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Proposition 6.15. (a) Let \( a, b \in K \) and let \( E \) be the curve given by the Weierstrass equation (6.11). Then \( E \) is nonsingular, and thus is an elliptic curve, if and only if \( \Delta \left( E\right) \neq 0 \) . | Proof. See [410, III §1]. | No |
Theorem 6.16. Let \( E/K \) be an elliptic curve defined over a field \( K \). (a) The addition law described above gives \( E = E\left( \bar{K}\right) \) the structure of an abelian group. (b) The group law is algebraic, in the sense that the addition and inversion maps, \[ E \times E\xrightarrow[]{\left( {P, Q}\right... | Proof. See [410, III §§2,3]. | No |
Proposition 6.17. (Elliptic Curve Group Law Algorithm) Let \( E \) be an elliptic curve given by a Weierstrass equation\n\n\[ E : {y}^{2} = {x}^{3} + {ax} + b, \]\n\nand let \( {P}_{1} = \left( {{x}_{1},{y}_{1}}\right) \) and \( {P}_{2} = \left( {{x}_{2},{y}_{2}}\right) \) be points on \( E \) .\n\nIf \( {x}_{1} = {x}_... | Proof. See [410, III.2.3] | No |
Proposition 6.18. Let \( E \) be an elliptic curve.\n\n(a) Every principal divisor on \( E \) has degree 0 .\n\n(b) \( A \) divisor \( D \in \operatorname{Div}\left( E\right) \) is principal if and only if both \( \deg \left( D\right) = 0 \) and \( \operatorname{sum}\left( D\right) = \mathcal{O} \) . | Proof. See [410, III.3.4 and III.3.5]. | No |
Theorem 6.21. An isogeny \( \psi : {E}_{1} \rightarrow {E}_{2} \) is a homomorphism of groups, i.e.,\n\n\[ \psi \left( {P + Q}\right) = \psi \left( P\right) + \psi \left( Q\right) \;\text{ for all }P, Q \in {E}_{1}\left( \bar{K}\right) . \] | Proof. See [410, III.4.8] | No |
Theorem 6.22. Let \( \psi : {E}_{1} \rightarrow {E}_{2} \) be an isogeny of degree \( d \) . Then there is a unique isogeny \( \widehat{\psi } : {E}_{2} \rightarrow {E}_{1} \), called the dual isogeny of \( \psi \), with the property that\n\n\[ \widehat{\psi }\left( {\psi \left( P\right) }\right) = \left\lbrack d\right... | Proof. See [410, III §6]. | No |
The elliptic curve \( E : {y}^{2} = {x}^{3} + x \) has CM, since the endomorphism\n\n\[ \psi : E \rightarrow E,\;\psi \left( {x, y}\right) = \left( {-x,{iy}}\right) ,\]\n\nis not in \( \mathbb{Z} \) . | An easy way to verify this assertion is to note that\n\n\[ {\psi }^{2}\left( {x, y}\right) = \left( {x, - y}\right) = - \left( {x, y}\right) \]\n\nso \( {\psi }^{2} = \left\lbrack {-1}\right\rbrack \) . This gives an embedding of the Gaussian integers \( \mathbb{Z}\left\lbrack i\right\rbrack \) into \( \operatorname{En... | Yes |
Proposition 6.25. Let \( E/K \) be an elliptic curve. Then the endomorphism ring of \( E \) is one of the following three kinds of rings:\n\n(a) \( \operatorname{End}\left( E\right) = \mathbb{Z} \).\n\n(b) \( \operatorname{End}\left( E\right) \) is an order in a quadratic imaginary field \( F \). This means that \( \op... | Proof. See [410, III §9]. | No |
Proposition 6.26. Let \( K \) be a field whose characteristic is not equal to 2 or 3 and let \( E/K \) be an elliptic curve. Then\n\n\[ \n\\operatorname{Aut}\\left( E\\right) = \\left\\{ \\begin{array}{ll} {\\mathbf{\\mu }}_{2} & \\text{ if }j\\left( E\\right) \\neq 0\\text{ and }j\\left( E\\right) \\neq {1728}, \\\\ {... | Proof. See [410, III §10]. | No |
Proposition 6.29. If \( E \) has good reduction, then the reduction modulo \( \mathfrak{p} \) map \( E\left( K\right) \rightarrow \widetilde{E}\left( k\right) \) is a homomorphism. | Proof. See [410, VII.2.1]. | No |
Theorem 6.31. Let \( E/K \) be an elliptic curve and assume that either \( K \) has characteristic 0 or else that \( K \) has characteristic \( p > 0 \) and \( p \nmid m \) . Then as an abstract group,\n\n\[ E\left\lbrack m\right\rbrack = \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z} \]\n\nIn other words, \( E\l... | Proof. See [410, III.6.4]. | No |
Theorem 6.32. Let \( K \) be a local field whose residue field has characteristic \( p \) , let \( E/K \) be an elliptic curve with good reduction, and let \( m \geq 1 \) be an integer with \( p \nmid m \) . Let \( E\left( K\right) \left\lbrack m\right\rbrack \) denote the subgroup of \( E\left\lbrack m\right\rbrack \)... | Proof. See [410, VII.3.1]. | No |
Corollary 6.33. Let \( K \) be a local field whose residue field has characteristic \( p \) , let \( E/K \) be an elliptic curve with good reduction, and let \( m \geq 1 \) be an integer with \( p \nmid m \) . Then the field \( K\left( {E\left\lbrack m\right\rbrack }\right) \) obtained by adjoining to \( K \) the coord... | Proof Sketch. Let \( {K}^{\prime } = K\left( {E\left\lbrack m\right\rbrack }\right) \), let \( {\mathfrak{p}}^{\prime } \) be the maximal ideal of the ring of integers of \( {K}^{\prime } \), and let \( {k}^{\prime } \) be the residue field. Suppose that \( \sigma \in \operatorname{Gal}\left( {{K}^{\prime }/K}\right) \... | Yes |
Theorem 6.35. Let \( E \) be an elliptic curve given by a Weierstrass equation and let \( {\omega }_{E} \) be the associated invariant differential on \( E \). (a) For any given point \( Q \in E \), let \( {\tau }_{Q} : E \rightarrow E \) be the translation-by- \( Q \) map defined by \( {\tau }_{Q}\left( P\right) = P +... | Proof. See [410, III §5]. | No |
Proposition 6.37. Let \( \Gamma \) be a nontrivial subgroup of \( \operatorname{Aut}\left( E\right) \). Then the quotient curve \( E/\Gamma \) is isomorphic to \( {\mathbb{P}}^{1} \) and the projection map \( \pi : E \rightarrow E/\Gamma \cong {\mathbb{P}}^{1} \) is given explicitly by\n\n\[ \pi \left( {x, y}\right) = ... | Proof. By definition, the quotient curve \( E/\Gamma \) is the curve whose function field is the subfield of \( K\left( E\right) = K\left( {x, y}\right) \) fixed by \( \Gamma \). Using the explicit description (6.13) of the action of \( \operatorname{Aut}\left( E\right) \) on the coordinates of \( E \), it is easy to f... | Yes |
Proposition 6.39. Let \( E/\mathbb{C} \) be an elliptic curve with complex multiplication, i.e., the endomorphism ring \( \operatorname{End}\left( E\right) \) is strictly larger than \( \mathbb{Z} \). Choose a lattice \( L \subset \mathbb{C} \) such that \( E\left( \mathbb{C}\right) \cong \mathbb{C}/L \), let\n\n\[ R =... | Proof. To describe the ring \( R \), we choose a basis for \( L \), say \( L = \mathbb{Z}{\omega }_{1} + \mathbb{Z}{\omega }_{2} \). We have \( R \neq \mathbb{Z} \) by assumption, so there exists an \( \alpha \in R \) with \( \alpha \notin \mathbb{Z} \). Write\n\n\[ \alpha {\omega }_{1} = a{\omega }_{1} + b{\omega }_{2... | Yes |
Proposition 6.40. Let \( F \) be a quadratic imaginary field with ring of integers \( {R}_{F} \) and ideal class group \( {\mathcal{C}}_{F} \), and let \( {h}_{F} = \# {\mathcal{C}}_{F} \) be the class number of \( F \) . Then with notation as above, the natural map\n\n\[ \n{\mathcal{C}}_{F} \rightarrow \mathcal{E}\ell... | Proof. See [412, II.1.2]. | No |
