continuity \( P\left\lbrack {u}_{1}\right\rbrack \left( z\right) \) approaches zero as \( z \) approaches a point in the interior of \( {C}_{2} \) . In proving that \( P\left\lbrack u\right\rbrack \) has limit \( u\left( {\mathrm{e}}^{\iota {\theta }_{0}}\right) \) at \( {\mathrm{e}}^{\iota {\theta }_{0}} \), we may assume that \( u\left( {\mathrm{e}}^{\iota {\theta }_{0}}\right) = 0 \) (if not replace \( u \) by \( u - u\left( {\mathrm{e}}^{\iota {\theta }_{0}}\right) \) ). Under this assumption, given an \( \epsilon > 0

1042_(GTM203)The Symmetric Group

Definition 4.4.1

Definition 4.4.1 Given a partition \( \lambda \), the associated Schur function is \[ {s}_{\lambda }\left( \mathbf{x}\right) = \mathop{\sum }\limits_{T}{\mathbf{x}}^{T} \] where the sum is over all semistandard \( \lambda \) -tableaux \( T \) . By way of illustration, if \( \lambda = \left( {2,1}\right) \), then some of the possible tableaux are \[ T : \begin{array}{l} 1 \\ 2 \end{array},\begin{array}{l} 1 \\ 2 \end{array},\begin{array}{l} 1 \\ 2 \end{array},\begin{array}{l} 1 \\ 3 \end{array},\begin{array}{l} 1 \\ 3 \end{array},\begin{array}{l} 1 \\ 3 \end{array},\ldots \begin{array}{l} 1 \\ 3 \end{array},\begin{array}{l} 1 \\ 2 \end{array},\begin{array}{l} 1 \\ 2 \end{array},\begin{array}{l} 1 \\ 2 \end{array},\begin{array}{l} 1 \\ 4 \end{array},\ldots , \] so \[ {s}_{\left( 2,1\right) }\left( \mathbf{x}\right) = {x}_{1}^{2}{x}_{2} + {x}_{1}{x}_{2}^{2} + {x}_{1}^{2}{x}_{3} + {x}_{1}{x}_{3}^{2} + \cdots + 2{x}_{1}{x}_{2}{x}_{3} + 2{x}_{1}{x}_{2}{x}_{4} + \cdots . \] Note that if \( \lambda = \left( n\right) \), then a one-rowed tableau is just a weakly increasing sequence of \( n \) positive integers, i.e., a partition with \( n \) parts (written backward), so \[ {s}_{\left( n\right) }\left( \mathbf{x}\right) = {h}_{n}\left( \mathbf{x}\right) \] (4.9) If we have only one column, then the entries must increase from top to bottom, so the partition must have distinct parts and thus \[ {s}_{\left( {1}^{n}\right) } = {e}_{n}\left( \mathbf{x}\right) \] (4.10) Finally, if \( \lambda \vdash n \) is arbitrary, then \[ \left\lbrack {{x}_{1}{x}_{2}\cdots {x}_{n}}\right\rbrack {s}_{\lambda }\left( \mathbf{x}\right) = {f}^{\lambda } \] since pulling out this coefficient merely considers the standard tableaux. Before we can show that the \( {s}_{\lambda } \) are a basis for \( {\Lambda }^{n} \), we must verify that they are indeed symmetric functions. We give two proofs of this fact, one based on our results from representation theory and one combinatorial (the latter being due to Knuth [Knu 70]). Proposition 4.4.2 The function \( {s}_{\lambda }\left( \mathbf{x}\right) \) is symmetric. Proof 1. By definition of the Schur functions and Kostka numbers, \[ {s}_{\lambda } = \mathop{\sum }\limits_{\mu }{K}_{\lambda \mu }{\mathbf{x}}^{\mu } \] (4.11) where the sum is over all compositions \( \mu \) of \( n \) . Thus it is enough to show that \[ {K}_{\lambda \mu } = {K}_{\lambda \widetilde{\mu }} \] (4.12) for any rearrangement \( \widetilde{\mu } \) of \( \mu \) . But in this case \( {M}^{\mu } \) and \( {M}^{\widetilde{\mu }} \) are isomorphic modules. Thus they have the same decomposition into irreducibles, and (4.12) follows from Young's rule (Theorem 2.11.2). Proof 2. It suffices to show that \[ \left( {i, i + 1}\right) {s}_{\lambda }\left( \mathbf{x}\right) = {s}_{\lambda }\left( \mathbf{x}\right) \] for each adjacent transposition. To this end, we describe an involution on semistandard \( \lambda \) -tableaux \[ T \rightarrow {T}^{\prime } \] such that the numbers of \( i \) ’s and \( \left( {i + 1}\right) \) ’s arc exchanged when passing from \( T \) to \( {T}^{\prime } \) (with all other multiplicities staying the same). Given \( T \), each column contains either an \( i, i + 1 \) pair; exactly one of \( i, i + 1 \) ; or neither. Call the pairs fixed and all other occurrences of \( i \) or \( i + 1 \) free. In each row switch the number of free \( i \) ’s and \( \left( {i + 1}\right) \) ’s; i.e., if the the row consists of \( k \) free \( i \) ’s followed by \( l \) free \( \left( {i + 1}\right) \) ’s then replace them by \( l \) free \( i \) ’s followed by \( k \) free \( \left( {i + 1}\right) \) ’s. To illustrate, if \( i = 2 \) and \[ T = \begin{array}{llllllllll} 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 3 \\ 2 & 2 & 3 & 3 & 3 & 3 & & & & \end{array}, \] then the twos and threes in columns 2 through 4 and 7 through 10 are free. So \[ {T}^{\prime } = \begin{array}{llllllllll} 1 & 1 & 1 & 1 & 2 & 2 & 2 & 3 & 3 & 3 \\ 2 & 2 & 2 & 3 & 3 & 3 & & & & \end{array}. \] The new tableau \( {T}^{\prime } \) is still semistandard by the definition of free. Since the fixed \( i \) ’s and \( \left( {i + 1}\right) \) ’s come in pairs, this map has the desired exchange property. It is also clearly an involution. - Using the ideas in the proof of Theorem 4.3.7, part 1, the following result guarantees that the \( {s}_{\lambda } \) are a basis. Proposition 4.4.3 We have \[ {s}_{\lambda } = \mathop{\sum }\limits_{{\mu \leq \lambda }}{K}_{\lambda \mu }{m}_{\mu } \] where the sum is over partitions \( \mu \) (rather than compositions) and \( {K}_{\lambda \lambda } = 1 \) . Proof. By equation (4.11) and the symmetry of the Schur functions, we have \[ {s}_{\lambda } = \mathop{\sum }\limits_{\mu }{K}_{\lambda \mu }{m}_{\mu } \] where the sum is over all partitions \( \mu \) . We can prove that \[ {K}_{\lambda \mu } = \left\{ \begin{array}{ll} 0 & \text{ if }\lambda \ntrianglerighteq \mu \\ 1 & \text{ if }\lambda = \mu \end{array}\right. \] in two different ways. One is to appeal again to Young's rule and Corollary 2.4.7. The other is combinatorial. If \( {K}_{\lambda \mu } \neq 0 \), then consider a \( \lambda \) -tableau \( T \) of content \( \mu \) . Since \( T \) is column-strict, all occurrences of the numbers \( 1,2,\ldots, i \) are in rows 1 through \( i \) . This implies that for all \( i \) , \[ {\mu }_{1} + {\mu }_{2} + \cdots + {\mu }_{i} \leq {\lambda }_{1} + {\lambda }_{2} + \cdots + {\lambda }_{i} \] i.e., \( \mu \trianglelefteq \lambda \) . Furthermore, if \( \lambda = \mu \), then by the same reasoning there is only one tableau of shape and content \( \lambda \), namely, the one where row \( i \) contains all occurrences of \( i \) . (Some authors call this tableau superstandard.) - Corollary 4.4.4 The set \( \left\{ {{s}_{\lambda } : \lambda \vdash n}\right\} \) is a basis for \( {\Lambda }^{n} \) . ∎ ## 4.5 The Jacobi-Trudi Determinants The determinantal formula (Theorem 3.11.1) calculated the number of standard tableaux, \( {f}^{\lambda } \) . Analogously, the Jacobi-Trudi determinants provide another expression for \( {s}_{\lambda } \) in terms of elementary and complete symmetric functions. Jacobi [Jac 41] was the first to obtain this result, and his student Trudi [Tru 64] subsequently simplified it. We have already seen the special \( 1 \times 1 \) case of these determinants in equations (4.9) and (4.10). The general result is as follows. Any symmetric function with a negative subscript is defined to be zero. Theorem 4.5.1 (Jacobi-Trudi Determinants) Let \( \lambda = \left( {{\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{l}}\right) \) . We have \[ {s}_{\lambda } = \left| {h}_{{\lambda }_{i} - i + j}\right| \] and \[ {s}_{{\lambda }^{\prime }} = \left| {e}_{{\lambda }_{i} - i + j}\right| \] where \( {\lambda }^{\prime } \) is the conjugate of \( \lambda \) and both determinants are \( l \times l \) . Proof. We prove this theorem using a method of Lindström [Lin 73] that was independently discovered and exploited by Gessel [Ges um] and Gessel-Viennot [G-V 85, G-V ip]. (See also Karlin [Kar 88].) The crucial insight is that one can view both tableaux and determinants as lattice paths. Consider the plane \( \mathbb{Z} \times \mathbb{Z} \) of integer lattice points. We consider (possibly infinite) paths in this plane \[ p = {s}_{1},{s}_{2},{s}_{3},\ldots \] where each step \( {s}_{i} \) is of unit length northward \( \left( N\right) \) or eastward \( \left( E\right) \) . Such a path is shown in the following figure.  Label the eastward steps of \( p \) using one of two labelings. The e-labeling assigns to each eastward \( {s}_{i} \) the label \[ L\left( {s}_{i}\right) = i \] The \( h \) -labeling gives \( {s}_{i} \) the label \[ \check{L}\left( {s}_{i}\right) = \text{(the number of northward}{s}_{j}\text{preceding}{s}_{i}\text{) +1 .} \] Intuitively, in the \( h \) -labeling all the eastward steps on the line through the origin of \( p \) are labeled 1, all those on the line one unit above are labeled 2, and so on. Labeling our example path with each of the two possibilities yields the next pair of diagrams.  \( h \) -labeling It is convenient to extend \( \mathbb{Z} \times \mathbb{Z} \) by the addition of some points at infinity. Specifically, for each \( x \in \mathbb{Z} \), add a point \( \left( {x,\infty }\right) \) above every point on the vertical line with coordinate \( x \) . We assume that a path can reach \( \left( {x,\infty }\right) \) only by ending with an infinite number of consecutive northward steps along this line. If \( p \) starts at a vertex \( u \) and ends at a vertex \( v \) (which may be a point at infinity), then we write \( u\overset{p}{ \rightarrow }v \) . There are two weightings of paths corresponding to the two labelings. If \( p \) has only a finite number of eastward steps, define \[ {\mathbf{x}}^{p} = \mathop{\prod }\limits_{{{s}_{i} \in p}}{x}_{L\left( {s}_{i}\right) } \] and \[ {\check{\mathbf{x}}}^{p} = \mathop{\prod }\limits_{{{s}_{i} \in p}}{x}_{\check{L}\left( {s}_{i}\right) } \] where each product is taken over the eastward \( {s}_{i} \) in \( p \) . Note that \( {\mathbf{x}}^{p} \) is always square-free and \( {\check{\mathbf{x}}}^{p} \) can be any monomial. So we have \[ {e}_{n}\left( \mathbf{x}\right) = \mathop{\sum }\limits_{p}{\mathbf{x}}^{p} \] and \[ {h}_{n}\left( \mathbf{x}\right) = \mathop{\sum }\limits_{p}{\check{\mathbf{x}}}^{p} \] where both sums are over all paths \( \left( {a, b}\right) \overset{p}{ \rightarrow }\left( {a + n,\infty }\right) \) for any fixed initial vertex \( \left( {a, b}\right) \) . Just as all paths between one pair of points describes a lone elementary or complete symmetric

