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Corollary 4.7. Let \( \phi \left( z\right) \in K\left( z\right) \) be a rational map of degree \( d \geq 2 \) . Then for all prime numbers \( \ell \) except possibly for \( d + 2 \) exceptions, the map \( \phi \) has a point of primitive period \( \ell \) . | Proof. We begin by discarding the finitely many primes \( \ell \) satisfying either of the following conditions:\n\n- \( K \) has characteristic \( \ell \) .\n\n- There is some \( Q \in \operatorname{Fix}\left( \phi \right) \) with \( \lambda \left( Q\right) \neq 1 \) and \( \lambda {\left( Q\right) }^{\ell } = 1 \) .\... | Yes |
Theorem 4.8. (I.N. Baker [19]) Let \( \phi \left( z\right) \in K\left( z\right) \) be a rational map of degree \( d \geq 2 \) defined over a field \( K \) of characteristic 0 . Suppose that \( \phi \) has no primitive \( n \) -periodic points. Then \( \left( {n, d}\right) \) is one of the pairs\n\n\[ \left( {2,2}\right... | Proof. For the proof, which is function-theoretic in nature, see [19] or \( \left\lbrack {{43},§{6.8}}\right\rbrack \) . | No |
It is easy to see that \( {X}_{1}\left( 1\right) \) and \( {X}_{1}\left( 2\right) \) are rational curves. Indeed, the projective closures of \( {Y}_{1}\left( 1\right) \) and \( {Y}_{1}\left( 2\right) \) are smooth conics, \[ {X}_{1}\left( 1\right) : {z}^{2} - {zw} + {yw} = 0\;\text{ and }\;{X}_{1}\left( 2\right) : {z}^... | It turns out that \( {X}_{1}\left( 3\right) \) is also rational, but this is less clear from the equation in Table 4.1. In order to parameterize \( {X}_{1}\left( 3\right) \), suppose that \( \phi \left( z\right) = A{z}^{2} + {Bz} + C \) is any quadratic polynomial with a periodic point of primitive period 3 . Conjugati... | Yes |
Theorem 4.11. Let \( K \) be a field of characteristic different from 2.\n\n(a) The map\n\n\[ \n{Y}_{1}\left( n\right) \rightarrow \operatorname{Formal}\left( n\right) ,\;\left( {c,\alpha }\right) \mapsto \left( {{z}^{2} + c,\alpha }\right) ,\n\]\n\n(4.14)\n\nis a bijection of sets. | Proof. (a) We have shown this earlier in this section. | No |
We have seen in Example 4.9 that \( {X}_{1}\left( 2\right) \) and \( {X}_{1}\left( 3\right) \) are rational curves, i.e., they are isomorphic to \( {\mathbb{P}}^{1} \), so their quotient curves \( {X}_{0}\left( 2\right) \) and \( {X}_{0}\left( 3\right) \) must also be rational curves. However, it is still of interest t... | The affine curve \( {Y}_{1}\left( 2\right) \) has equation\n\n\[ \n{Y}_{1}\left( 2\right) : {z}^{2} + z + y + 1 = 0, \n\] \n\nwhere a point \( \left( {c,\alpha }\right) \in {Y}_{1}\left( 2\right) \) corresponds to the quadratic map \( {\phi }_{c}\left( z\right) = {z}^{2} + c \) and point \( \alpha \in {\operatorname{Pe... | No |
Theorem 4.17. (a) The affine curve \( {Y}_{1}\left( n\right) \) defined by the equation\n\n\[ \n{Y}_{1}\left( n\right) : {\Phi }_{n,\phi }^{ * }\left( {y, z}\right) = 0 \n\]\n\nis nonsingular. | Proof. The properties of \( {Y}_{1}\left( n\right) \) and \( {X}_{1}\left( n\right) \) were originally proven by Bousch [83], with subsequent proofs by Lau and Schleicher [261] using analytic methods and Morton [307] via algebraic arguments. The latter two papers give various generalizations, including results for the ... | No |
Proposition 4.19. The Mandelbrot set is contained in the disk of radius 2, \[ \mathcal{M} \subset \{ c \in \mathbb{C} : \left| c\right| \leq 2\} \] | Proof. Suppose that \( \left| c\right| > 2 \) and let \( {z}_{n} = {\phi }_{c}^{n}\left( 0\right) \) . Then \[ \left| {z}_{n + 1}\right| \geq \left| {z}_{n}^{2}\right| - \left| c\right| = \left| {z}_{n}\right| \cdot \left( {\left| {z}_{n}\right| - 1}\right) + \left( {\left| {z}_{n}\right| - \left| c\right| }\right) . \... | Yes |