Let \( E : {y}^{2} = {x}^{3} + {ax} + b \) be an elliptic curve. Then the classical formula for \( x\left( {2P}\right) \) (Proposition 6.17) and the isomorphism \( x : E/\{ \pm 1\} \rightarrow {\mathbb{P}}^{1} \) yield the Lattès map | \[ \phi \left( x\right) = x\left( {2P}\right) = \frac{{x}^{4} - {2a}{x}^{2} - {8bx} + {a}^{2}}{4{x}^{3} + {4ax} + {4b}}. \] | Yes |
Let \( E \) be the elliptic curve \( E : {y}^{2} = {x}^{3} + {ax} \) with \( j\left( E\right) = {1728} \) and again let \( \psi \left( P\right) = \left\lbrack 2\right\rbrack P \) be the doubling map. If we take \( \pi \left( {x, y}\right) = x \), then we are in the \( b = 0 \) case of Example 6.41, and we obtain the La... | Note that the map \( \pi \left( {x, y}\right) = {x}^{2} \) corresponds to taking the quotient of \( E \) by its automorphism group \( \operatorname{Aut}\left( E\right) \cong {\mathbf{\mu }}_{4} \) via the association described in Remark 6.27. | Yes |
Theorem 6.46. Let \( K \) be an algebraically closed field of characteristic not equal to 2 and let \( \phi \) and \( {\phi }^{\prime } \) be Lattès maps defined over \( K \) that are associated, respectively, to elliptic curves \( E \) and \( {E}^{\prime } \) . Assume further that the projection maps \( \pi \) and \( ... | Proof. Let \( f \in {\operatorname{PGL}}_{2}\left( K\right) \) be a linear fractional transformation conjugating \( {\phi }^{\prime } \) to \( \phi \) . Then we have a commutative diagram\n\n\n\nWe let \( {\pi }^{\pr... | Yes |
We saw in Example 6.41 that the Lattès function associated to the duplication map \( \psi \left( P\right) = \left\lbrack 2\right\rbrack \left( P\right) \) on the elliptic curve \( E : {y}^{2} = {x}^{3} + {ax} + b \) is given by the formula \[ {\phi }_{a, b}\left( x\right) = x\left( {2P}\right) = \frac{{x}^{4} - {2a}{x}... | More precisely, the set of maps \( {\phi }_{a, b} \) is a two-dimensional algebraic family of points in the space \( {\operatorname{Rat}}_{4} \), given explicitly by \[ {\mathbb{A}}^{2} \rightarrow {\operatorname{Rat}}_{4} \subset {\mathbb{P}}^{9},\;\left( {a, b}\right) \mapsto \left\lbrack {1,0, - {2a}, - {8b},{a}^{2}... | Yes |
The elliptic curve \( E : {y}^{2} = {x}^{3} + a{x}^{2} + {bx} \) has the 2-torsion point \( T = \left( {0,0}\right) \). To compute the Lattès function \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) associated to the translated duplication map \( \psi \left( P\right) = \left\lbrack 2\right\rbrack \left( P\ri... | we first use the classical duplication formula to compute\n\n\[ \n{2P} = \left( {\frac{{x}^{4} - {2b}{x}^{2} + {b}^{2}}{4{y}^{2}},\frac{{x}^{6} + {2a}{x}^{5} + {5b}{x}^{4} - 5{b}^{2}{x}^{2} - {2a}{b}^{2}x - {b}^{3}}{8{y}^{3}}}\right) .\n\]\n\nThen the addition formula and some algebra yield\n\n\[ \n\phi \left( x\right)... | Yes |
Proposition 6.51. Let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a flexible Lattès map whose associated map \( \psi : E \rightarrow E \) has the form \( \psi \left( P\right) = \left\lbrack m\right\rbrack P + T \) .\n\n(a) The map \( \phi \) has degree \( {m}^{2} \) .\n\n(b) The point \( T \) satisfie... | Proof. (a) The commutativity of the diagram (6.22) tells us that\n\n\[ \deg \left( \phi \right) \deg \left( \pi \right) = \deg \left( \pi \right) \deg \left( \psi \right) \]\n\nThe map \( \psi \) has degree \( {m}^{2} \), since multiplication-by- \( m \) has degree \( {m}^{2} \) and translation-by- \( T \) has degree 1... | Yes |
Proposition 6.52. Let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a flexible Lattès map and assume that \( T = \mathcal{O} \) , so \( \psi \left( P\right) = \left\lbrack m\right\rbrack \left( P\right) \) . (See Exercise 6.18 for the case \( T \neq \mathcal{O} \) .)\n\n(a) The set of \( n \) -periodic ... | Proof. (a) Let \( \zeta \in {\mathbb{P}}^{1} \) be a fixed point of \( \phi \) and choose a point \( P \in E \) with \( \pi \left( P\right) = \zeta \) . Note that there are generally two choices for \( P \), so we simply choose either one of them. Then\n\n\[ \n\pi \left( P\right) = \zeta = \phi \left( \zeta \right) = \... | Yes |
Theorem 6.57. Let \( K \) be a field of characteristic 0 and let \( \phi \) be a Lattès map defined over \( K \) . Then there exists a commutative diagram of the form (6.36) such that the map \( \pi \) has the form\n\n\[ \pi : E \rightarrow E/\Gamma \overset{ \sim }{ \rightarrow }{\mathbb{P}}^{1} \]\n\nfor some nontriv... | Proof. For a proof over \( \mathbb{C} \), see [300, Theorem 3.1]. The general case for characteristic-0 fields follows by the Lefschetz principle, cf. [410, VI §6]. | No |
Corollary 6.58. Let \( \phi \) be a Lattès map given by a reduced diagram (6.37). Then the point \( \psi \left( \mathcal{O}\right) \) is fixed by every element of \( \Gamma \), so in particular, \( \psi \left( \mathcal{O}\right) \in {E}_{\text{tors }} \) . If further \( j\left( E\right) \neq 0 \) and \( j\left( E\right... | Proof. We defer the proof that \( \psi \left( \mathcal{O}\right) \) is fixed by every \( \xi \in \Gamma \) until Proposition \( {6.77}\left( \mathrm{\;b}\right) \), where we prove it in a much more general setting. (Cf. the proof for flexible Lattès maps in Proposition 6.51(b).) To see that \( \psi \left( \mathcal{O}\r... | No |
Proposition 6.61. Let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a Lattès map and fix a reduced Lattès diagram (6.37) for \( \phi \) . We assume that \( j\left( E\right) \neq 0 \) and \( j\left( E\right) \neq {1728} \) . We further assume that \( \psi \) is an isogeny, i.e., with our usual notation \... | Proof. (a) We have \( \pi \left( P\right) \in \operatorname{Fix}\left( \phi \right) \) if and only if\n\n\[ \pi \left( P\right) = \phi \left( {\pi \left( P\right) }\right) = \pi \left( {\psi \left( P\right) }\right) . \]\n\nOur assumption on \( j\left( E\right) \) means that \( \Gamma = \operatorname{Aut}\left( E\right... | No |
Theorem 6.63. (Mazur-Kamienny-Merel) For all integers \( D \geq 1 \) there is a constant \( B\left( D\right) \) such that for all number fields \( K/\mathbb{Q} \) of degree at most \( D \) and all elliptic curves \( E/K \) we have\n\n\[ \n\# E{\left( K\right) }_{\text{tors }} \leq B\left( D\right) \n\] | Discussion. This deep result was first proven by Mazur [292] for \( K = \mathbb{Q} \), then by Kamienny [225] for \( \left\lbrack {K : \mathbb{Q}}\right\rbrack = 2 \), and then was extended to various specific larger degrees before the proof was completed for all degrees by Merel [297]. The proof uses the theory of mod... | No |