1074_(GTM232)An Introduction to Number Theory

Definition 2.2

Definition 2.2. A commutative ring \( R \) is Euclidean if there is a function \[ N : R \smallsetminus \{ 0\} \rightarrow \mathbb{N} \] with the following properties: (1) \( N\left( {ab}\right) = N\left( a\right) N\left( b\right) \) for all \( a, b \in R \), and (2) for all \( a, b \in R \), if \( b \neq 0 \), then there exist \( q, r \in R \) such that \[ a = {bq} + r\text{ and }r = 0\text{ or }N\left( r\right) < N\left( b\right) . \] Such a function is called a norm on \( R \) . Much of what follows can be done with weaker conditions. In particular, one does not need such a strong property as (1). However, in many cases, the norm does have this property, so we assume it to allow a speedier and more natural development of the argument. Example 2.3. The following are examples of Euclidean rings. (1) Let \( R = \mathbb{Z}\left\lbrack \mathrm{i}\right\rbrack \) denote the Gaussian integers, so \[ R = \{ x + \mathrm{i}y \mid x, y \in \mathbb{Z}\} \] where \( {\mathrm{i}}^{2} = - 1 \) . Setting \( N\left( {x + \mathrm{i}y}\right) = {x}^{2} + {y}^{2} \) shows that \( R \) is a Euclidean ring. (2) Let \( \mathbb{F} \) denote any field and let \( R = \mathbb{F}\left\lbrack x\right\rbrack \) be the ring of polynomials with coefficients in \( \mathbb{F} \) . Define \( N\left( f\right) = {2}^{\deg \left( f\right) } \), where \( \deg \left( f\right) \) is the degree of \( f \) in \( \mathbb{F}\left\lbrack x\right\rbrack \), which is defined for all nonzero elements of \( R \) . We prove the first of these; the second is an exercise. Proof THAT ℤ[i] Is Euclidean. Condition (1) of Definition 2.2 is easily verified by direct computation. For property (2), let \( a, b \neq 0 \in R \) and write \( a{b}^{-1} = p + \mathrm{i}q \) with \( p, q \in \mathbb{Q} \) . Now define \( m, n \in \mathbb{Z} \) by \[ m \in \lbrack p - 1/2, p + 1/2), n \in \lbrack q - 1/2, q + 1/2). \] Let \( q = m + \mathrm{i}n \in R \) and \( r = a - b\left( {m + \mathrm{i}n}\right) \) . For \( r \neq 0 \) , \[ N\left( r\right) = N\left( {\left( {a{b}^{-1} - m - \mathrm{i}n}\right) b}\right) \] \[ = N\left( {p + \mathrm{i}q - m - \mathrm{i}n}\right) N\left( b\right) \] \[ = N\left( {p - m + \mathrm{i}\left( {q - n}\right) }\right) N\left( b\right) \leq \left( {\frac{1}{4} + \frac{1}{4}}\right) N\left( b\right) < N\left( b\right) , \] showing property (2). Exercise 2.3. When \( R = \mathbb{Z} \), for any fixed \( a \) and \( b \), the values of \( q \) and \( r \) in Definition 2.2(2) are uniquely determined. Is the same true when \( R = \mathbb{Z}\left\lbrack \mathrm{i}\right\rbrack \) ? In any ring, we define greatest common divisors in exactly the same way as before. A greatest common divisor is defined up to multiplication by units (invertible elements). In any Euclidean ring, the function \( N \) can be used to define a Euclidean Algorithm, which can be used to find the greatest common divisor just as for the integers. Definition 2.4. In a ring \( R \) , (1) \( \alpha \) divides \( \beta \), written \( \alpha \mid \beta \), if there is an element \( \gamma \in R \) with \( \beta = {\alpha \gamma } \) ; (2) \( u \) is a unit if \( u \) divides 1 ; (3) \( \pi \) (not equal to zero nor to a unit) is prime if for all \( \alpha ,\beta \in R \) , \[ \pi \left| {{\alpha \beta } \Rightarrow \pi }\right| \alpha \text{ or }\pi \mid \beta \] (4) a non-unit \( \mu \) is irreducible if \[ \mu = {\alpha \beta } \Rightarrow \alpha \text{ or }\beta \text{ is a unit. } \] Notice that \( u \in R \) is a unit if and only if there is some \( \mu \) with \( {u\mu } = 1 \) . We write \( U\left( R\right) \) or \( {R}^{ * } \) for the units in the commutative ring \( R \) ; this is an Abelian group under multiplication. If the recent clutch of definitions are new to you, we recommend the following exercise. Exercise 2.4. (a) Show that, in any commutative ring, every prime element is irreducible. (b) Show that, in a Euclidean ring, \( u \) is a unit if and only if \( N\left( u\right) = 1 \) . (c) Show that there are infinitely many units in \( \mathbb{Z}\left\lbrack \sqrt{3}\right\rbrack \) . (d) Show that \( 3 + \sqrt{-2} \) is an irreducible element of \( \mathbb{Z}\left\lbrack \sqrt{-2}\right\rbrack \) . (e) Let \( \xi = \frac{-1 + \sqrt{-3}}{2} \) and \( R = \mathbb{Z}\left\lbrack \xi \right\rbrack \) . Prove that \( R \) is a Euclidean domain with respect to the norm \( N\left( {a + {b\xi }}\right) = {a}^{2} - {ab} + {b}^{2} = \left( {a + {b\xi }}\right) \left( {a + b\bar{\xi }}\right) \) and find all the units in \( R \) . Exercise 2.5. Prove the Remainder Theorem: For a polynomial \( f \in \mathbb{F}\left\lbrack x\right\rbrack ,\mathbb{F} \) a field, \( f\left( a\right) = 0 \) if and only if \( \left( {x - a}\right) \mid f\left( x\right) \) . Exercise 2.6. Give a different proof of Lemma 1.17 on p. 31 using group theory by considering the multiplicative group of units \( U\left( {\mathbb{Z}/{F}_{n}\mathbb{Z}}\right) = {\left( \mathbb{Z}/{F}_{n}\mathbb{Z}\right) }^{ * } \) . Exercise 2.7. Prove that \( \mathbb{Z}\left\lbrack x\right\rbrack \) does not have a Euclidean Algorithm by showing that the equation \( {2f}\left( x\right) + {xg}\left( x\right) = 1 \) has no solution for \( f, g \in \mathbb{Z}\left\lbrack x\right\rbrack \), but 2 and \( x \) have no common divisor in \( \mathbb{Z}\left\lbrack x\right\rbrack \) . Despite the conclusion of Exercise 2.7, the ring \( \mathbb{Z}\left\lbrack x\right\rbrack \) does have unique factorization into irreducibles. We will say that a ring has the Fundamental Theorem of Arithmetic if either of the following properties hold. (FTA1) Every irreducible element is prime. (FTA2) Every nonzero non-unit can be factorized uniquely up to order and multiplication by units. Theorem 2.5. Every Euclidean ring has the Fundamental Theorem of Arithmetic. Proof. Clearly, every irreducible \( \mu \) has \( N\left( \mu \right) \geq 2 \) . Arguing as we did in \( \mathbb{Z} \) shows we cannot keep factorizing into irreducibles forever, so the existence part is easy. To complete the argument, we just need to show that every irreducible is prime. This follows easily from Theorem 1.23. Let \( \mu \) be an irreducible and suppose that \( \mu \) divides \( {\alpha \beta } \) but \( \mu \) does not divide \( \alpha \) . Clearly, the greatest common divisor of \( \mu \) and \( \alpha \) is 1 because \( \mu \) admits only itself and units as divisors and \( \mu \) does not divide \( \alpha \), so we can write \[ {\mu x} + {\alpha y} = 1 \] for some \( x, y \in R \) by Theorem 1.23. Multiply through by \( \beta \) to obtain \[ {\mu x\beta } + {\alpha \beta y} = \beta . \] Since \( \mu \) divides both terms on the left-hand side, it must divide the right-hand side, and this completes the proof. ## 2.3 Sums of Squares The resolution of the Pythagorean equation ( Equation (2.1)) is an elementary and well-known result. We are now going to show how the Fundamental Theorem of Arithmetic in other contexts can yield solutions to less tractable Diophantine equations. Consider the following problem: Which integers can be represented as the sum of two squares? That is, what are the solutions to the Diophantine problem \[ n = {x}^{2} + {y}^{2}? \] When \( n \) is a prime, experimenting with a few small values suggests the following. Theorem 2.6. The prime \( p \) can be written as the sum of two squares if and only if \( p = 2 \) or \( p \) is congruent to 1 modulo 4 . To prove this, we are going to use the Fundamental Theorem of Arithmetic in the ring of Gaussian integers \( R = \mathbb{Z}\left\lbrack \mathrm{i}\right\rbrack \) with norm function \( N : R \rightarrow \mathbb{N} \) defined by \( N\left( {x + \mathrm{i}y}\right) = {x}^{2} + {y}^{2} \) as in Example 2.3(1). Lemma 2.7. If \( p \) is 2 or a prime congruent to 1 modulo 4, then the congruence \[ {T}^{2} + 1 \equiv 0\;\left( {\;\operatorname{mod}\;p}\right) \] is solvable in integers. Proof. This is clear for \( p = 2 \) so suppose \( p = {4n} + 1 \) for some integer \( n > 0 \) . Using al-Haytham's Theorem (Theorem 1.19), \[ \left( {p - 1}\right) ! = \left( {p - 1}\right) \left( {p - 2}\right) \cdots 3 \cdot 2 \cdot 1 \equiv - 1\;\left( {\;\operatorname{mod}\;p}\right) . \] Now \[ {4n} = p - 1 \equiv - 1\;\left( {\;\operatorname{mod}\;p}\right) \] \[ {4n} - 1 = p - 2 \equiv - 2\;\left( {\;\operatorname{mod}\;p}\right) \] \[ \vdots \] \[ {2n} + 1 = p - {2n} \equiv - {2n}\;\left( {\;\operatorname{mod}\;p}\right) . \] It follows that \[ \left( {-1}\right) \left( {-2}\right) \cdots \left( {-{2n}}\right) \left( {2n}\right) \left( {{2n} - 1}\right) \cdots 3 \cdot 2 \cdot 1 = \left( {2n}\right) !{\left( -1\right) }^{2n} \equiv - 1\;\left( {\;\operatorname{mod}\;p}\right) . \] Thus \( T = \left( {2n}\right) \) ! has \( {T}^{2} + 1 \equiv 0 \) modulo \( p \), proving the lemma. Proof of Theorem 2.6. The case \( p = 2 \) is trivial. The case when \( p \) is congruent to 3 modulo 4 is also dealt with easily; no integer that is congruent to 3 modulo 4 can be the sum of two squares because squares are 0 or 1 modulo 4. Assume that \( p \) is a prime congruent to 1 modulo 4 . By Lemma 2.7, we can write \[ {cp} = {T}^{2} + 1 = \left( {T + \mathrm{i}}\right) \left( {T - \mathrm{i}}\right) \text{ in }R = \mathbb{Z}\left\lbrack \mathrm{i}\right\rbrack \] for some integers \( T \) and \( c \) . Suppose (for a contradiction) that \( p \) is irreducible in \( R \) . Then since \( \mathbb{Z}\left\lbrack \mathrm{i}\right\rbrack \) has the Fundamental Theorem of Arithmetic, \( p \) is prime. Hence \( p \) must divide one of \( T \pm \mathrm{i} \) in \( R \) since it divides their product, and this is impossible because \( p \) does not divide the coefficient of i. It follows that \( p \) cannot be irreducible in \( R \) , so \[ p = {\mu \nu } \] is a product of two non-units in \( R \) . Taking the norm of both sides shows that \[ {p}^{2} = N\left( {\mu \nu