Corollary 6.64. For all integers \( n \geq 1 \), all number fields \( K/\mathbb{Q} \), and all elliptic curves \( E/K \) we have\n\n\[ \n\# \left( {\mathop{\bigcup }\limits_{{\left\lbrack {L : K}\right\rbrack \leq n}}E{\left( L\right) }_{\text{tors }}}\right) \leq B{\left( n\left\lbrack K : \mathbb{Q}\right\rbrack \rig... | Proof. To ease notation, we let \( D = \left\lbrack {K : \mathbb{Q}}\right\rbrack \) . Every field \( L \) appearing in the union in (6.41) satisfies\n\n\[ \n\left\lbrack {L : \mathbb{Q}}\right\rbrack = \left\lbrack {L : K}\right\rbrack \left\lbrack {K : \mathbb{Q}}\right\rbrack \leq {nD},\n\]\n\nso Theorem 6.63 tells ... | Yes |
Theorem 6.65. Let \( D \geq 1 \) be an integer. There is a constant \( C\left( D\right) \) such that for all number fields \( K/\mathbb{Q} \) of degree \( D \) and all Lattès maps \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) defined over \( K \) we have\n\n\[ \n\# \operatorname{PrePer}\left( {\phi ,{\math... | Proof. Without loss of generality we fix a reduced Lattès diagram (6.37) for \( \phi \) . Then Proposition 6.26 says that the projection map \( \pi : E \rightarrow {\mathbb{P}}^{1} \) has degree at most 6, and indeed if \( j\left( E\right) \neq 0 \) and \( j\left( E\right) \neq {1728} \), then \( \deg \left( \pi \right... | Yes |
Theorem 6.67. Let \( K \) be a number field of degree \( D \geq 2 \), let \( E/K \) be an elliptic curve whose \( j \) -invariant is an algebraic integer, and let \( \phi \) be a Lattès map associated to \( E \) . Then there is an absolute constant \( c \) such that\n\n\[ \n\# \operatorname{PrePer}\left( {\phi ,{\mathb... | Proof. The assumption that the elliptic curve \( E \) has integral \( j \) -invariant means that it has everywhere potential good reduction. Replacing \( K \) by an extension of bounded degree, we may assume that \( E \) has everywhere good reduction. (In fact, it suffices to go to the field \( K\left( {E\left\lbrack 3... | Yes |
Theorem 6.70. Let \( E/\mathbb{Q} \) be an elliptic curve given by a Weierstrass equation with integer coefficients, let \( m \geq 2 \) be an integer, and let \( \phi \left( z\right) \in \mathbb{Q}\left( z\right) \) be the Lattès map satisfying\n\n\[ \phi \left( {x\left( P\right) }\right) = x\left( {\left\lbrack m\righ... | Proof. Write the given Weierstrass equation for \( E \) as\n\n\[ E : {y}^{2} = {x}^{3} + {ax} + b\;\text{ with }a, b \in \mathbb{Z}. \]\n\nWe begin by showing that the minimality assumption (6.44) implies that there are no primes \( p \) with \( {p}^{2}\left| {a\text{and}{p}^{3}}\right| b \) . The rational function \( ... | Yes |
Proposition 6.75. Let \( \psi : G \rightarrow G \) be an affine morphism of an algebraic group \( G \) , so \( \psi \) has the form \( \psi \left( z\right) = a \cdot \alpha \left( z\right) \) for some \( \alpha \in \operatorname{End}\left( G\right) \) and some \( a \in G \) .\n\n(a) The endomorphism \( \alpha \) and tr... | Proof. The definition of affine morphism tells us that there are an element \( a \in G \) and an endomorphism \( \alpha \) of \( G \) such that the map \( \psi \) has the form \( \psi \left( z\right) = {a\alpha }\left( z\right) \) . Evaluating at the identity element \( e \in G \) yields \( \psi \left( e\right) = {a\al... | Yes |
Proposition 6.77. Let \( \phi : V \rightarrow V \) be a dynamically affine map and let \( \psi : G \rightarrow G \) and \( \Gamma \subset \operatorname{Aut}\left( G\right) \) be the associated quantities fitting into the commutative diagram (6.48).\n\n(a) For every \( \xi \in \Gamma \) there exists a unique \( {\xi }^{... | Proof. (a) The uniqueness is clear, since if \( \psi \circ \xi = {\xi }_{1} \circ \psi = {\xi }_{2} \circ \psi \), then \( {\xi }_{1} = {\xi }_{2} \) because the finite map \( \psi : G \rightarrow G \) is surjective.\n\nWe now prove the existence. The commutativity of (6.48) tells us that for all \( z \in \) \( G \) an... | Yes |
Theorem 6.79. (Ritt and Erëmenko) Let \( \phi ,\psi \in \mathbb{C}\left( z\right) \) be rational maps of degree at least 2 with the property that \( \phi \circ \psi = \psi \circ \phi \) . Then one of the following two conditions is true:\n\n(a) There are integers \( m, n \geq 1 \) such that \( {\phi }^{n} = {\psi }^{m}... | Proof. The first part of the theorem, in somewhat different language, is due to Ritt [371]. See Erëmenko's paper [152] for a proof of both parts of the theorem and some additional geometric dynamical properties shared by commuting \( \phi \) and \( \psi \) . A higher-dimensional analogue is discussed in [135]. We remar... | Yes |
Theorem 6.80. Let \( K \) be a field, let \( \phi \left( z\right) \in K\left\lbrack z\right\rbrack \) be a polynomial of degree \( d \geq 2 \) , and let \( \psi \left( z\right) \in K\left( z\right) \) be a nonconstant rational map. We assume that both \( \phi \) and \( \psi \) are separable, i.e., neither of the deriva... | Proof. The proof is an application of ramification theory and the Riemann-Hurwitz formula (Theorem 1.1). By assumption, the map \( \phi \) is a polynomial, so \( \infty \) is a totally ramified fixed point of \( \phi \) . Suppose that \( \psi \left( z\right) \) is not a polynomial. This means that we can find a point \... | Yes |
The rational map \[ \phi : {\mathbb{P}}^{2} \rightarrow {\mathbb{P}}^{2},\;\phi \left( \left\lbrack {{X}_{0},{X}_{1},{X}_{2}}\right\rbrack \right) = \left\lbrack {{X}_{0}^{2},{X}_{0}{X}_{1},{X}_{2}^{2}}\right\rbrack \] is not a morphism, since it is not defined at the point \( \left\lbrack {0,1,0}\right\rbrack \) . | Notice that if we discard \( \left\lbrack {0,1,0}\right\rbrack \), then \( \phi \) fixes every point on the line \( {X}_{0} = {X}_{2} \), and \( \phi \) sends every point on the line \( {X}_{0} = 0 \) to the single point \( \left\lbrack {0,0,1}\right\rbrack \) . This kind of behavior is not possible for morphisms \( {\... | Yes |
Consider the map \( \phi \left( {x, y}\right) = \left( {x, y + {x}^{2}}\right) \). It has degree 2 and is an automorphism, since it has the inverse \( {\phi }^{-1}\left( {x, y}\right) = \left( {x, y - {x}^{2}}\right) \). The composition \( {\phi }^{2} \) is | \[ {\phi }^{2}\left( {x, y}\right) = \phi \left( {x, y + {x}^{2}}\right) = \left( {x, y + 2{x}^{2}}\right) ,\] so \( \deg \left( {\phi }^{2}\right) = 2 = \deg \left( \phi \right) \). More generally, \( {\phi }^{n}\left( {x, y}\right) = \left( {x, y + n{x}^{2}}\right) \) has degree 2, so the degree of \( {\phi }^{n} \) ... | Yes |
Let \( a \in {K}^{ * } \) and let \( f\left( y\right) \in K\left\lbrack y\right\rbrack \) be a polynomial of degree \( d \geq 2 \) . The map \[ \phi : {\mathbb{A}}^{2} \rightarrow {\mathbb{A}}^{2},\;\phi \left( {x, y}\right) = \left( {y,{ax} + f\left( y\right) }\right) ,\] is called a Hénon map. It is an automorphism o... | It is an automorphism of \( {\mathbb{A}}^{2} \), since one easily checks that it has an inverse \( {\phi }^{-1} \) given by \[ {\phi }^{-1} : {\mathbb{A}}^{2} \rightarrow {\mathbb{A}}^{2},\;{\phi }^{-1}\left( {x, y}\right) = \left( {{a}^{-1}y - {a}^{-1}f\left( x\right), x}\right) . \] | Yes |