1359_[陈省身] Lectures on Differential Geometry

Definition 2.2

Definition 2.2. Suppose \( M \) is a connected Riemannian manifold, and \( p, q \) are two arbitrary points in \( M \) . Let \[ \rho \left( {p, q}\right) = \inf \overset{⏜}{pq} \] \( \left( {2.47}\right) \) where \( \overset{⏜}{pq} \) denotes the arc length of a curve connecting \( p \) and \( q \) with measurable arc length. Then \( \rho \left( {p, q}\right) \) is called the distance between points \( p \) and \( q \) . Because \( M \) is connected, there always exists a curve connecting \( p \) and \( q \) with measurable arc length. Therefore (2.47) is always meaningful, and defines a real function on \( M \times M \) . Theorem 2.6. The function \( \rho : M \times M \rightarrow \mathbb{R} \) has the following properties: 1) for any \( p, q \in M,\rho \left( {p, q}\right) \geq 0 \), and the equality holds only when \( p = q \) ; 2) \( \rho \left( {p, q}\right) = \rho \left( {q, p}\right) \) ; 3) for any three points \( p, q, r \in M \) we have \[ \rho \left( {p, q}\right) + \rho \left( {q, r}\right) \geq \rho \left( {p, r}\right) \] Therefore \( \rho \) becomes a distance function on \( M \) and makes \( M \) a metric space. The topology of \( M \) as a metric space and the original topology of \( M \) as a manifold are equivalent. Proof. According to definition (2.47), the above properties are obvious. We need only show that \( \rho \left( {p, q}\right) > 0 \) whenever \( p \neq q \) . Suppose \( p, q \) are any two points in \( M, p \neq q \) . Since \( M \) is a Hausdorff space, there exists a neighborhood \( U \) of \( \mathrm{p} \) such that \( q \notin U \) . By Theorem 2.4, there must exist a normal coordinate neighborhood \( W \subset U \) of \( p \) such that its normal coordinates are \( {u}^{i} = {\alpha }^{i}s \), where \( \mathop{\sum }\limits_{{i = 1}}^{\widetilde{m}}{\left( {\alpha }^{i}\right) }^{2} = 1 \) and \( 0 \leq s \leq {s}_{0} \) . Choose \( \delta \) such that \( 0 < \delta < {s}_{0} \) . Then the hypersurface \( {\sum }_{\delta } \subset W \) . Suppose \( \gamma \) is a measurable curve connecting \( p \) and \( q \) . Then the length of \( \gamma \) is at least \( \delta \), that is \[ \rho \left( {p, q}\right) \geq \delta > 0 \] By Theorem 2.5, the interior of \( {\sum }_{\delta } \) is precisely the set \[ \{ q \in M \mid \rho \left( {p, q}\right) < \delta \} \] that is, the interior of \( {\sum }_{\delta } \) is a \( \delta \) -ball neighborhood of \( p \) when \( M \) is viewed as a metric space. Thus the topology of \( M \) viewed as a metric space and the original topology of \( M \) are equivalent. We note that if \( W \) is a ball-shaped normal coordinate neighborhood at the point \( O \) constructed as in Theorem 2.4, then for any point \( p \in W \) the unique geodesic curve connecting \( O \) and \( p \) in \( W \) has length \( \rho \left( {O, p}\right) \) . Theorem 2.7. There exists a \( \eta \) -ball neighborhood \( W \) at any point \( p \) in a Riemannian manifold \( M \), where \( \eta \) is a sufficiently small positive number, such that any two points in \( W \) can be connected by a unique geodesic curve. Any neighborhood satisfying the above property is called a geodesic convex neighborhood. Thus the theorem states that there exists a geodesic convex neighborhood at every point in a Riemannian manifold. Proof. Suppose \( p \in M \) . By Theorem 2.4 there exists a ball-shaped normal coordinate neighborhood \( U \) of \( p \) with radius \( \epsilon \) such that for any point \( q \) in \( U \) there is a normal coordinate neighborhood \( {V}_{q} \) that contains \( U \) . We may assume that \( \epsilon \) also satisfies the requirements of Theorem 2.5. Choose a positive number \( \eta \leq \frac{1}{4}\epsilon \) . Then the \( \eta \) -ball neighborhood \( W \) of \( p \) is a geodesic convex neighborhood of \( p \) . Choose any \( {q}_{1},{q}_{2} \in W \) . Then \[ \rho \left( {{q}_{1},{q}_{2}}\right) \leq \rho \left( {p,{q}_{1}}\right) + \rho \left( {p,{q}_{2}}\right) < {2\eta } \leq \frac{\epsilon }{2}. \] (2.48) Suppose \( U\left( {{q}_{1};\epsilon /2}\right) \) is an \( \epsilon /2 \) -ball neighborhood of \( {q}_{1} \) . Then the above formula indicates that \( {q}_{2} \in U\left( {{q}_{1};\epsilon /2}\right) \) . For any \( q \in U\left( {{q}_{1};\epsilon /2}\right) \) we have \[ \rho \left( {p, q}\right) \leq \rho \left( {p,{q}_{1}}\right) + \rho \left( {{q}_{1}, q}\right) < \frac{3\epsilon }{4}. \] Hence \[ U\left( {{q}_{1};\frac{\epsilon }{2}}\right) \subset U \subset {V}_{{q}_{1}} \] (2.49) that is, the \( \epsilon /2 \) -ball neighborhood of \( {q}_{1} \) is contained in the normal coordinate neighborhood of \( {q}_{1} \) . By Theorem 2.4 and the statement immediately following the proof of Theorem 2.6, there exists a unique geodesic curve \( \gamma \) in \( U\left( {{q}_{1};\epsilon /2}\right) \) connecting \( {q}_{1} \) and \( {q}_{2} \), whose length is precisely \( \rho \left( {{q}_{1},{q}_{2}}\right) \) . In particular, if \( r \in \gamma \) , then \[ \rho \left( {{q}_{1}, r}\right) \leq \rho \left( {{q}_{1},{q}_{2}}\right) \] \( \left( {2.50}\right) \) Finally we prove that the geodesic curve \( \gamma \) lies inside \( W \) . Since \( \gamma \subset \) \( U\left( {{q}_{1};\epsilon /2}\right) \subset U \), the function \( \rho \left( {p, q}\right) \left( {q \in \gamma }\right) \) is bounded. If \( \gamma \) does not lie inside \( W \) completely, and \( {q}_{1},{q}_{2} \in W \), then the function \( \rho \left( {p, q}\right) \left( {q \in \gamma }\right) \) must attain its maximum at an interior point \( {q}_{0} \) of \( \gamma \) . Let \( \delta = \rho \left( {p,{q}_{0}}\right) \) . Then \( \delta < \epsilon \), and the hypersphere \( {\sum }_{\delta } \) is tangent to \( \gamma \) at \( {q}_{0} \) . By Theorem 2.5, \( \gamma \) lies completely outside \( {\sum }_{\delta } \) near \( {q}_{0} \), which contradicts the fact that \( \rho \left( {p, q}\right) \left( {q \in \gamma }\right) \) attains its maximum at \( {q}_{0} \) . Hence \( \gamma \subset W \) . ## §5-3 Sectional Curvature Suppose \( M \) is an \( m \) -dimensional Riemannian manifold whose curvature tensor \( R \) is a covariant tensor of rank 4, and \( {u}^{i} \) is a local coordinate system in \( M \) . Then \( R \) can be expressed as \[ R = {R}_{ijkl}d{u}^{i} \otimes d{u}^{j} \otimes d{u}^{k} \otimes d{u}^{l}, \] (3.1) where \( {R}_{ijkl} \) is defined as in (1.50). A covariant tensor of rank 4 can be viewed as a linear function on the space of contravariant tensors of rank 4 (see \( §2 - 2 \) ), so at every point \( p \in M \) we have a multilinear function \( R : {T}_{p}\left( M\right) \times {T}_{p}\left( M\right) \times \) \( {T}_{p}\left( M\right) \times {T}_{p}\left( M\right) \rightarrow R \), defined by \[ R\left( {X, Y, Z, W}\right) = \langle X \otimes Y \otimes Z \otimes W, R\rangle , \] (3.2) where the notation \( \langle \) , \( \rangle {isdefinedasin}\left( {2.17}\right) {ofChapter2}.{Ifwelet} \) \[ X = {X}^{i}\frac{\partial }{\partial {u}^{i}},\;Y = {Y}^{i}\frac{\partial }{\partial {u}^{i}},\;Z = {Z}^{i}\frac{\partial }{\partial {u}^{i}},\;W = {W}^{i}\frac{\partial }{\partial {u}^{i}}, \] (3.3) then \[ R\left( {X, Y, Z, W}\right) = {R}_{ijkl}{X}^{i}{Y}^{j}{Z}^{k}{W}^{l}. \] (3.4) In particular, \[ {R}_{ijkl} = R\left( {\frac{\partial }{\partial {u}^{i}},\frac{\partial }{\partial {u}^{j}},\frac{\partial }{\partial {u}^{k}},\frac{\partial }{\partial {u}^{l}}}\right) . \] (3.5) In \( §4 - 2 \), we have already interpreted the curvature tensor of a connection \( D \) as a curvature operator: for any given \( Z, W \in {T}_{p}\left( M\right), R\left( {Z, W}\right) \) is a linear map from \( {T}_{p}\left( M\right) \) to \( {T}_{p}\left( M\right) \) defined by \[ R\left( {Z, W}\right) X = {R}_{ikl}^{j}{X}^{i}{Z}^{k}{W}^{l}\frac{\partial }{\partial {u}^{j}}. \] (3.6) If \( D \) is the Levi-Civita connection of a Riemannian manifold \( M \), then we have \[ R\left( {X, Y, Z, W}\right) = \left( {R\left( {Z, W}\right) X}\right) \cdot Y \] (3.7) where the notation "." on the right hand side is the inner product defined by (1.4). By Theorem 1.4, the 4-linear function \( R\left( {X, Y, Z, W}\right) \) has the following properties: 1) \( R\left( {X, Y, Z, W}\right) = - R\left( {X, Y, W, Z}\right) = - R\left( {Y, X, Z, W}\right) \) ; 2) \( R\left( {X, Y, Z, W}\right) + R\left( {X, Z, W, Y}\right) + R\left( {X, W, Y, Z}\right) = 0 \) ; 3) \( R\left( {X, Y, Z, W}\right) = R\left( {Z, W, X, Y}\right) \) . Using the fundamental tensor \( G \) of \( M \), we can also define a 4-linear function as follows: \[ G\left( {X, Y, Z, W}\right) = G\left( {X, Z}\right) G\left( {Y, W}\right) - G\left( {X, W}\right) G\left( {Y, Z}\right) . \] (3.8) Obviously the function defined above is linear with respect to every variable, and also has the same properties 1)-3) as \( R\left( {X, Y, Z, W}\right) \) . If \( X, Y \in {T}_{p}\left( M\right) \), then \[ G\left( {X, Y, X, Y}\right) = {\left| X\right| }^{2} \cdot {\left| Y\right| }^{2} - {\left( X \cdot Y\right) }^{2} = {\left| X\right| }^{2} \cdot {\left| Y\right| }^{2} \cdot {\sin }^{2}\angle \left( {X, Y}\right) . \] (3.9) Therefore, when \( X, Y \) are linearly independent, \( G\left( {X, Y, X, Y}\right) \) is precisely the square of the area of the parallelogram determined by the tangent vectors \( X \) and \( Y \) . Hence \( G\left( {X, Y, X, Y}\right) \neq 0 \) . Suppose \( {X}^{\prime },{Y}^{\prime } \) are another two linearly independent tangent vectors at the point \( p \), and that they span the same 2-dimensional tangent subspace \( E \) as that spanned by \( X \) and \( Y \) . Then we may assume that \[ {X}^{\prime } = {aX} + {bY},\;{Y}^{\prime } = {cX} + {dY} \] where \( {ad} - {bc} \neq 0 \) . By properties 1)-3) we have \[ R\left( {{X}^{\prime },{Y}^{\prime },{X}^{\prime },{Y}^{\prime }}\right) = {\left( ad - bc\right) }^{2}R\left( {X, Y, X, Y}\right) , \] \[ G\