Consider the very simple Hénon map\n\n\[ \phi \left( {x, y}\right) = \left( {y, - x + {y}^{2}}\right) . \]\n\nThe extension \( \bar{\phi } = \left\lbrack {{X}_{0}^{2},{X}_{0}{X}_{2}, - {X}_{0}{X}_{1} + {X}_{2}^{2}}\right\rbrack \) of \( \phi \) to \( {\mathbb{P}}^{2} \) has degree 2, but it is not a morphism, since it ... | We can see this by noting that\n\n\[ \bar{\phi }\left( \left\lbrack {b, a, b}\right\rbrack \right) = \left\lbrack {{b}^{2},{b}^{2}, - {ab} + {b}^{2}}\right\rbrack = \left\lbrack {b, b, - a + b}\right\rbrack , \]\n\nso if \( a, b, \in \mathbb{Z} \) with \( \gcd \left( {a, b}\right) = 1 \) and \( b > a > 0 \), then \( \l... | Yes |
More generally, if \( \phi : {\mathbb{A}}^{N} \rightarrow {\mathbb{A}}^{N} \) is an affine automorphism, then it is not possible to have simultaneous estimates of the form\n\n\[ h\left( {\phi \left( P\right) }\right) \geq \left( {1 + \epsilon }\right) h\left( P\right) + O\left( 1\right) \]\n\n\[ h\left( {{\phi }^{-1}\l... | To see this, suppose that (7.2) were true. Then we would have for all \( P \in {\mathbb{A}}^{N}\left( K\right) \),\n\n\[ h\left( P\right) = h\left( {\phi \left( {{\phi }^{-1}\left( P\right) }\right) }\right) \geq \left( {1 + \epsilon }\right) h\left( {{\phi }^{-1}\left( P\right) }\right) + O\left( 1\right) \geq {\left(... | Yes |
Lemma 7.7. Let \( \phi : {\mathbb{A}}^{N} \rightarrow {\mathbb{A}}^{N} \) be an affine automorphism of degree at least 2 and denote the hyperplane at infinity by \( {H}_{0} = \left\{ {{X}_{0} = 0}\right\} = {\mathbb{P}}^{N} \smallsetminus {\mathbb{A}}^{N} \). Then\n\n\[ \bar{\phi }\left( {{H}_{0} \smallsetminus Z\left(... | Proof. Let\n\n\[ \Phi = \left( {{X}_{0}^{d},{\bar{F}}_{1},{\bar{F}}_{2},\ldots ,{\bar{F}}_{N}}\right) \;\text{ and }\;{\Phi }^{-1} = \left( {{X}_{0}^{e},{\bar{G}}_{1},{\bar{G}}_{2},\ldots ,{\bar{G}}_{N}}\right) \]\n\nbe the lifts of \( \bar{\phi } \) and \( {\bar{\phi }}^{-1} \), respectively. The fact that \( \phi \) ... | Yes |
Lemma 7.8. Let \( \phi : {\mathbb{A}}^{N} \rightarrow {\mathbb{A}}^{N} \) and \( \psi : {\mathbb{A}}^{N} \rightarrow {\mathbb{A}}^{N} \) be affine morphisms, and let \( {H}_{0} = \left\{ {{X}_{0} = 0}\right\} = {\mathbb{P}}^{N} \smallsetminus {\mathbb{A}}^{N} \) be the usual hyperplane at infinity. Then \[ \deg \left( ... | Proof. Let \( d = \deg \left( \phi \right) \), let \( e = \deg \left( \psi \right) \), and let \( \Phi \) and \( \Psi \) be lifts of \( \bar{\phi } \) and \( \bar{\psi } \) , respectively. We write \( \Phi \) explicitly as \[ \Phi = \left( {{X}_{0}^{d},{\bar{F}}_{1},{\bar{F}}_{2},\ldots ,{\bar{F}}_{N}}\right) \] The co... | Yes |
Let \( \phi \) be the map \( \phi \left( {x, y}\right) = \left( {x, y + {x}^{2}}\right) \) that we studied in Example 7.3. Dehomogenizing \( \phi \) yields \[ \bar{\phi }\left( \left\lbrack {{X}_{0},{X}_{1},{X}_{2}}\right\rbrack \right) = \left\lbrack {{X}_{0}^{2},{X}_{0}{X}_{1},{X}_{0}{X}_{2} + {X}_{1}^{2}}\right\rbra... | Notice that \[ \bar{\phi }\left( \left\lbrack {0,{X}_{1},{X}_{2}}\right\rbrack \right) = \left\lbrack {0,0,{X}_{1}^{2}}\right\rbrack = \left\lbrack {0,0,1}\right\rbrack \in Z\left( \phi \right) . \] Hence \( \bar{\phi }\left( {{H}_{0} \smallsetminus Z\left( \phi \right) }\right) = Z\left( \phi \right) \), so Lemma 7.8 ... | Yes |
Let \( \phi : {\mathbb{A}}^{3} \rightarrow {\mathbb{A}}^{3} \) be given by\n\n\[ \phi \left( {x, y, z}\right) = \left( {y, z + {y}^{2}, x + {z}^{2}}\right) . \]\n\nOne can check that the inverse of \( \phi \) is\n\n\[ {\phi }^{-1}\left( {x, y, z}\right) = \left( {z - {\left( y - {x}^{2}\right) }^{2}, x, y - {x}^{2}}\ri... | Homogenizing \( x = {X}_{1}/{X}_{0}, y = {X}_{2}/{X}_{0}, z = {X}_{3}/{X}_{0} \), we have the formulas\n\n\[ \bar{\phi } = \left\lbrack {{X}_{0}^{2},{X}_{0}{X}_{2},{X}_{0}{X}_{3} + {X}_{2}^{2},{X}_{0}{X}_{1} + {X}_{3}^{2}}\right\rbrack \]\n\n\[ {\bar{\phi }}^{-1} = \left\lbrack {{X}_{0}^{4},{X}_{0}^{3}{X}_{3} - {\left(... | Yes |
Theorem 7.15. Let \( {\phi }_{1} : {\mathbb{A}}^{N} \rightarrow {\mathbb{A}}^{N} \) and \( {\phi }_{2} : {\mathbb{A}}^{N} \rightarrow {\mathbb{A}}^{N} \) be affine morphisms with the property that\n\n\[ Z\left( {\phi }_{1}\right) \cap Z\left( {\phi }_{2}\right) = \varnothing \]\n\n(We say that \( {\phi }_{1} \) and \( ... | Proof of Theorem 7.15. Write the rational functions \( {\mathbb{P}}^{N} \rightarrow {\mathbb{P}}^{N} \) induced by \( {\phi }_{1} \) and \( {\phi }_{2} \) as\n\n\[ {\bar{\phi }}_{1} = \left\lbrack {{X}_{0}^{{d}_{1}},{\bar{F}}_{1},{\bar{F}}_{2},\ldots ,{\bar{F}}_{N}}\right\rbrack \;\text{ and }\;{\bar{\phi }}_{2} = \lef... | Yes |
Lemma 7.17. Let \( u,{a}_{1},\ldots ,{a}_{N},{b}_{1},\ldots ,{b}_{N} \in \overline{\mathbb{Q}} \) with \( u \neq 0 \) . Then\n\n\[ h\left( \left\lbrack {u,{a}_{1},\ldots ,{a}_{N},{b}_{1},\ldots ,{b}_{N}}\right\rbrack \right) \leq h\left( \left\lbrack {u,{a}_{1},\ldots ,{a}_{N}}\right\rbrack \right) + h\left( \left\lbra... | Proof. Let \( {\alpha }_{i} = {a}_{i}/u \) and \( {\beta }_{i} = {b}_{i}/u \) for \( 1 \leq i \leq N \) . Then for any absolute value \( v \) we have the trivial estimate\n\n\[ \max \left\{ {1,{\left| {\alpha }_{1}\right| }_{v},\ldots ,{\left| {\alpha }_{N}\right| }_{v},{\left| {\beta }_{1}\right| }_{v},\ldots ,{\left|... | Yes |
A prime divisor \( W \) of \( {\mathbb{P}}^{N} \) is the zero set of an irreducible homogeneous polynomial \( F \in K\left\lbrack {{X}_{0},\ldots ,{X}_{N}}\right\rbrack \) . We define the degree of \( W \) to be the degree of the polynomial \( F \) and extend this to obtain a homomorphism\n\n\[ \deg : \operatorname{Div... | It is not hard to see that a divisor on \( {\mathbb{P}}^{N} \) is principal if and only if it has degree 0, so the degree map gives an isomorphism\n\n\[ \deg : \operatorname{Pic}\left( {\mathbb{P}}^{N}\right) \overset{ \sim }{ \rightarrow }\mathbb{Z} \] | Yes |
A prime divisor of \( {\mathbb{P}}^{N} \times {\mathbb{P}}^{M} \) is the zero set of an irreducible bihomo-geneous polynomial \( F \in K\left\lbrack {{X}_{0},\ldots ,{X}_{N},{Y}_{0},\ldots ,{Y}_{M}}\right\rbrack \) . We say that \( F \) and \( W \) have bidegree \( \left( {d, e}\right) \) if \( F \) satisfies\n\n\[ F\l... | The bidegree map can be extended linearly to give an isomorphism\n\n\[ \text{ bideg } : \operatorname{Pic}\left( {{\mathbb{P}}^{N} \times {\mathbb{P}}^{M}}\right) \overset{ \sim }{ \rightarrow }\mathbb{Z} \times \mathbb{Z}\text{. } \] | Yes |