1088_(GTM245)Complex Analysis

Definition 9.32

Definition 9.32. Let \( D \) be a domain in \( \mathbb{C} \) . A Perron family \( \mathcal{F} \) in \( D \) is a nonempty collection of subharmonic functions in \( D \) such that (a) If \( u, v \) are in \( \mathcal{F} \), then so is \( \max \{ u, v\} \) . (b) If \( u \) is in \( \mathcal{F} \), then so is \( {u}_{U} \) for every disc \( U \) with cl \( U \subset D \) . The following result, due to Perron, is useful for constructing harmonic functions. Theorem 9.33 (Perron’s Principle). If \( \mathcal{F} \) is a uniformly bounded from above Perron family in \( D \), then the function defined for \( z \in D \) by \[ V\left( z\right) = \sup \{ u\left( z\right) : u \in \mathcal{F}\} \] (9.16) is harmonic in \( D \) . Proof. First note that by definition a Perron family is never empty. Since we are assuming that there exists a constant \( M \) such that \( u\left( z\right) < M \) for all \( z \) in \( D \) and all \( u \) in \( \mathcal{F} \), the function \( V \) is clearly well defined and real-valued. Let \( U \) be any disc such that \( \operatorname{cl}U \subset D \) . It is enough to show that \( V \) is harmonic in \( U \) . For any point \( {z}_{0} \) in \( U \), there exists a sequence \( \left\{ {{u}_{j} : j \in \mathbb{N}}\right\} \) of functions in \( \mathcal{F} \) such that \[ \mathop{\lim }\limits_{{j \rightarrow \infty }}{u}_{j}\left( {z}_{0}\right) = V\left( {z}_{0}\right) \] (9.17) Without loss of generality, we may assume \( {u}_{j + 1} \geq {u}_{j} \) for all \( j \) in \( \mathbb{N} \), since if \( \left\{ {u}_{j}\right\} \) is any sequence in \( \mathcal{F} \) satisfying (9.17), then the new sequence given by \( {v}_{1} = {u}_{1} \) and \( {v}_{j + 1} = \max \left\{ {{u}_{j + 1},{v}_{j}}\right\} \) for \( j \geq 1 \) is also contained in \( \mathcal{F} \), satisfies (9.17) (with \( {u}_{j} \) replaced by \( {v}_{j} \), of course), and is nondecreasing, as needed. The sequence \( \left\{ {{w}_{j} = {\left( {u}_{j}\right) }_{U}}\right\} \) of harmonizations of the \( {u}_{j} \) in \( U \) consists of subharmonic functions with the following properties: 1. \( {w}_{j} \geq {u}_{j} \) for all \( j \) 2. \( {w}_{j} \leq {w}_{j + 1} < M \) for all \( j \) , since the two inequalities clearly hold outside \( U \) and on the boundary of \( U \), from which it follows that they also hold in \( U \) . Thus the sequence \( \left\{ {w}_{j}\right\} \) lies in \( \mathcal{F} \), is nondecreasing, and satisfies \( \mathop{\lim }\limits_{{j \rightarrow \infty }}{w}_{j}\left( {z}_{0}\right) = \) \( V\left( {z}_{0}\right) \) . It follows from the Harnack's convergence Theorem 9.14 that the function defined by \[ \Phi \left( z\right) = \mathop{\lim }\limits_{{j \rightarrow \infty }}{w}_{j}\left( z\right) = \sup \left\{ {{w}_{j}\left( z\right) : j \in \mathbb{N}}\right\} \] is harmonic in \( U \) . We will now show that \( \Phi = V \) in \( U \) . Let \( c \) denote any point in \( U \) . As before, we can find a nondecreasing sequence \( \left\{ {s}_{j}\right\} \) in \( \mathcal{F} \) such that \( V\left( c\right) = \mathop{\lim }\limits_{{j \rightarrow \infty }}{s}_{j}\left( c\right) \) . By setting \( {t}_{1} = \max \left\{ {{s}_{1},{w}_{1}}\right\} \) and \( {t}_{j + 1} = \max \left\{ {{s}_{j + 1},{w}_{j + 1},{t}_{j}}\right\} \) for all \( j \geq 1 \), we obtain a nondecreasing sequence \( \left\{ {t}_{j}\right\} \) in \( \mathcal{F} \) such that \( {t}_{j} \geq {w}_{j} \) for all \( j \), and such that \( \mathop{\lim }\limits_{{j \rightarrow \infty }}{t}_{j}\left( z\right) = V\left( z\right) \) for \( z = c \) and \( z = {z}_{0}. \) The harmonizations of the \( {t}_{j} \) in \( U \) give a nondecreasing sequence \( \left\{ {{r}_{j} = {\left( {t}_{j}\right) }_{U}}\right\} \) in \( \mathcal{F} \) satisfying \( M > {r}_{j} \geq {t}_{j} \geq {w}_{j} \) for all \( j \) . As before, the function defined by \[ \Psi \left( z\right) = \sup \left\{ {{r}_{j}\left( z\right) : j \in \mathbb{N}}\right\} \] is harmonic in \( U \), and coincides with \( V \) at \( c \) and \( {z}_{0} \) . But \( \Psi \geq \Phi \), since \( {r}_{j} \geq {w}_{j} \) for all \( j \), and hence \( \Psi - \Phi \) is a nonnegative harmonic function in \( U \) . Since it is equal to zero at \( {z}_{0} \), by the minimum principle for harmonic functions, it is identically zero in \( U \), and the result follows. ## 9.8 The Dirichlet Problem (Revisited) This section has two parts. The first describes a method for obtaining the solution to the Dirichlet problem, provided it is solvable. In the second part, we offer a solution. Recall that the Dirichlet problem for a bounded region \( D \) in \( \mathbb{C} \) and a function \( f \in \) \( \mathbf{C}\left( {\partial D}\right) \) is to find a continuous function \( U \) on the closure of \( D \) whose restriction to \( D \) is harmonic and which agrees with \( f \) on the boundary of \( D \) . Under these conditions, let \( \mathcal{F} \) denote the family of all continuous functions \( u \) on cl \( D \) such that \( u \) is subharmonic in \( D \) and \( u \leq f \) on \( \partial D \) . Then \( \mathcal{F} \) is a Perron family of functions uniformly bounded from above. Note that the constant function \( u = \min \{ f\left( z\right) : z \in \) \( \partial D\} \) belongs to \( \mathcal{F} \), hence \( \mathcal{F} \) is nonempty. The other conditions for \( \mathcal{F} \) to be a Perron family are also easily verified. Therefore, by Theorem 9.33, the function \( V \) defined by (9.16) is harmonic in \( D \) . Now, if we assume that there is a solution \( U \) to the Dirichlet problem for \( D \) and \( f \), then we can show that \( U = V \) . Indeed, for each \( u \) in \( \mathcal{F} \) the function \( u - U \) is subharmonic in \( D \), and satisfies \( u - U = u - f \leq 0 \) on \( \partial D \), from where it follows that \( u - U \leq 0 \) in \( D \), and hence \( V \leq U \) in \( D \) . But \( U \) belongs to \( \mathcal{F} \), and it follows that \( U \leq V \), and therefore \( U = V \) . The Dirichlet problem does not always have a solution. A very simple example is given by considering the domain \( D = \{ 0 < \left| z\right| < 1\} \) and the function \[ f\left( z\right) = \left\{ \begin{array}{ll} 0, & \text{ if }\left| z\right| = 1 \\ 1, & \text{ if }z = 0 \end{array}\right. \] The corresponding function \( V \) given by Theorem 9.33 is harmonic in the punctured disc \( D \) . If the Dirichlet problem were solvable in our case, then \( V \) would extend to a continuous function on \( \left| z\right| \leq 1 \) that is harmonic in \( \left| z\right| < 1 \) (see Exercise 9.17). But then the maximum principle would imply that \( V \) is identically zero, a contradiction. To solve the Dirichlet problem, we start with a bounded domain \( D \subset \mathbb{C} \), with boundary \( \partial D \), and the following definition. Definition 9.34. A function \( \beta \) is a barrier at \( {z}_{0} \in \partial D \), and \( {z}_{0} \) is a regular point for the Dirichlet problem provided there exists an open neighborhood \( N \) of \( {z}_{0} \) in \( \mathbb{C} \) such that (1) \( \beta \in \mathbf{C}\left( {\operatorname{cl}D \cap N}\right) \) . (2) \( - \beta \) is subharmonic in \( D \cap N \) . (3) \( \beta \left( z\right) > 0 \) for \( z \neq {z}_{0},\beta \left( {z}_{0}\right) = 0 \) . (4) \( \beta \left( z\right) = 1 \) for \( z \notin N \) . Remark 9.35. A few observations are in order. 1. Condition (4) is easily satisfied by adjusting a function \( \beta \) that satisfies the other three conditions for being a barrier. To see this we may assume that \( N \) is relatively compact in \( \mathbb{C} \), and choose a smaller neighborhood \( {N}_{0} \) of \( {z}_{0} \) with cl \( {N}_{0} \subset N \) . Then let \[ m = \min \left\{ {\beta \left( z\right) ;z \in \operatorname{cl}\left( {N - {N}_{0}}\right) \cap \operatorname{cl}D}\right\} \] note that \( m > 0 \), and define \[ {\beta }_{1}\left( z\right) = \left\{ \begin{array}{ll} \min \{ m,\beta \left( z\right) \} & \text{ for }z \in N \cap D, \\ m & \text{ for }z \in \operatorname{cl}\left( {D - N}\right) . \end{array}\right. \] Finally set \( {\beta }_{2} = \frac{{\beta }_{1}}{m} \), and observe that \( {\beta }_{2} \) satisfies all the conditions for being a barrier at \( {z}_{0} \) . Thus, to prove the existence of a barrier, it suffices to produce a function that satisfies the first three conditions. 2. The existence of barriers is a local property. If a point \( {z}_{0} \in \partial D \) can be reached by an analytic arc (a curve that is the image of \( \left\lbrack {0,1}\right\rbrack \) under an injective analytic map defined in a neighborhood of \( \left\lbrack {0,1}\right\rbrack ) \) with no points in common with cl \( D - \) \( \left\{ {z}_{0}\right\} \), then a barrier exists at this point. To establish this we may, without loss of generality, assume that \( {z}_{0} = 0 \), that the closure of \( D \) lies in the right half plane, and that the analytic arc consists of the negative real axis including the origin. Using polar coordinates \( z = r{\mathrm{e}}^{\iota \theta } \), we see that \( \beta \left( z\right) = {r}^{\frac{1}{2}}\cos \frac{\theta }{2}, - \pi < \theta < \pi \) , satisfies the first three conditions for a barrier function. Definition 9.36. Let \( D \) be a nonempty domain in \( \mathbb{C} \) . A solution \( u \) to the Dirichlet problem for \( f \in {\mathbf{C}}_{\mathbb{R}}\left( {\partial D}\right) \) is proper provided \[ \inf \{ f\left( w\right) ;w \in \partial D\} \leq u\left( z\right) \leq \sup \{ f\left( w\right) ;w \in \partial D\} \] for all \( z \) in \( D \) . A far-reaching generalization of Schwartz's Theorem 9.15 is provided by our next result. Theorem 9.37. Let \( D \) be a nonempty domain in \( \mathbb{C} \) . There exists a proper solution to the Dirichlet problem for \( D \) for every bounded continuous real-valued function on \( \partial D \) if and

113_Topological Groups

Definition 8.22

Definition 8.22. Let \( \mathcal{P} = \left( {n, c, P}\right) \) be a sentential language. Members of \( {}^{P}2 \) are called models of \( \mathcal{P} \) . (Intuitively,0 means falsity,1 means truth, and a function \( f \in {}^{P}2 \) is just an assignment of a truth value to each sentence of P.) Using the recursion principle for sentences, we can associate with each \( f \in {}^{P}2 \) a function \( {f}^{ + } : \) Sent \( \mathcal{P} \rightarrow 2 \) such that for any \( s \in P \) and any \( \varphi ,\psi \in {\text{Sent}}_{\mathcal{P}} \) \[ {f}^{ + }\langle s\rangle = {fs} \] \[ {f}^{ + }\neg \varphi = 1\;\text{ if }{f}^{ + }\varphi = 0, \] \[ {f}^{ + }\neg \varphi = 0\;\text{ if }{f}^{ + }\varphi = 1, \] \[ {f}^{ + }\left( {\varphi \rightarrow \psi }\right) = 0\;\text{ iff }{f}^{ + }\varphi = 1\text{ and }{f}^{ + }\varphi = 0. \] \( \left( {f}^{ + }\right. \) intuitively tells us about the truth or falsity of any sentence of \( \mathcal{P} \) , given the truth or falsity of members of \( P \) .) We say that \( f \) is a model of \( \varphi \) if \( {f}^{ + }\varphi = 1 \) ; \( f \) is a model of a set \( \Gamma \) of sentences iff \( {f}^{ + }\varphi = 1 \) for all \( \varphi \in \Gamma \) . We write \( \Gamma { \vDash }_{\mathcal{P}}\varphi \) iff every model of \( \Gamma \) is a model of \( \varphi \), and we write \( { \vDash }_{\mathcal{P}}\varphi \) instead of \( 0{ \vDash }_{\mathcal{P}}\varphi \) . Sentences \( \varphi \) with \( { \vDash }_{\mathcal{P}}\varphi \) are called tautologies. Whether or not a sentence \( \varphi \) is a tautology can be decided by the familiar truth table method: one writes in rows all possible \( f \in {}^{P}2 \) and for each such \( f \) calculates \( {f}^{ + }\varphi \) from inside out. Of course instead of all \( f \in {}^{P}2 \) it suffices to list only the \( f \in {}^{Q}2 \), where \( Q \) is the set of \( s \in P \) which occur in \( \varphi \) . For example, the following table shows that \( \left\langle {s}_{1}\right\rangle \rightarrow \left( {\left\langle {s}_{2}\right\rangle \rightarrow \left\langle {s}_{1}\right\rangle }\right) \) is a tautology: <table><thead><tr><th>\( {s}_{1} \)</th><th>\( {s}_{2} \)</th><th>\( \left\langle {s}_{2}\right\rangle \rightarrow \left\langle {s}_{1}\right\rangle \)</th><th>\( \left\langle {s}_{1}\right\rangle \rightarrow \left( {\left\langle {s}_{2}\right\rangle \rightarrow \left\langle {s}_{1}\right\rangle }\right) \)</th></tr></thead><tr><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>1</td><td>0</td><td>1</td><td>1</td></tr><tr><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><td>0</td><td>0</td><td>1</td><td>1</td></tr></table> The following table shows that \( \neg \left\langle {s}_{1}\right\rangle \rightarrow \left( {\neg \left\langle {s}_{1}\right\rangle \rightarrow \left\langle {s}_{1}\right\rangle }\right) \) is not a tautology: <table><thead><tr><th>\( {s}_{1} \)</th><th>\( \neg \left\langle {s}_{1}\right\rangle \)</th><th>\( \neg \left\langle {s}_{1}\right\rangle \rightarrow \left\langle {s}_{1}\right\rangle \)</th><th>\( \neg \left\langle {s}_{1}\right\rangle \rightarrow \left( {\neg \left\langle {s}_{1}\right\rangle \rightarrow \left\langle {s}_{1}\right\rangle }\right) \)</th></tr></thead><tr><td>1</td><td>0</td><td>1</td><td>1</td></tr><tr><td>0</td><td>1</td><td>0</td><td>0</td></tr></table> Clearly this truth table procedure provides an effective procedure for determining whether or not a sentence is a tautology. This statement could be made precise for sentential languages \( \mathcal{P} = \left( {n, c, P}\right) \) with \( P \) countable by the usual procedure of Gödel numbering. (See 10.19-10.22, where this is done in detail for first-order languages.) In practice, to check that a statement is or is not a tautology it is frequently better to argue informally, assuming the given sentence is not true and trying to infer a contradiction from this. For example, if \( \left\langle {s}_{1}\right\rangle \rightarrow \left( {\left\langle {s}_{2}\right\rangle \rightarrow \left\langle {s}_{1}\right\rangle }\right) \) is false, then \( {s}_{1} \) is true and \( \left\langle {s}_{2}\right\rangle \rightarrow \left\langle {s}_{1}\right\rangle \) is false; but this is impossible; \( \left\langle {s}_{2}\right\rangle \rightarrow \left\langle {s}_{1}\right\rangle \) is true since \( {s}_{1} \) is true. Thus \( \left\langle {s}_{1}\right\rangle \rightarrow \) \( \left( {\left\langle {s}_{2}\right\rangle \rightarrow \left\langle {s}_{1}\right\rangle }\right) \) is a tautology. We are going to show shortly that the relations \( \vdash \) and \( \vDash \) are identical. To do this we need some preliminary statements. Lemma 8.23. If \( \Gamma \vdash \varphi \), then \( \Gamma \vDash \varphi \) . Proof. Let \( \Delta = \{ \varphi \) : every model of \( \Gamma \) is a model of \( \varphi \} \) . It is easy to check, using truth tables for the logical axioms, that \( \Gamma \subseteq \Delta \), every logical axiom is in \( \Delta \), and \( \Delta \) is closed under detachment. Hence all \( \Gamma \) -theorems are in \( \Delta \) . The lemma follows. Definition 8.24. \( \Gamma \) is consistent iff \( \Gamma \nvdash \varphi \) for some \( \varphi \) . Theorem 8.25. The following conditions are equivalent: (i) \( \Gamma \) is inconsistent. (ii) \( \Gamma \vdash \neg \left( {\varphi \rightarrow \varphi }\right) \) for every sentence \( \varphi \) . (iii) \( \Gamma \vdash \neg \left( {\varphi \rightarrow \varphi }\right) \) for some sentence \( \varphi \) . Proof. Obviously \( \left( i\right) \Rightarrow \left( {ii}\right) \Rightarrow \left( {iii}\right) \) . Now suppose \( \Gamma \vdash \neg \left( {\varphi \rightarrow \varphi }\right) \) for a certain sentence \( \varphi \) . Let \( \psi \) be any sentence. By \( \mathrm{{Al}},\Gamma \vdash \left( {\varphi \rightarrow \varphi }\right) \rightarrow \) \( \left\lbrack {\neg \psi \rightarrow \left( {\varphi \rightarrow \varphi }\right) }\right\rbrack \) ; from 8.10 we infer that \( \Gamma \vdash \neg \psi \rightarrow \left( {\varphi \rightarrow \varphi }\right) \), and then 8.17 yields \( \Gamma \vdash \neg \left( {\varphi \rightarrow \varphi }\right) \rightarrow \neg \neg \psi \) . Hence \( \Gamma \vdash \neg \neg \psi \) . So by \( {8.16},\Gamma \vdash \psi : \psi \) being any sentence, \( \Gamma \) is inconsistent. Theorem 8.26. \( \Gamma \cup \{ \varphi \} \) is inconsistent iff \( \Gamma \vdash \neg \varphi \) . Proof. \( \Rightarrow \) : Since \( \Gamma \cup \{ \varphi \} \vdash \psi \) for any sentence \( \psi \), we have \( \Gamma \cup \{ \varphi \} \vdash \neg \varphi \), so by the deduction theorem \( \Gamma \vdash \varphi \rightarrow \neg \varphi \) . By 8.19, \( \Gamma \vdash \neg \varphi \) . \( \Leftarrow : \Gamma \cup \{ \varphi \} \vdash \neg \varphi \) and \( \Gamma \cup \{ \varphi \} \vdash \varphi \), so by 8.14, \( \Gamma \cup \{ \varphi \} \vdash \psi \) for any sentence \( \psi \) . Theorem 8.27. 0 is consistent. Proof. Since \( \neg \left( {\varphi \rightarrow \varphi }\right) \) always receives the value 0 under any model, for any sentence \( \varphi \), by 8.23 we have not \( \left( { \vdash \neg \left( {\varphi \rightarrow \varphi }\right) }\right) \) . Theorem 8.28 (Extended completeness theorem). Every consistent set of sentences has a model. Proof. Let \( \Gamma \) be a consistent set of sentences. Let \( \mathcal{A} = \{ \Delta : \Gamma \subseteq \Delta ,\Delta \) is consistent \( \} \) . Since \( \Gamma \in \mathcal{A},\mathcal{A} \) is nonempty. Suppose \( \mathcal{B} \) is a subset of \( \mathcal{A} \) simply ordered by inclusion, \( \mathcal{B} \neq 0 \) . Then \( \Gamma \subseteq \bigcup \mathcal{B} \) . Also, \( \bigcup \mathcal{B} \) is consistent, for, if not, there would be, by 8.25, a sentence \( \varphi \) such that \( \bigcup \mathcal{B} \vdash \neg \left( {\varphi \rightarrow \varphi }\right) \) . Then by \( {8.9},\left\{ {{\psi }_{0},\ldots ,{\psi }_{m - 1}}\right\} \vdash \neg \left( {\varphi \rightarrow \varphi }\right) \) for some finite subset \( \left\{ {{\psi }_{0},\ldots ,{\psi }_{m - 1}}\right\} \) of \( \bigcup \mathcal{B} \) . Say \( {\psi }_{0} \in {\Delta }_{0} \in \mathcal{B},\ldots ,{\psi }_{m - 1} \in {\Delta }_{m - 1} \in \mathcal{B} \) . Since \( \mathcal{B} \) is simply ordered, there is an \( i < m \) such that \( {\Delta }_{j} \subseteq {\Delta }_{i} \) for all \( j < m \) . Thus \( {\psi }_{0} \in {\Delta }_{i},\ldots ,{\psi }_{m - 1} \in {\Delta }_{i} \) , so \( {\Delta }_{i} \vdash \neg \left( {\varphi \rightarrow \varphi }\right) \) . Thus \( {\Delta }_{i} \) is inconsistent by 8.25, contradicting \( {\Delta }_{i} \in \mathcal{B} \) . Thus \( \cup \mathcal{B} \) is consistent. Hence we may apply Zorn’s lemma to obtain a member \( \Delta \) of \( \mathcal{A} \) maximal under inclusion. Now we establish some important properties of \( \Delta \) . (1) \[ \Delta \vdash \varphi \text{implies that}\varphi \in \Delta \text{.} \] For, if \( \Delta \vdash \varphi \) and \( \varphi \notin \Delta \), then \( \Delta \cup \{ \varphi \} \) is inconsistent, so by \( {8.26},\Delta \vdash \neg \varphi \) . Then by \( {8.14},\Delta \) is inconsistent, contradiction. (2) \[ \text{if}\varphi \in \text{Sent, then}\varphi \in \Delta \text{or}\neg \varphi \in \Delta \text{.} \] For, suppose \( \varphi \notin \Delta \) . Then \( \Delta \cup \{ \varphi \} \) is inconsistent, so by \( {8.26\Delta } \vdash \neg \varphi \), and (1) yields \( \neg \varphi \in \Delta \) . (3) \[ \varphi \rightarrow \psi \in \Delta \text{iff}\neg \varphi \in \Delta \text{or}\psi \in \Delta \text{.} \] To prove this, first suppose \( \neg \varphi \in \Delta \) . By 8.15 and (1), \( \varphi \rightarrow \psi \in \Delta \) . If \( \psi \in \Delta \) , then \( \varphi \rightarrow \psi \in \Delta \) by A1 and (1). Thus \( \Leftarrow \) in (3) holds. Now suppose \( \neg \varphi \notin \Delta \) and \( \psi \notin \Delta \) . By (2)