Let \( {p}_{1}^{ * }{H}_{1} \) and \( {p}_{2}^{ * }{H}_{2} \) be the generators of \( \operatorname{Pic}\left( {{\mathbb{P}}^{N} \times {\mathbb{P}}^{M}}\right) \) described in Example 7.27. Then \( {p}_{1}^{ * }{H}_{1} + {p}_{2}^{ * }{H}_{2} \) is a very ample divisor on \( {\mathbb{P}}^{N} \times {\mathbb{P}}^{M} \) ... | The associated embedding is called the Segre embedding. It is given explicitly by the formula\n\n\[ \n{\mathbb{P}}^{N} \times {\mathbb{P}}^{M}\; \rightarrow \;{\mathbb{P}}^{{NM} + N + M} \n\]\n\n\[ \n\left( {\left\lbrack {{X}_{0},\ldots ,{X}_{N}}\right\rbrack ,\left\lbrack {{Y}_{0},\ldots ,{Y}_{M}}\right\rbrack }\right... | Yes |
Theorem 7.29. (Weil Height Machine) For every nonsingular variety \( V/\overline{\mathbb{Q}} \) there exists a map\n\n\[ \n{h}_{V} : \operatorname{Div}\left( V\right) \rightarrow \{ \text{ functions }V\left( \overline{\mathbb{Q}}\right) \rightarrow \mathbb{R}\} ,\;D \mapsto {h}_{V, D},\n\]\n\nwith the following propert... | Proof. See [76, Chapter 2], [205, Theorem B.3.2], or [256, Chapter 4]. | No |
Let \( \phi : {\mathbb{P}}^{N} \rightarrow {\mathbb{P}}^{N} \) be a morphism of degree \( d \) and let \( H \in \operatorname{Div}\left( {\mathbb{P}}^{N}\right) \) be a hyperplane. Then \( {\phi }^{ * }H \sim {dH} \) | so Theorem 7.29 allows us to compute\n\n\[\n\begin{matrix} {h}_{{\mathbb{P}}^{N}, H}\left( {\phi \left( P\right) }\right) = {h}_{{\mathbb{P}}^{N},{\phi }^{ * }H}\left( P\right) + O\left( 1\right) = {h}_{{\mathbb{P}}^{N},{dH}}\left( P\right) + O\left( 1\right) = d{h}_{{\mathbb{P}}^{N}, H}\left( P\right) + O\left( 1\righ... | No |
Let \( V \) be a subvariety of \( {\mathbb{P}}^{N} \times {\mathbb{P}}^{M} \), say \( \phi : V \hookrightarrow {\mathbb{P}}^{N} \times {\mathbb{P}}^{M} \). Continuing with the notation from Examples 7.27 and 7.28, the height of a point \( P = \left\lbrack {\mathbf{x},\mathbf{y}}\right\rbrack \in V \) with respect to th... | \[ {h}_{V,{\phi }^{ * }{p}_{1}^{ * }{H}_{1}}\left( P\right) = {h}_{{\mathbb{P}}^{N},{H}_{1}}\left( {{p}_{1}\phi \left( P\right) }\right) = h\left( \mathbf{x}\right) ,\] \[ {h}_{V,{\phi }^{ * }{p}_{2}^{ * }{H}_{2}}\left( P\right) = {h}_{{\mathbb{P}}^{M},{H}_{2}}\left( {{p}_{2}\phi \left( P\right) }\right) = h\left( \mat... | Yes |
Let \( E \) be an elliptic curve given by a Weierstrass equation. Then the \( x \) - coordinate on \( E \), considered as a map \( x : E \rightarrow {\mathbb{P}}^{1} \), satisfies\n\n\[ \n{x}^{ * }\left( \infty \right) = 2\left( \mathcal{O}\right) \n\]\n\nso we have\n\n\[ \n{h}_{E,\left( \mathcal{O}\right) }\left( P\ri... | Note that the height \( {h}_{{\mathbb{P}}^{1},\left( \infty \right) } \) is just the usual height on \( {\mathbb{P}}^{1} \) from Theorem 7.29(a). | No |
We illustrate the involutions on \( {S}_{\mathbf{A},\mathbf{B}} \) using the example\n\n\[ L\left( {\mathbf{x},\mathbf{y}}\right) = {x}_{0}{y}_{0} + {x}_{1}{y}_{1} + {x}_{2}{y}_{2} \]\n\n\[ Q\left( {\mathbf{x},\mathbf{y}}\right) = {x}_{0}^{2}{y}_{0}^{2} + 4{x}_{0}^{2}{y}_{0}{y}_{1} - {x}_{0}^{2}{y}_{1}^{2} + 7{x}_{0}^{... | Thus\n\n\[ L\left( {\left\lbrack {1,0,0}\right\rbrack ,\mathbf{y}}\right) = {y}_{0} = 0\text{ and }Q\left( {\left\lbrack {1,0,0}\right\rbrack ,\mathbf{y}}\right) = {y}_{0}^{2} + 4{y}_{0}{y}_{1} - {y}_{1}^{2} + 7{y}_{1}{y}_{2} = 0, \]\n\nso the solutions are \( \mathbf{y} = \left\lbrack {0,{y}_{1},{y}_{2}}\right\rbrack ... | Yes |
We illustrate Proposition 7.39 using the surface described in Example 7.36. The polynomials \( {G}_{k}^{ * } \) and \( {H}_{ij}^{ * } \) for this example are given in Table 7.1. Proposition 7.39 says that \( {\iota }_{1} \) is defined at \( P = \left\lbrack {\mathbf{a},\mathbf{b}}\right\rbrack \) provided that at least... | \[ {G}_{0}^{x}\left( \mathbf{a}\right) = {G}_{1}^{x}\left( \mathbf{a}\right) = {G}_{2}^{x}\left( \mathbf{a}\right) = {H}_{01}^{x}\left( \mathbf{a}\right) = {H}_{02}^{x}\left( \mathbf{a}\right) = {H}_{12}^{x}\left( \mathbf{a}\right) = 0. \] Our first observation is that \[ {G}_{2}^{x}\left( {0,{x}_{1},{x}_{2}}\right) = ... | Yes |
Proposition 7.41. There is a proper Zariski closed set \( Z \subset {\mathbb{P}}^{8} \times {\mathbb{P}}^{35} \) such that if \( \left( {\mathbf{A},\mathbf{B}}\right) \notin Z \), then the involutions\n\n\[ \n{\iota }_{1} : {S}_{\mathbf{A},\mathbf{B}} \rightarrow {S}_{\mathbf{A},\mathbf{B}}\;\text{ and }\;{\iota }_{2} ... | Proof. According to Proposition 7.39, the involution \( {\iota }_{1} \) is defined on all of \( {S}_{\mathbf{A},\mathbf{B}} \) provided that the system of equations\n\n\[ \n{G}_{0}^{x}\left( \mathbf{x}\right) = {G}_{1}^{x}\left( \mathbf{x}\right) = {G}_{2}^{x}\left( \mathbf{x}\right) = {H}_{01}^{x}\left( \mathbf{x}\rig... | No |
Proposition 7.43. Let \( {D}_{1} = {p}_{1}^{ * }H \) and \( {D}_{2} = {p}_{2}^{ * }H \). The involutions \( {\iota }_{1} \) and \( {\iota }_{2} \) act on the subspace of \( \operatorname{Pic}\left( {S}_{\mathbf{A},\mathbf{B}}\right) \) generated by \( {D}_{1} \) and \( {D}_{2} \) according to the following rules:\n\n\[... | Proof. The involution \( {\iota }_{1} \) switches the sheets of the projection \( {p}_{1} \), so it is clear that \( {p}_{1} \circ {\iota }_{1} = {p}_{1} \). This allows us to compute\n\n\[ \n{\iota }_{1}^{ * }{D}_{1} = {\iota }_{1}^{ * }{p}_{1}^{ * }H = {\left( {p}_{1} \circ {\iota }_{1}\right) }^{ * }H = {p}_{1}^{ * ... | Yes |
Proposition 7.49. Let \( {S}_{\mathbf{A},\mathbf{B}} \) be defined over a number field \( K \), let \( {\widehat{h}}^{ + } \) and \( {\widehat{h}}^{ - } \) be the canonical height functions constructed in Proposition 7.47, and let\n\n\[ \widehat{h} = {\widehat{h}}^{ + } + {\widehat{h}}^{ - } \]\n\n(a) The set\n\n\[ \le... | Proof. (a) Using the properties of \( {\widehat{h}}^{ + } \) and \( {\widehat{h}}^{ - } \), we find that\n\n\[ \widehat{h} = {\widehat{h}}^{ + } + {\widehat{h}}^{ - } \]\nby definition of \( \widehat{h} \) ,\n\n\[ = \left( {-{h}_{{D}_{1}} + \alpha {h}_{{D}_{2}}}\right) + \left( {\alpha {h}_{{D}_{1}} - {h}_{{D}_{2}}}\ri... | Yes |
Proposition 1.8. Let \( S \) be a set with a multiplication and let \( X \) be a subset of \( S \) . If every element of \( S \) is a product of elements of \( X \), and every element of \( X \) passes Light’s test, then every element of \( S \) passes Light’s test (and the operation on \( S \) is associative). | In Example 1.7, \( {d}^{2} = c,{dc} = a \), and \( {da} = b \), so that \( a, b, c, d \) all are products of \( d \) ’s; since \( d \) passes Light’s test, Example 1.7 is associative. | No |