1074_(GTM232)An Introduction to Number Theory

Definition 10.7

Definition 10.7. Let \( G \) be a finite Abelian group. A character of \( G \) is a homomorphism \[ \chi : G \rightarrow \left( {{\mathbb{C}}^{ * }, \cdot }\right) \] The multiplicative group \( {\mathbb{C}}^{ * } \) is \( \mathbb{C} \smallsetminus \{ 0\} \) equipped with the usual multiplication. By convention, we will write all finite groups multiplicatively in this section - hence the identity will be written as \( {1}_{G} \) or 1 . For any group, the map \[ {\chi }_{0} : G \rightarrow {\mathbb{C}}^{ * },{\chi }_{0}\left( g\right) = 1, \] is a character called the trivial character. Lemma 10.8. Let \( G \) be a finite Abelian group, and let \( \chi \) be a character of \( G \) . Then \( \chi \left( {1}_{G}\right) = 1 \) and \( \chi \left( g\right) \) is a root of unity for any \( g \in G \) . In particular, \( \left| {\chi \left( g\right) }\right| = 1 \) . Thus \( \chi \left( g\right) \) lies on the unit circle in \( \mathbb{C} \) . Proof. Clearly \[ \chi \left( {1}_{G}\right) = \chi \left( {{1}_{G} \cdot {1}_{G}}\right) = \chi \left( {1}_{G}\right) \chi \left( {1}_{G}\right) \] so \( \chi \left( {1}_{G}\right) = 1 \) since \( \chi \left( {1}_{G}\right) \neq 0 \) . As to the second statement, we use the fact that for every \( g \in G \) there exists \( n \in \mathbb{N} \) such that \( {g}^{n} = {1}_{G} \) . This implies that \[ \chi {\left( g\right) }^{n} = \chi \left( {g}^{n}\right) = \chi \left( {1}_{G}\right) = 1. \] Example 10.9. Let \( G = {C}_{k} = \langle g\rangle \), a cyclic group of order \( k \) . Now \[ {g}^{k} = 1 \] so \[ \chi {\left( g\right) }^{k} = 1 \] and therefore \( \chi \left( g\right) \) must be a \( k \) th root of unity. Any of the \( k \) different \( k \) th roots of unity can occur as \( \chi \left( g\right) \), and of course \( \chi \left( g\right) \) determines all the values of \( \chi \) on \( G \) since \( G \) is generated by \( g \), so there are \( k \) distinct characters of \( G \) . We can label the characters of \( G \) with labels \( 0,1,\ldots, n - 1 \) as follows: \( {\chi }_{j} \) is determined by \( {\chi }_{j}\left( g\right) = {e}^{{2\pi }\mathrm{i}j/k} \), so \( {\chi }_{j}\left( {g}^{m}\right) = {e}^{{2\pi }\mathrm{i}{jm}/k} \) . Theorem 10.10. Let \( G \) be a finite Abelian group. Then the characters of \( G \) form a group with respect to the multiplication \[ \left( {\chi \cdot \psi }\right) \left( g\right) = \chi \left( g\right) \psi \left( g\right) \] denoted \( \widehat{G} \) . The identity in \( \widehat{G} \) is the trivial character. The group \( \widehat{G} \) is isomorphic to \( G \) . In particular, any finite Abelian group \( G \) of order \( n \) has exactly \( n \) distinct characters. This theorem is the first intimation of an entire dual world, a mirror image to the familiar world of finite Abelian groups. This duality extends to a larger class of Abelian groups and in that wider class takes subgroups to quotient groups, quotient groups to subgroups, and products to sums. Exercise 10.3. What happens if the same construction is made for other groups? (a) Describe the group \[ \widehat{\mathbb{Z}} = \left\{ {\text{ homomorphisms }\mathbb{Z} \rightarrow {\mathbb{S}}^{1}}\right\} \] (b) For nondiscrete groups \( G \), we need to restrict to continuous characters. Find \[ \widehat{{\mathbb{S}}^{1}} = \left\{ {\text{continuous homomorphisms}{\mathbb{S}}^{1} \rightarrow {\mathbb{S}}^{1}}\right\} . \] \( {\left( \mathrm{c}\right) }^{ * }\mathrm{\;A} \) more challenging problem is to describe the group \( \widehat{\mathbb{Q}} \) . Proof of Theorem 10.10. Use the structure theorem for finite Abelian groups, which says that \( G \) is isomorphic to a product of cyclic groups, \[ G \cong \mathop{\prod }\limits_{{j = 1}}^{k}{C}_{{n}_{j}} \] Choose a generator \( {g}_{j} \) for each of the factors \( {C}_{{n}_{j}} \) and define characters on \( G \) by \[ {\chi }^{\left( j\right) }\left( {*,\ldots ,*,{g}_{j},*,\ldots , * }\right) = {e}^{{2\pi }\mathrm{i}/{n}_{j}}, \] that is, ignore all entries except the \( j \) th, and there use the same definition as in Example 10.9. Then the characters \( {\chi }^{\left( 1\right) },\ldots ,{\chi }^{\left( k\right) } \) generate a subgroup of \( \widehat{G} \) that is isomorphic to \( G \) : Each \( {\chi }^{\left( j\right) } \) generates a cyclic group of order \( {n}_{j} \) , and this group has a trivial intersection with the span of all the other \( {\chi }^{\left( i\right) }\mathrm{s} \) since all characters in the latter have value 1 at \( {g}_{j} \) . Likewise, for any given character of \( G \), it is easy to write down a product of powers of the \( {\chi }^{\left( j\right) } \) that coincides with \( \chi \) on the generators \( {g}_{j} \) and hence on all of \( G \) . Corollary 10.11. Let \( G \) be a finite Abelian group. For any \( 1 \neq g \in G \), there exists \( \chi \in \widehat{G} \) such that \( \chi \left( g\right) \neq 1 \) . Proof. Looking again at the proof of Theorem 10.10, we may write \[ g = \left( {*,\ldots ,*,{g}_{j}^{r},*,\ldots , * }\right) \] with some entry \( {g}_{j}^{r} \neq 1,0 < r < {n}_{j} \) . Then \( {\chi }^{\left( j\right) }\left( g\right) = {e}^{{2\pi }\mathrm{i}r/{n}_{j}} \neq 1 \) . Theorem 10.12. Let \( G \) be a finite Abelian group. Then, for any element \( h \in \) \( G \) and any character \( \psi \in \widehat{G} \) , \[ \mathop{\sum }\limits_{{g \in G}}\psi \left( g\right) = \left\{ \begin{matrix} \left| G\right| & \text{ if }\psi = {\chi }_{0} \\ 0 & \text{ if }\psi \neq {\chi }_{0} \end{matrix}\right. \] (10.13) \[ \mathop{\sum }\limits_{{\chi \in \widehat{G}}}\chi \left( h\right) = \left\{ \begin{matrix} \left| G\right| & \text{ if }h = 1 \\ 0 & \text{ if }h \neq 1 \end{matrix}\right. \] (10.14) These identities are known as the orthogonality relations for finite Abelian group characters. Proof. Consider Equation (10.13) first. The case \( \psi = {\chi }_{0} \) is trivial, so assume \( \psi \neq {\chi }_{0} \) . There is an element \( h \in G \) such that \( \psi \left( h\right) \neq 1 \) . Then \[ \psi \left( h\right) \mathop{\sum }\limits_{{g \in G}}\psi \left( g\right) = \mathop{\sum }\limits_{{g \in G}}\psi \left( {gh}\right) = \mathop{\sum }\limits_{{g \in G}}\psi \left( g\right) \] because multiplication by \( h \) only permutes the summands. This equation can only be true if \( \mathop{\sum }\limits_{{g \in G}}\psi \left( g\right) = 0 \) . For Equation (10.14), assume \( h \neq 1 \) . By Corollary 10.11, there exists some character \( \psi \in \widehat{G} \) such that \( \psi \left( h\right) \neq 1 \) . We now use the dual of the argument above, \[ \psi \left( h\right) \mathop{\sum }\limits_{{\chi \in \widehat{G}}}\chi \left( h\right) = \mathop{\sum }\limits_{{\chi \in \widehat{G}}}\left( {\psi \cdot \chi }\right) \left( h\right) = \mathop{\sum }\limits_{{\chi \in \widehat{G}}}\chi \left( h\right) \] since multiplication by \( \psi \) only permutes the elements of \( \widehat{G} \), and again this can only be true if \( \mathop{\sum }\limits_{{\chi \in \widehat{G}}}\chi \left( h\right) = 0 \) . Corollary 10.13. For all \( g, h \in G \), we have \[ \mathop{\sum }\limits_{{\chi \in \widehat{G}}}\chi \left( g\right) \overline{\chi \left( h\right) } = \left\{ \begin{array}{ll} \left| G\right| & \text{ if }g = h \\ 0 & \text{ if }g \neq h \end{array}\right. \] Proof. Note that \[ \chi \left( {h}^{-1}\right) = \chi {\left( h\right) }^{-1} = \overline{\chi \left( h\right) } \] since \( \chi \left( h\right) \) is on the unit circle in \( \mathbb{C} \) . Then use Theorem 10.12 with \( g{h}^{-1} \) in place of \( h \) . This is the gadget in its ultimate form. Character theory allows us to construct functions that will extract any desired residue class. As an example, take \( G = U\left( {\mathbb{Z}/5\mathbb{Z}}\right) \cong {C}_{4} \) . Table 10.1 shows all the characters on \( G \) . Table 10.1. Characters on \( U\left( {\mathbb{Z}/5\mathbb{Z}}\right) \) . \[ \begin{array}{lllll} 1 & {\chi }_{0} & {\chi }_{1} & {\chi }_{2} & {\chi }_{3} \\ 1 & 1 & 1 & 1 & 1 \\ 2 & 1 & i & - 1 & - i \\ 4 & 1 & - 1 & 1 & - 1 \\ 3 & 1 & - i & - 1 & i \end{array} \] Note that we have written the elements of \( U\left( {\mathbb{Z}/5\mathbb{Z}}\right) \) in Table 10.1 in an unusual ordering \( {2}^{0},{2}^{1},{2}^{2},{2}^{3} \), adapted to the generator 2 . The character values behave likewise. Note also \( {\chi }_{1}^{2} = {\chi }_{2} \) and \( {\chi }_{1}^{3} = {\chi }_{3} = {\chi }_{1}^{-1} \) . We used earlier \[ {\chi }_{0}\left( n\right) + {\chi }_{1}\left( n\right) + {\chi }_{2}\left( n\right) + {\chi }_{3}\left( n\right) = 4{\mathrm{c}}_{1}\left( n\right) \] which is just the case \( h = 1 \) of Corollary 10.13. We asked then,"What about \( {\mathrm{c}}_{2}\left( n\right) \), which is 1 if \( n \) is congruent to 2 and 0 otherwise?" The corollary suggests that we take \( h = 2 \), and we get \[ {\chi }_{0}\left( n\right) - i{\chi }_{1}\left( n\right) - {\chi }_{2}\left( n\right) + i{\chi }_{3}\left( n\right) = 4{\mathrm{c}}_{2}\left( n\right) . \] This can be checked simply by going through the possible cases. If you compare the ideas used here with Fourier analysis, much is familiar. The expression \[ \frac{1}{\left| G\right| }\mathop{\sum }\limits_{{h \in G}}\mathrm{f}\left( h\right) \overline{\mathrm{g}\left( h\right) } \] is an inner product on the vector space of all functions on \( G \), and the characters form a complete orthonormal set. There are no difficulties about convergence because the group is finite. In particular, any complex function on \( G \) can be written as a linear combination of the characters. ## 10.4 Dirichlet Characters and L-Functions Definition 10.14. Given \( 1 < q \in \mathbb{N} \), let \( G = U\left( {\mathbb{Z}/q\mathbb{Z}}\right) \) and fix a character \( \chi \) in \( \widehat{G} \) . Extend \( \chi \) to a function \( X \) on \( \mathbb{N} \) by setting \[ X\left( n\right) = \lef