Proposition 2.1. In a group, written multiplicatively, the cancellation laws hold: \( {xy} = {xz} \) implies \( y = z \), and \( {yx} = {zx} \) implies \( y = z \) . Moreover, the equations \( {ax} = b,{ya} = b \) have unique solutions \( x = {a}^{-1}b, y = b{a}^{-1} \) . | Proof. \( {xy} = {xz} \) implies \( y = {1y} = {x}^{-1}{xy} = {x}^{-1}{xz} = {1z} = z \), and similarly for \( {yx} = {zx} \) . The equation \( {ax} = b \) has at most one solution \( x = {a}^{-1}{ax} = {a}^{-1}b \) , and \( x = {a}^{-1}b \) is a solution since \( a{a}^{-1}b = {1b} = b \) . The equation \( {ya} = b \) ... | Yes |
Proposition 2.2. In a group, written multiplicatively, \( {\left( {x}^{-1}\right) }^{-1} = x \) and \( {\left( {x}_{1}{x}_{2}\cdots {x}_{n}\right) }^{-1} = {x}_{n}^{-1}\cdots {x}_{2}^{-1}{x}_{1}^{-1} \) . | Proof. In a group, \( {uv} = 1 \) implies \( v = {1v} = {u}^{-1}{uv} = {u}^{-1} \) . Hence \( {x}^{-1}x = 1 \) implies \( x = {\left( {x}^{-1}\right) }^{-1} \) . We prove the second property when \( n = 2 \) and leave the general case to our readers: \( {xy}{y}^{-1}{x}^{-1} = {x1}{x}^{-1} = 1 \) ; hence \( {y}^{-1}{x}^... | No |
Proposition 2.3. In a group \( G \) (written multiplicatively) the following properties hold for all \( a \in S \) and all integers \( m, n \) :\n\n(1) \( {a}^{0} = 1,{a}^{1} = a \) ;\n\n(2) \( {a}^{m}{a}^{n} = {a}^{m + n} \) ;\n\n(3) \( {\left( {a}^{m}\right) }^{n} = {a}^{mn} \) ;\n\n(4) \( {\left( {a}^{n}\right) }^{-... | The proof makes an awful exercise, inflicted upon readers for their own good. | No |
Corollary 2.4. In a finite group, the inverse of an element is a positive power of that element. | Proof. Let \( G \) be a finite group and let \( x \in G \) . Since \( G \) is finite, the powers \( {x}^{n} \) of \( x, n \in \mathbb{Z} \), cannot be all distinct; there must be an equality \( {x}^{m} = {x}^{n} \) with, say, \( m < n \) . Then \( {x}^{n - m} = 1, x{x}^{n - m - 1} = 1 \), and \( {x}^{-1} = {x}^{n - m -... | Yes |
Proposition 3.1. A subset \( H \) of a group \( G \) is a subgroup if and only if \( H \neq \varnothing \) and \( x, y \in H \) implies \( x{y}^{-1} \in H \) . | Proof. These conditions are necessary by (1), (2), and (3). Conversely, assume that \( H \neq \varnothing \) and \( x, y \in H \) implies \( x{y}^{-1} \in H \) . Then there exists \( h \in H \) and \( 1 = h{h}^{-1} \in H \) . Next, \( x \in H \) implies \( {x}^{-1} = 1{x}^{-1} \in H \) . Hence \( x, y \in H \) implies ... | Yes |
Proposition 3.2. A subset \( H \) of a finite group \( G \) is a subgroup if and only if \( H \neq \varnothing \) and \( x, y \in H \) implies \( {xy} \in H \) . | Proof. If \( H \neq \varnothing \) and \( x, y \in H \) implies \( {xy} \in H \), then \( x \in H \) implies \( {x}^{n} \in H \) for all \( n > 0 \) and \( {x}^{-1} \in H \), by 2.4; hence \( x, y \in H \) implies \( {y}^{-1} \in H \) and \( x{y}^{-1} \in H \), and \( H \) is a subgroup by 3.1. Conversely, if \( H \) i... | Yes |
Proposition 3.3. Let \( G \) be a group and let \( X \) be a subset of \( G \) . The set of all products in \( G \) (including the empty product and one-term products) of elements of \( X \) and inverses of elements of \( X \) is a subgroup of \( G \) ; in fact, it is the smallest subgroup of \( G \) that contains \( X... | Proof. Let \( H \subseteq G \) be the set of all products of elements of \( X \) and inverses of elements of \( X \) . Then \( H \) contains the empty product \( 1;h \in H \) implies \( {h}^{-1} \in H \) , by 2.2; and \( h, k \in H \) implies \( {hk} \in H \), since the product of two products of elements of \( X \) an... | Yes |
Corollary 3.4. In a finite group \( G \), the subgroup \( \langle X\rangle \) of \( G \) generated by a subset \( X \) of \( G \) is the set of all products in \( G \) of elements of \( X \). | Proof. This follows from 3.3: if \( G \) is finite, then the inverses of elements of \( X \) are themselves products of elements of \( X \), by 2.4. \( ▱ \) | No |
Let \( G \) be a group and let \( a \in G \) . The set of all powers of a is a subgroup of \( G \) ; in fact, it is the subgroup generated by \( \{ a\} \) . | That the powers of \( a \) constitute a subgroup of \( G \) follows from the parts \( {a}^{0} = 1,{\left( {a}^{n}\right) }^{-1} = {a}^{-n} \), and \( {a}^{m}{a}^{n} = {a}^{m + n} \) of 2.3. Also, nonnegative powers of \( a \) are products of \( a \) ’s, and negative powers of \( a \) are products of \( {a}^{-1} \) ’s, ... | No |
Proposition 3.6. Every subgroup of \( \mathbb{Z} \) is cyclic, generated by a unique nonnegative integer. | Proof. The proof uses integer division. Let \( H \) be a subgroup of (the additive group) \( \mathbb{Z} \) . If \( H = 0\left( { = \{ 0\} }\right) \), then \( H \) is cyclic, generated by 0 . Now assume that \( H \neq 0 \), so that \( H \) contains an integer \( m \neq 0 \) . If \( m < 0 \), then \( - m \in H \) ; henc... | Yes |
Proposition 3.8. Every intersection of subgroups of a group \( G \) is a subgroup of \( G \) . | The proofs are exercises. | No |
Proposition 3.10. If \( H \) is a subgroup of a group, then \( {HH} = {Ha} = {aH} = H \) for every \( a \in H \) . | Proof. In the group \( H \), the equation \( {ax} = b \) has a solution for every \( b \in H \) . Therefore \( H \subseteq {aH} \) . But \( {aH} \subseteq H \) since \( a \in H \) . Hence \( {aH} = H \) . Similarly, \( {Ha} = H \) . Finally, \( H \subseteq {aH} \subseteq {HH} \subseteq H \) . \( ▱ \) | Yes |
Proposition 3.11. Let \( H \) be a subgroup of a group \( G \) . The left cosets of \( H \) constitute a partition of \( G \) ; the right cosets of \( H \) constitute a partition of \( G \) . | Proof. Define a binary relation \( \mathcal{R} \) on \( G \) by\n\n\[ x\mathcal{R}y\text{if and only if}x{y}^{-1} \in H\text{.} \]\n\nThe relation \( \mathcal{R} \) is reflexive, since \( x{x}^{-1} = 1 \in H \) ; symmetric, since \( x{y}^{-1} \in H \) implies \( y{x}^{-1} = {\left( x{y}^{-1}\right) }^{-1} \in H \) ; an... | Yes |
Proposition 3.12. The number of left cosets of a subgroup is equal to the number of its right cosets. | Proof. Let \( G \) be a group and \( H \leqq G \) . Let \( a \in G \) . If \( y \in {aH} \), then \( y = {ax} \) for some \( x \in H \) and \( {y}^{-1} = {x}^{-1}{a}^{-1} \in H{a}^{-1} \) . Conversely, if \( {y}^{-1} \in H{a}^{-1} \) , then \( {y}^{-1} = t{a}^{-1} \) for some \( t \in H \) and \( y = a{t}^{-1} \in {aH}... | Yes |
Corollary 3.14 (Lagrange’s Theorem). In a finite group \( G \), the order and index of a subgroup divide the order of \( G \) . | Proof. Let \( H \leqq G \) and let \( a \in G \) . By definition, \( {aH} = \{ {ax} \mid x \in H\} \) , and the cancellation laws show that \( x \mapsto {ax} \) is a bijection of \( H \) onto \( {aH} \) . Therefore \( \left| {aH}\right| = \left| H\right| \) : all left cosets of \( H \) have order \( \left| H\right| \) ... | Yes |
Proposition 4.1. If \( \varphi : A \rightarrow B \) and \( \psi : B \rightarrow C \) are homomorphisms of groups, then so is \( \psi \circ \varphi : A \rightarrow C \) . Moreover, the identity mapping \( {1}_{G} \) on a group \( G \) is a homomorphism. | Homomorphisms preserve identity elements, inverses, and powers, as readers will gladly verify. In particular, homomorphisms of groups preserve the constant and unary operation as well as the binary operation. | No |
Proposition 4.7. Let \( N \) be a normal subgroup of a group \( G \). The cosets of \( N \) constitute a group under the multiplication of subsets, and the mapping \( x \mapsto {xN} = {Nx} \) is a surjective homomorphism, whose kernel is \( N \). | Proof. Let \( S \) temporarily denote the set of all cosets of \( N \). Multiplication of subsets of \( G \) is associative and induces a binary operation on \( S \), since \( {xNyN} = {xyNN} = {xyN} \). The identity element is \( N \), since \( {NxN} = {xNN} = {xN} \). The inverse of \( {xN} \) is \( {x}^{-1}N \), sin... | Yes |
Proposition 4.8. \( {\mathbb{Z}}_{n} \) is a cyclic group of order \( n \), with elements \( \overline{0},\overline{1},\ldots ,\overline{n - 1} \) and addition | Proof. The proof uses integer division. For every \( x \in \mathbb{Z} \) there exist unique \( q \) and \( r \) such that \( x = {qn} + r \) and \( 0 \leqq r < n \) . Therefore every coset \( \bar{x} = x + \mathbb{Z}n \) is the coset of a unique \( 0 \leqq r < n \) . Hence \( {\mathbb{Z}}_{n} = \{ \overline{0},\overlin... | Yes |
Proposition 4.9. Let \( N \) be a normal subgroup of a group \( G \) . Every subgroup of \( G/N \) is the quotient \( H/N \) of a unique subgroup \( H \) of \( G \) that contains \( N \) . | Proof. Let \( \pi : G \rightarrow G/N \) be the canonical projection and let \( B \) be a subgroup of \( G/N \) . By 4.3,\n\n\[ A = {\pi }^{-1}\left( B\right) = \{ a \in G \mid {aN} \in B\} \]\n\nis a subgroup of \( G \) and contains \( {\pi }^{-1}\left( 1\right) = \operatorname{Ker}\pi = N \) . Now, \( N \) is a subgr... | Yes |
Proposition 4.10. Let \( N \) be a normal subgroup of a group \( G \) . Direct and inverse image under the canonical projection \( G \rightarrow G/N \) induce a one-to-one correspondence, which preserves inclusion and normality, between subgroups of \( G \) that contain \( N \) and subgroups of \( G/N \) . | Proof. Let \( \mathcal{A} \) be the set of all subgroups of \( G \) that contain \( N \) ; let \( \mathcal{B} \) be the set of all subgroups of \( G/N \) ; let \( \pi : G \rightarrow G/N \) be the canonical projection. By 4.16 and its proof, \( A \mapsto A/N \) is a bijection of \( \mathcal{A} \) onto \( \mathcal{B} \)... | No |
Theorem 5.1 (Factorization Theorem). Let \( N \) be a normal subgroup of a group \( G \) . Every homomorphism of groups \( \varphi : G \rightarrow H \) whose kernel contains \( N \) factors uniquely through the canonical projection \( \pi : G \rightarrow G/N \) (there exists a homomorphism \( \psi : G/N \rightarrow H \... | Proof. We use the formal definition of a mapping \( \psi : A \rightarrow B \) as a set of ordered pairs \( \left( {a, b}\right) \) with \( a \in A, b \in B \), such that (i) for every \( a \in A \) there exists \( b \in B \) such that \( \left( {a, b}\right) \in \psi \), and (ii) if \( \left( {{a}_{1},{b}_{1}}\right) \... | Yes |
Theorem 5.2 (Homomorphism Theorem). If \( \varphi : A \rightarrow B \) is a homomorphism of groups, then\n\n\[ A/\operatorname{Ker}\varphi \cong \operatorname{Im}\varphi \]\n\nin fact, there is an isomorphism \( \theta : A/\operatorname{Ker}f \rightarrow \operatorname{Im}f \) unique such that \( \varphi = \iota \circ \... | Proof. Let \( \psi : A \rightarrow \operatorname{Im}\varphi \) be the same mapping as \( \varphi \) (the same set of ordered pairs) but viewed as a homomorphism of \( A \) onto \( \operatorname{Im}\varphi \) . Then \( \operatorname{Ker}\psi = \) Ker \( \varphi \) ; by 5.1, \( \psi \) factors through \( \pi : \psi = \th... | Yes |
Corollary 5.3. Let \( \varphi : A \rightarrow B \) be a homomorphism. If \( \varphi \) is injective, then \( A \cong \operatorname{Im}\varphi \) . If \( \varphi \) is surjective, then \( B \cong A/\operatorname{Ker}\varphi \) . | Proof. If \( \varphi \) is injective, then \( \operatorname{Ker}\varphi = 1 \) and \( A \cong A/\operatorname{Ker}\varphi \cong \operatorname{Im}\varphi \) . If \( \varphi \) is surjective, then \( B = \operatorname{Im}\varphi \cong A/\operatorname{Ker}\varphi \) . \( ▱ \) | Yes |
Proposition 5.4. Let \( G \) be a group and let \( a \in G \) . If \( {a}^{m} \neq 1 \) for all \( m \neq 0 \), then \( \langle a\rangle \cong \mathbb{Z} \) ; in particular, \( \langle a\rangle \) is infinite. Otherwise, there is a smallest positive integer \( n \) such that \( {a}^{n} = 1 \), and then \( {a}^{m} = 1 \... | Proof. The power map \( p : m \mapsto {a}^{m} \) is a homomorphism of \( \mathbb{Z} \) into \( G \) . By 5.1, \( \langle a\rangle = \operatorname{Im}p \cong \mathbb{Z}/\operatorname{Ker}p \) . By 3.6, Ker \( p \) is cyclic, Ker \( p = \mathbb{Z}n \) for some unique nonnegative integer \( n \) . If \( n = 0 \), then \( ... | Yes |
Corollary 5.6. Every subgroup of a cyclic group is cyclic. | This follows from Propositions 5.4 and 3.6; the details make a pretty exercise. | No |
Theorem 5.8 (First Isomorphism Theorem). Let \( A \) be a group and let \( B, C \) be normal subgroups of \( A \) . If \( C \subseteq B \), then \( C \) is a normal subgroup of \( B, B/C \) is a normal subgroup of \( A/C \), and \[ A/B \cong \left( {A/C}\right) /\left( {B/C}\right) ; \] in fact, there is a unique isomo... | Proof. By 5.1, \( \rho \) factors through \( \pi : \rho = \sigma \circ \pi \) for some homomorphism \( \sigma : A/C \rightarrow A/B \) ; namely, \( \sigma : {aC} \mapsto {aB} \) . Like \( \rho ,\sigma \) is surjective. We show that \( \operatorname{Ker}\sigma = B/C \) . First, \( C \leqq B \), since \( C \leqq A \) . I... | Yes |
Theorem 5.9 (Second Isomorphism Theorem). Let \( A \) be a subgroup of a group \( G \), and let \( N \) be a normal subgroup of \( G \) . Then \( {AN} \) is a subgroup of \( G, N \) is a normal subgroup of \( {AN}, A \cap N \) is a normal subgroup of \( A \), and\n\n\[ \n{AN}/N \cong A/\left( {A \cap N}\right) \n\]\n\n... | Proof. We show that \( {AN} \leqq G \) . First, \( 1 \in {AN} \) . Since \( N \leqq G,{NA} = {AN} \) ; hence \( {an} \in {AN} \) (with \( a \in A, n \in N \) ) implies \( {\left( an\right) }^{-1} = {n}^{-1}{a}^{-1} \in {NA} = {AN} \) . Finally, \( {ANAN} = {AANN} = {AN} \) .\n\nNow, \( N \leqq {AN} \) . Let \( \varphi ... | Yes |