1112_(GTM267)Quantum Theory for Mathematicians

Definition 23.21

Definition 23.21 A smooth, complex-valued function \( f \) on \( N \) is quantizable with respect to \( P \) if \( {Q}_{\text{pre }}\left( f\right) \) preserves the space of smooth sections that are polarized with respect to \( P \) . The following definition will provide a natural geometric condition guaranteeing quantizability of a function. Definition 23.22 A possibly complex vector field \( X \) preserves a polarization \( P \) if for every vector field \( Y \) lying in \( P \), the vector field \( \left\lbrack {X, Y}\right\rbrack \) also lies in \( P \) . Note that if \( X \) lies in \( P \), then \( X \) preserves \( P \), by the integrability assumption on \( P \) . There will typically be, however, many vector fields that do not lie in \( P \) but nevertheless preserve \( P \) . If \( X \) is a real vector field, then \( \left\lbrack {X, Y}\right\rbrack \) is the same as the Lie derivative \( {\mathcal{L}}_{X}\left( Y\right) \) . It is then not hard to show that \( X \) preserves \( P \) if and only if the flow generated by \( X \) preserves \( P \), that is, if and only if \( {\left( {\Phi }_{t}\right) }_{ * }\left( {P}_{z}\right) = {P}_{{\Phi }_{t}\left( z\right) } \) for all \( z \) and \( t \), where \( \Phi \) is the flow of \( X \) . Furthermore, if \( X \) is real, then \( X \) preserves \( P \) if and only if \( X \) preserves \( \bar{P} \) . Example 23.23 If \( N = {T}^{ * }M \) for some manifold \( M \) and \( P \) is the vertical polarization on \( N \), then a Hamiltonian vector field \( {X}_{f} \) preserves \( P \) if and only if \( f = {f}_{1} + {f}_{2} \), where \( {f}_{1} \) is constant on each fiber and \( {f}_{2} \) is linear on each fiber. Proof. In local coordinates \( \left\{ {{x}_{j},{p}_{j}}\right\} \), a vector field \( X \) lying in \( P \) has the form \( X = {g}_{j}\partial /\partial {p}_{j} \) . Thus, \[ \left\lbrack {{X}_{f}, X}\right\rbrack = \left\lbrack {\frac{\partial f}{\partial {p}_{j}}\frac{\partial }{\partial {x}_{j}},{g}_{k}\frac{\partial }{\partial {p}_{k}}}\right\rbrack - \left\lbrack {\frac{\partial f}{\partial {x}_{j}}\frac{\partial }{\partial {p}_{j}},{g}_{k}\frac{\partial }{\partial {p}_{k}}}\right\rbrack . \] This commutator will consist of three "good" terms, which involve only \( p \) -derivatives, along with the following "bad" term: \[ - {g}_{k}\frac{{\partial }^{2}f}{\partial {p}_{k}\partial {p}_{j}}\frac{\partial }{\partial {x}_{j}}. \] If \( {\partial }^{2}f/\partial {p}_{k}\partial {p}_{j} \) is 0 for all \( j \) and \( k \), then the bad term vanishes and \( \left\lbrack {{X}_{f}, X}\right\rbrack \) again lies in \( P \) . Conversely, if we want the bad term to vanish for each choice of the coefficient functions \( {g}_{j} \), we must have \( {\partial }^{2}f/\partial {p}_{k}\partial {p}_{j} = 0 \) for all \( j \) and \( k \) . Thus, for each fixed value of \( x, f \) must contain only terms that are independent of \( p \) and terms that are linear in \( p \) . ∎ We now identify the condition for quantizability of functions. Theorem 23.24 For any smooth, complex-valued function \( f \) on \( N \), if the Hamiltonian vector field \( {X}_{f} \) preserves \( \bar{P} \), then \( f \) is quantizable. Since we do not assume that \( f \) is real-valued, the condition that \( {X}_{f} \) preserve \( \bar{P} \) is not equivalent to the condition that \( {X}_{f} \) preserve \( P \) . Proof. Given a polarized section \( s \), we apply \( {Q}_{\text{pre }}\left( f\right) \) to \( s \) and then test whether \( {Q}_{\text{pre }}\left( f\right) s \) is still polarized, by applying \( {\nabla }_{X} \) for some vector field \( X \) lying in \( \bar{P} \) . To this end, it is useful to compute the commutator of \( {\nabla }_{X} \) and \( {Q}_{\text{pre }}\left( f\right) \), as follows: \[ \left\lbrack {{\nabla }_{X},{Q}_{\mathrm{{pre}}}\left( f\right) }\right\rbrack = i\hslash \left\lbrack {{\nabla }_{X},{\nabla }_{{X}_{f}}}\right\rbrack + \left\lbrack {{\nabla }_{X}, f}\right\rbrack \] \[ = i\hslash \left( {{\nabla }_{\left\lbrack X,{X}_{f}\right\rbrack } - \frac{i}{\hslash }\omega \left( {X,{X}_{f}}\right) }\right) + X\left( f\right) \] \[ = i\hslash {\nabla }_{\left\lbrack X,{X}_{f}\right\rbrack } \] (23.12) where we have used that \[ \omega \left( {X,{X}_{f}}\right) = - \omega \left( {{X}_{f}, X}\right) = - {df}\left( X\right) = - X\left( f\right) , \] by Definition 21.6. Since \( {X}_{f} \) preserves \( \bar{P} \), the vector field \( \left\lbrack {X,{X}_{f}}\right\rbrack \) again lies in \( \bar{P} \) and, thus, \[ {\nabla }_{X}\left( {{Q}_{\mathrm{{pre}}}\left( f\right) s}\right) = {Q}_{\mathrm{{pre}}}\left( f\right) {\nabla }_{X}s + i\hslash {\nabla }_{\left\lbrack X,{X}_{f}\right\rbrack }s = 0, \] for every polarized section \( s \), showing that \( {Q}_{\text{pre }}\left( f\right) s \) is again polarized. ∎ The converse of Theorem 23.24 is false in general. After all, as we will see in the following subsections, for a given polarization, there may not be any nonzero globally defined polarized sections, in which case, any function is quantizable. On the other hand, it can be shown that if \( {Q}_{\text{pre }}\left( f\right) \) preserves the space of locally defined polarized sections, then the Hamiltonian flow generated by \( f \) must preserve \( \bar{P} \) . This result follows by the same reasoning as in the proof of Theorem 23.24, once we know that there are sufficiently many locally defined polarized sections. We will establish such an existence result for purely real and purely complex polarizations in the following subsections; for the general case, see the discussion following Definition 9.1.1 in [45]. A special case of Theorem 23.24 is provided by "polarized functions," that is, functions \( f \) for which \( X\left( f\right) = 0 \) for all vector fields \( X \) lying in \( \bar{P} \) . For such an \( f \), the action of \( {Q}_{\text{pre }}\left( f\right) \) on the quantum space is simply multiplication by \( f \), as we anticipated in the introductory discussion in Sect. 23.4. Proposition 23.25 If \( f \) is a smooth, complex-valued function on \( N \) and the derivatives of \( f \) in the \( \bar{P} \) directions are zero, then \( {Q}_{\mathrm{{pre}}}\left( f\right) \) preserves the space \( P \) -polarized sections, and the restriction of \( {Q}_{\mathrm{{pre}}}\left( f\right) \) to this space is simply multiplication by \( f \) . We have already seen special cases of this result in the \( {\mathbb{R}}^{2n} \) case; see the discussion following Proposition 22.11. Proof. If the derivatives of \( f \) in the direction of \( \bar{P} \) are zero, then for \( X \in \bar{P} \) , we have \[ 0 = X\left( f\right) = {df}\left( X\right) = \omega \left( {{X}_{f}, X}\right) \] meaning that \( {X}_{f} \) is in the \( \omega \) -orthogonal complement of \( \bar{P} \) . But since \( \bar{P} \) is Lagrangian, this complement is just \( \bar{P} \) . Thus, \( {X}_{f} \) belongs to \( \bar{P} \) and, in particular, \( {X}_{f} \) preserves \( \bar{P} \), so that \( f \) is quantizable, by Theorem 23.24. Furthermore, \( {\nabla }_{{X}_{f}}s = 0 \) for any \( P \) -polarized section \( s \), leaving only the \( {fs} \) term in the formula for \( {Q}_{\text{pre }}\left( f\right) s \) . ## 23.5.2 The Real Case In the \( {\mathbb{R}}^{2n} \) case, we have already computed the space of polarized sections for the vertical polarization in Proposition 22.8. As we observed there, there are no nonzero polarized sections that are square integrable over \( {\mathbb{R}}^{2n} \) . The same difficulty is easily seen to arise for the vertical polarization on any cotangent bundle \( N = {T}^{ * }M \) . In Sect. 23.6, we will introduce half-forms to deal with this failure of square integrability. We now examine properties of general real polarizations. We will see that polarized sections always exist locally, but not always globally. Proposition 23.26 If \( P \) is a purely real polarization on \( N \), then for any \( {z}_{0} \in N \), there exist a neighborhood \( U \) of \( {z}_{0} \) and a \( P \) -polarized section \( s \) of \( L \) defined over \( U \) such that \( s\left( {z}_{0}\right) \neq 0 \) . Proof. According to the local form of the Frobenius theorem, we can find a neighborhood \( U \) of \( {z}_{0} \) and a diffeomorphism \( \Phi \) of \( U \) with a neighborhood \( V \) of the origin in \( {\mathbb{R}}^{n} \times {\mathbb{R}}^{n} \) such that under \( \Phi \), the polarization \( P \) looks like the vertical polarization. That is to say, for each \( z \in U \), the image of \( {P}_{z} \) under \( {\Phi }_{ * }\left( z\right) \) is just the span of the vectors \( \partial /\partial {y}_{1},\ldots ,\partial /\partial {y}_{n} \), where the \( y \) ’s are the coordinates on the second copy of \( {\mathbb{R}}^{n} \) . By shrinking \( U \) if necessary, we can assume that \( L \) can be trivialized over \( U \) and that the open set \( V \) is the product of a ball \( {B}_{1} \) centered at the origin in the first copy of \( {\mathbb{R}}^{n} \) with a ball \( {B}_{2} \) centered at the origin in the second copy of \( {\mathbb{R}}^{n} \) . Let \( \theta \) be the connection 1 -form for an isometric trivialization of \( L \) over \( U \) and let \( \widetilde{\theta } = {\left( {\Phi }^{-1}\right) }^{ * }\left( \theta \right) \) . Since the subspaces \( {P}_{z} \) are Lagrangian, the restriction of \( \widetilde{\theta } \) to the each set of the form \( \{ \mathbf{x}\} \times {B}_{2} \) is closed. Since \( {B}_{2} \) is simply connected, there exists, for each \( \mathbf{x} \in {B}_{1} \), a function \( {f}_{\mathbf{x}} \) on \( {B}_{2} \) such that the restriction of \( \widetilde{\theta } \) to \( \{ \mathbf{x}\} \times {B}_{2} \) equals \( d{f}_{\mathbf{x}} \) . If we assume that \( {f}_{\mathbf{x}}\left( 0\right) = 0 \), then \( {f}_{\mathbf{x}}\left( \mathbf{y}\right) \) will be smooth as a function of \( \left( {\mathbf{x},\mathbf{y}