Lemma 6.1. For every word \( a \in W \) there is a reduction \( a \rightarrow b \) to a reduced word \( b \) . | Proof. By induction on the length of \( a \) . If \( a \) is reduced, then \( b = a \) serves. Otherwise, \( a\overset{1}{ \rightarrow }c \) for some \( c \in W, c \rightarrow b \) for some reduced \( b \in W \) since \( c \) is shorter than \( a \), and then \( a \rightarrow b \) . \( ▱ \) | Yes |
Lemma 6.2. If \( a\overset{1}{ \rightarrow }b \) and \( a\overset{1}{ \rightarrow }c \neq b \), then \( b\overset{1}{ \rightarrow }d, c\overset{1}{ \rightarrow }d \) for some \( d \) . | Proof. Let \( a = \left( {{a}_{1},{a}_{2},\ldots ,{a}_{n}}\right) \) . We have \( {a}_{i + 1} = {a}_{i}^{\prime } \) and\n\n\[ b = \left( {{a}_{1},\ldots ,{a}_{i - 1},{a}_{i + 2},\ldots ,{a}_{n}}\right) ,\]\n\nfor some \( 1 \leqq i < n \) ; also, \( {a}_{j + 1} = {a}_{j}^{\prime } \) and\n\n\[ c = \left( {{a}_{1},\ldot... | Yes |
Lemma 6.3. If \( a \rightarrow b \) and \( a \rightarrow c \), then \( b \rightarrow d \) and \( c \rightarrow d \) for some \( d \) . | Proof. Say \( a\overset{k}{ \rightarrow }b \) and \( a\overset{\ell }{ \rightarrow }c \) . The result is trivial if \( k = 0 \) or if \( \ell = 0 \) .\n\nWe first prove 6.3 when \( \ell = 1 \), by induction on \( k \) . We have \( a\overset{1}{ \rightarrow }c \) . If \( k \leqq 1 \) , then 6.3 holds, by 6.2. Now let \(... | Yes |
Lemma 6.4. For every word \( a \in W \) there is a unique reduced word \( b \) such that \( a \rightarrow b \) . | Proof. If \( a \rightarrow b \) and \( a \rightarrow c \), with \( b \) and \( c \) reduced, then, in Lemma 6.3, \( b \rightarrow d \) and \( c \rightarrow d \) imply \( b = d = c \) . \( ▱ \) | Yes |
Proposition 6.5. Under the operation \( a \cdot b = \operatorname{red}\left( {ab}\right) \), the set \( {F}_{X} \) of all reduced words in \( X \) is a group. | Proof. If \( a\overset{1}{ \rightarrow }b \), then \( {ac}\overset{1}{ \rightarrow }{bc} \) and \( {ca}\overset{1}{ \rightarrow }{cb} \) for all \( c \in W \) . Hence \( a \rightarrow b \) implies \( {ac} \rightarrow {bc} \) and \( {ca} \rightarrow {cb} \) for all \( c \in W \) . If now \( a, b, c \in W \) are reduced,... | Yes |
Proposition 6.6. If \( a = \left( {{a}_{1},{a}_{2},\ldots ,{a}_{n}}\right) \) is a reduced word in \( X \), then\n\n\[ a = \eta \left( {a}_{1}\right) \cdot \eta \left( {a}_{2}\right) \cdot \cdots \cdot \eta \left( {a}_{n}\right) . \]\n\nIn particular, \( {F}_{X} \) is generated by \( \eta \left( X\right) \) . | Proof. If \( a = \left( {{a}_{1},{a}_{2},\ldots ,{a}_{n}}\right) \) is reduced, then concatenating the one-letter words \( \left( {a}_{1}\right) ,\left( {a}_{2}\right) ,\ldots ,\left( {a}_{n}\right) \) yields a reduced word; hence\n\n\[ a = \left( {a}_{1}\right) \cdot \left( {a}_{2}\right) \cdot \cdots \cdot \left( {a}... | Yes |
Theorem 6.7. Let \( \eta : X \rightarrow {F}_{X} \) be the canonical injection. For every mapping \( f \) of \( X \) into a group \( G \), there is a homomorphism \( \varphi \) of \( {F}_{X} \) into \( G \) unique such that \( f = \varphi \circ \eta \), namely\n\n\[ \varphi \left( {{a}_{1},{a}_{2},\ldots ,{a}_{n}}\righ... | Proof. We show uniqueness first. Let \( \varphi : {F}_{X} \rightarrow G \) be a homomorphism such that \( f = \varphi \circ \eta \) . Extend \( f \) to \( {X}^{\prime } \) so that \( f\left( {x}^{\prime }\right) = f{\left( x\right) }^{-1} \) for all \( x \in X \) . For every \( x \in X \), we have \( \varphi \left( {\e... | Yes |
Corollary 6.8. If the group \( G \) is generated by a subset \( X \), then there is a surjective homomorphism of \( {F}_{X} \) onto \( G \) . | Proof. By Theorem 6.7, there is a homomorphism \( \varphi : {F}_{X} \rightarrow G \) such that \( \varphi \circ \eta \) is the inclusion mapping \( X \rightarrow G \) ; then \( \operatorname{Im}\varphi = G \), since \( \operatorname{Im}\varphi \) contains every generator \( x = \varphi \left( {\eta \left( x\right) }\ri... | Yes |
Proposition 7.1. Let \( R \) be a set of group relations between elements of a set \( X \) . Every relation \( \left( {u, v}\right) \in R \) holds in \( \langle X \mid R\rangle \) via the canonical mapping \( \iota : X \rightarrow \langle X \mid R\rangle \) ; moreover, \( \langle X \mid R\rangle \) is generated by \( \... | Proof. The canonical projection \( \pi : {F}_{X} \rightarrow \langle X \mid R\rangle \) is a homomorphism that extends \( \iota \) to \( {F}_{X} \), since \( \pi \circ \eta = \iota \) ; therefore it is the homomorphism that extends \( \iota \) to \( {F}_{X} \). If \( \left( {u, v}\right) \in R \), then \( u{v}^{-1} \in... | Yes |
Theorem 7.2 (Dyck [1882]). Let \( R \) be a set of group relations between elements of a set \( X \) . If \( f \) is a mapping of \( X \) into a group \( G \), and every relation \( \left( {u, v}\right) \in R \) holds in \( G \) via \( f \), then there exists a homomorphism \( \psi : \langle X \mid R\rangle \rightarrow... | Proof. Let \( N \) be the smallest normal subgroup of \( {F}_{X} \) that contains all \( u{v}^{-1} \) with \( \left( {u, v}\right) \in R \) . By 6.7 there is a unique homomorphism \( \varphi : {F}_{X} \rightarrow G \) that extends \( f \) . Since every \( \left( {u, v}\right) \in R \) holds in \( G \) via \( f \), we h... | Yes |
List the elements and construct a multiplication table of the quaternion group\n\n\[ \nQ = \left\langle {a, b \mid {a}^{4} = 1,{b}^{2} = {a}^{2},{ba}{b}^{-1} = {a}^{-1}}\right\rangle .\n\] | Solution. As in the case of \( {D}_{n} \), the elements of \( Q \) are products of \( a \) ’s and \( b \) ’s, which can be rewritten, using the relation \( {ba} = {a}^{3}b \), so that all \( a \) ’s precede all \( b \) ’s. Since \( {a}^{4} = 1 \) and \( {b}^{2} = {a}^{2} \), at most three \( a \) ’s and at most one \( ... | No |
Lemma 8.2. If \( x\overset{1}{ \rightarrow }y \) and \( x\overset{1}{ \rightarrow }z \neq y \), then \( y\overset{1}{ \rightarrow }t, z\overset{1}{ \rightarrow }t \) for some \( t \) . | Proof. By definition, \( x = \left( {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right) ,{x}_{i},{x}_{i + 1} \in A \) or \( {x}_{i},{x}_{i + 1} \in B \) for some \( i \) ,\n\n\[ y = \left( {{x}_{1},\ldots ,{x}_{i - 1},{x}_{i}{x}_{i + 1},{x}_{i + 2},\ldots ,{x}_{n}}\right) ,\]\n\n\( {x}_{j},{x}_{j + 1} \in A \) or \( {x}_{j},{x}_{... | Yes |
Proposition 8.5. If \( A \cap B = 1 \), then the set \( A \coprod B \) of all reduced nonempty words in \( A \cup B \) is a group under the operation \( x \cdot y = \operatorname{red}\left( {xy}\right) \) . | Proof. Associativity is proved as in Proposition 6.5. The one-letter word \( 1 = \left( 1\right) \) is reduced and is the identity element of \( A \coprod B \), since \( 1 \cdot x = \operatorname{red}\left( {1x}\right) = \) \( \operatorname{red}x = x \) and \( x \cdot 1 = \operatorname{red}x = x \) when \( x \) is redu... | Yes |
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