1143_(GTM48)General Relativity for Mathematicians

Definition 6.4.4

Definition 6.4.4. A simple cosmological model is a cosmological model \( \left( {M,\mathcal{M}, z}\right) \) such that: (a) \( \left( {M, g}\right) \) is a simple cosmological spacetime,(b) \( M = {\mathbb{R}}^{3} \times \left( {0,\infty }\right) \) and \( R \rightarrow 0 \) as \( {u}^{4} \rightarrow 0 \) ; (c) \( {u}^{4}z = {10}^{10} \) years. Here the motivations for, and limitations of, assumption (a) have already been outlined (Sections 6.2.3 and 6.2.8). In (b), taking \( \mathcal{F} = \left( {0, a}\right), a \in (0,\infty \rbrack \) , involves no further loss of generality given (a) and the assumptions (Section 6.2.3) of a cosmological model (Exercise 6.2.13b). The remaining parts of (b) are that \( a = \infty \) and that \( R \rightarrow 0 \) as \( {u}^{4} \rightarrow 0 \) . Both of these can usually be proved once a sufficiently detailed matter model is chosen (cf. Proposition 6.2.7 and Exercise 6.2.11 for two examples). Thus (b) in effect acts as an extremely general assumption on \( \mathcal{M} \), which replaces the very specific requirement that \( \mathcal{M} \) be a dust. For a perfect fluid that obeys the requirements of Lemma 6.4.3, our assumption (b) implies that the energy density \( \rho \rightarrow \infty \) as \( {u}^{4} \rightarrow 0 \) . For this (and other) reasons we can again regard \( {u}^{4} \rightarrow 0 \) as approach to a big bang and expect that, qualitatively speaking, matter is becoming ever denser in this limit. Then, in view of Exercise 6.2.13, \( {u}^{4} \) can again be interpreted as maximum proper time since the big bang. Thus the motivations for, and limitations of, the final assumption \( {u}^{4}z = {10}^{10} \) years are just as before (Section 6.2.9). Let \( \left( {M,\mathcal{M}, z}\right) \) be a simple cosmological model. We again designate \( {u}^{4} \) as the cosmological time. Exercise 6.2.13 shows that the comoving reference frame is \( {\partial }_{4} \), that \( {\partial }_{4} \) is expanding, and that \( \dot{R} > 0 \) . We can now use a simple cosmological model to cure the particular disease indicated by Proposition 6.4.2. The resulting model, detailed in Proposition 6.4.5, is superior to the Einstein-de Sitter model or the modified Einstein-de Sitter model of Section 6.4.1 in that the microwave radiation is built into this model from the beginning in the form of a rest-mass zero perfect fluid. In particular, as a consequence of the influence of this rest-mass zero perfect fluid on spacetime (via the Einstein field equation), the resulting spacetime is no longer the Einstein-de Sitter spacetime. Of course, even this model has its own limitations (Sections 6.2.1a, b and 6.6). Proposition 6.4.5. Let \( \left( {M,\mathcal{M}, z}\right) \) be a simple cosmological model such that: (a) \( \mathcal{M} \) is a superposition of a dust \( {\mathcal{M}}_{1} = \left( {{\rho }_{1},{\partial }_{4}}\right) \) and a rest-mass zero perfect fluid \( {\mathcal{M}}_{2} = \left( {{\rho }_{2},{\rho }_{2}/3,{\partial }_{4}}\right) \) ; (b) the stress-energy tensor \( {\widehat{T}}_{2} \) of \( {\mathcal{M}}_{2} \) obeys \( \operatorname{div}{\widehat{\mathbf{T}}}_{2} = 0 \) ; (c) \( {\partial }_{\mu }{\rho }_{2} = 0 \), for \( \mu = 1,2,3 \) . Let a be the positive number defined by \( {\rho }_{2}z = a{\rho }_{1}z \) . Then, up to equivalence (Exercise 6.0.18) the following hold: (d) \( R \) is the function determined implicitly by \( u = {\left\lbrack R\left( u\right) + b\right\rbrack }^{1/2} \times \) \( \left\lbrack {R\left( u\right) - {2b}}\right\rbrack + 2{b}^{3/2} \), where \( b = a{\left\{ \left( {u}^{4}z\right) /\left\lbrack \left( 1 - 2a\right) {\left( a + 1\right) }^{1/2} + 2{a}^{3/2}\right\rbrack \right\} }^{2/3} \) ; (e) \( {\rho }_{1} = \left( {4{R}^{-3}/3}\right) \circ {u}^{4} \) ; and (f) \( {\rho }_{2} = \left( {{4b}{R}^{-4}/3}\right) \circ {u}^{4} \) . 6.4.6 Remarks Before giving the proof we make a few comments. (a) The interpretation of interest here is that \( {\mathcal{M}}_{1} \) models the matter in galaxies and \( {\mathcal{M}}_{2} \), as mentioned above, models the microwave photons. In this context, \( a \) is about \( {10}^{-4} \) and all three assumptions in Proposition 6.4.5. have been motivated earlier in this section. However, the proposition has other applications (cf. Section 6.6.4). (b) Note that conditions 6.4.5d-f determine \( \left( {M,\mathcal{M}, z}\right) \) uniquely if it exists. We are thus presenting our model in the form of a uniqueness theorem; Exercise 6.4.10 states the corresponding existence result and gives hints on its proof. (c) Note that as \( a \rightarrow 0 \), one has \( b \rightarrow 0,{\rho }_{2} \rightarrow 0 \), and \( R\left( u\right) \rightarrow {u}^{2/3} \) ; thus the whole model approaches the Einstein-de Sitter model, as one would expect. Proof of Proposition 6.4.5. Let \( \left( {M, z}\right) \) be a simple cosmological model such that Assumptions 6.4.5a-c hold. The Einstein field equation \( G = {T}_{1} + {T}_{2} \) implies div \( {\widehat{T}}_{1} = 0 \), because of (b) and div \( \widehat{G} = 0 \) ; it also implies \( {\partial }_{\mu }{\rho }_{1} = 0\forall \mu \in \left( {1,2,3}\right) \), because of (c) and Lemma 6.2.6b. Thus Lemma 6.4.3 implies both \( {\rho }_{1} = \left( {4{b}_{1}/3}\right) {\left( R \circ {u}^{4}\right) }^{-3} \) and \( {\rho }_{2} = \) \( \left( {4/3}\right) b{\left( R \circ {u}^{4}\right) }^{-4} \) for some \( {b}_{1}, b \in \left( {0,\infty }\right) \) . Using the diffeomorphism determined by \( {u}^{4} \rightarrow {u}^{4},{u}^{u} \rightarrow {\left( {b}_{1}\right) }^{-1/3}{u}^{u} \) for \( \mu = 1,2,3 \), we can and shall assume \( {b}_{1} = 1 \) without loss of generality (cf. Exercise 6.0.18a). Thus (e) and (f) hold. Using Lemma 6.2.6b again we now get \( 3{\left( \dot{R}/R\right) }^{2} = \) \( \left( {4/3}\right) \left( {{R}^{-3} + b{R}^{-4}}\right) \) . Since \( R \rightarrow 0 \) as \( {u}^{4} \rightarrow 0 \) we can integrate to obtain \( {u}^{4} = 3/2{\int }_{0}^{R}{vdv}/{\left( v + b\right) }^{1/2} = {\left( R + b\right) }^{1/2}\left( {R - {2b}}\right) + 2{b}^{3/2} \) . To complete the proof of (d), we note from (e) and (f) that \( {\rho }_{2}z = a{\rho }_{1}z \) iff \( b = {aR}\left( {{u}^{4}z}\right) \) iff \( b = a{\left\{ {10}^{10}/\left\lbrack {\left( 1 + a\right) }^{1/2}\left( 1 - 2a\right) + 2{a}^{3/2}\right\rbrack \right\} }^{2/3}. \) Now that we have a model (Proposition 6.4.5) that takes into account the influence of the microwave photons on spacetime, we must next make sure that, if we abandon the Einstein-de Sitter model we are not throwing out the baby with the bath water-that for the observations analyzed in Section 6.3 the model (Proposition 6.4.5) is no worse than the Einstein-de Sitter model. In fact such is the case. The observations mentioned in Section 6.3 concern only red shifts less than nine, in most cases very much less. Thus we are concerned with at most the \( {u}^{4} \) range determined by \( \left( {1/{10}}\right) R\left( {{u}^{4}z}\right) \leq R\left( {u}^{4}\right) \leq \) \( R\left( {{u}^{4}z}\right) \) (Exercise 6.0.17); here \( R \) is given by Proposition 6.4.5d with \( a \cong {10}^{-4} \) and \( {u}^{4}z = {10}^{10} \) years. Numerical estimates (using Taylor series) show that throughout this entire \( {u}^{4} \) range \( R\left( {u}^{4}\right) \) differs from \( {\left( {u}^{4}\right) }^{2/3} \) (the Einstein-de Sitter behavior) by less than \( 1\% \) . The same then holds for all the predicted effects in Section 6.3 and such small changes are negligible compared to the empirical uncertainties. As an example, note from Proposition 6.4.5d that \( a = {10}^{-4} \) gives \( b \cong {10}^{-4} \times {10}^{{20}/3} \) and thus \( R\left( {10}^{10}\right) \cong \left( {1 + {10}^{-4}}\right) {10}^{{20}/3} \) . The predicted value of the Hubble time is now \( \mathrm{t} = \left( {R/\dot{R}}\right) \left( {10}^{10}\right) \) (Exercise 6.3.17c). The model gives \( R/\dot{R} = R\left( {d{u}^{4}/{dR}}\right) = \left( {3/2}\right) {R}^{2}{\left( R + b\right) }^{-1/2} \) (Proposition 6.4.5d). Thus \( t \cong \left( {3/2}\right) \left( {1 + \frac{3}{2} \cdot {10}^{-4}}\right) \times {10}^{10} \) -that is, the predicted Hubble time differs from that predicted by the Einstein-de Sitter model by a bit more than \( {0.01}\% \) . Compared to the \( {15}\% \) observational inaccuracy of the Hubble constant (Section 6.1.7), and the still larger ## 6 Cosmology uncertainty in choosing \( {u}^{4}z = {10}^{10} \) years (Section 6.2.9), a discrepancy of \( \left( {3/2}\right) \times {10}^{-4} \) is grotesquely small. Thus the model (Proposition 6.4.5) shares the crude but important virtues of the Einstein-de Sitter model without having the disease diagnosed at the start of this section. Why didn't we use it ab initio? Because it is much clumsier than the Einstein-de Sitter model, is no better near here-now and, as we shall see shortly, has diseases of its own. ## EXERCISE 6.4.7. THE BIG BANG Let \( \left( {M,\mathcal{M}, z}\right) \) be a simple cosmological model such that \( \mathcal{M} \) is a superposition of perfect fluids \( \left\{ {\left( {{\rho }_{A},{p}_{A},{\partial }_{A}}\right) \mid A = 1,\ldots, N}\right\} \) . (a) Generalize Lemma 6.4.3 by showing \( \mathop{\sum }\limits_{{A = 1}}^{N}{\rho }_{A} \rightarrow \infty \) as \( {u}^{4} \rightarrow 0 \) . (Intuitively: "overall, matter gets denser as we approach the big bang".) (b) Use Example 3.12.1 to construct a case where \( N = 2 \) and \( {\rho }_{2} \rightarrow 0 \) as \( {u}^{4} \rightarrow 0 \) . (b) Shows that not every matter component need be present at the big bang " \( {u}^{4} = 0 \) ": some may be made later. In fact probably helium and deuterium are made later (Section 6.5). There is some indication that when quantum models are used for very early times in some general models (a) can fail: conceivably all forms of matter were made, by quantum process, after the big bang. S