The set of Misiurewicz points is a set of bounded (absolute) height in \( \overline{\mathbb{Q}} \) . More precisely, the height of a Misiurewicz point \( \gamma \) satisfies \( H\left( \gamma \right) \leq 2 \) . Hence there are only finitely many Misiurewicz points defined over any given number field. | A Misiurewicz point \( c = \gamma \) is the root of a polynomial of the form\n\n\[ \n{M}_{m, n}\left( c\right) = {\phi }_{c}^{n + m}\left( 0\right) - {\phi }_{c}^{m}\left( 0\right) \n\]\n\nfor some \( m \geq 1 \) and \( n \geq 1 \) . These are monic polynomials with coefficients in \( \mathbb{Z} \) , so not only is \( ... | Yes |
Theorem 4.23. (Douady-Hubbard) There is a conformal isomorphism from the exterior of the unit disk to the complement of the Mandelbrot set,\n\n\[ \n\\theta : \\{ w \\in \\mathbb{C} : \\left| w\\right| > 1\\} \\overset{ \\sim }{ \\rightarrow }\\mathbb{C} \\smallsetminus \\mathcal{M}.\n\] | Proof. It is not hard to show that for all sufficiently large \( z \) (depending on \( c \) ) there is a consistent way to choose square roots so that the limit\n\n\[ \n{\\psi }_{c}\\left( z\\right) = \\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\sqrt[{2}^{n}]{{\\phi }_{c}^{n}\\left( z\\right) }\n\]\n\nconverge... | No |
Proposition 4.27. The variety \( {\operatorname{Rat}}_{d} \) is an affine variety defined over \( \mathbb{Q} \) . The ring of regular functions \( \mathbb{Q}\left\lbrack {\operatorname{Rat}}_{d}\right\rbrack \) of \( {\operatorname{Rat}}_{d} \) is given explicitly by\n\n\[ \mathbb{Q}\left\lbrack {\operatorname{Rat}}_{d... | Proof. We remind the reader that in general, if \( F \in K\left\lbrack {{X}_{0},\ldots ,{X}_{r}}\right\rbrack \) is a homogeneous polynomial of degree \( n \), then the complement of the zero set of \( F \), \n\n\[ V = {\mathbb{P}}^{r} \smallsetminus \{ F = 0\} \]\n\n is an affine variety of dimension \( r \) . (See [1... | Yes |
Let\n\n\\[ \n\\rho \\left( {\\mathbf{a},\\mathbf{b}}\\right) = {a}_{2}^{2}{b}_{0}^{2} - {a}_{1}{a}_{2}{b}_{0}{b}_{1} + {a}_{0}{a}_{2}{b}_{1}^{2} + {a}_{1}^{2}{b}_{0}{b}_{2} - 2{a}_{0}{a}_{2}{b}_{0}{b}_{2} - {a}_{0}{a}_{1}{b}_{1}{b}_{2} + {a}_{0}^{2}{b}_{2}^{2} \n\\]\n\nbe the resultant of \\( {a}_{0}{X}^{2} + {a}_{1}{X... | Of course, this is not the smallest affine space into which \\( {\\operatorname{Rat}}_{2} \\) can be embedded. Projecting onto appropriately chosen hy-perplanes, there is certainly an affine embedding of the 5-dimensional space \\( {\\operatorname{Rat}}_{2} \\) into \\( {\\mathbb{A}}^{11} \\) ; see [198, Exercise IV.3.... | Yes |
Proposition 4.31. Let \( V \) be an algebraic variety and let\n\n\[ \psi : {\mathbb{P}}_{V}^{1} \rightarrow {\mathbb{P}}_{V}^{1} \]\n\nbe a morphism over \( V \) of degree \( d \) . Then there is a unique morphism\n\n\[ \lambda : V \rightarrow {\operatorname{Rat}}_{d} \]\n\nsuch that the induced map \( \lambda : {\math... | Proof. Let \( U \subset V \) be an affine open subset and write \( K\left\lbrack U\right\rbrack \) for its affine coordinate ring. The fact that \( \psi \) is a morphism over \( V \) implies that it restricts to give a morphism \( \psi : {\mathbb{P}}_{U}^{1} \rightarrow {\mathbb{P}}_{U}^{1} \) over \( U \) . This restr... | Yes |
Proposition 4.33. The map\n\n\[ \n{\mathrm{{PGL}}}_{2} \times {\mathrm{{Rat}}}_{d} \rightarrow {\mathrm{{Rat}}}_{d},\;\left( {f,\phi }\right) \mapsto {\phi }^{f} = {f}^{-1}{\phi f}, \n\]\n\n(4.24)\n\nis an algebraic group action of \( {\mathrm{{PGL}}}_{2} \) on \( {\operatorname{Rat}}_{d} \) and is defined over \( \mat... | Proof. The proof is mostly a matter of unsorting the definitions. Let\n\n\[ \nf = \left\lbrack {{\alpha X} + {\beta Y},{\gamma X} + {\delta Y}}\right\rbrack \in {\mathrm{{PGL}}}_{2} \subset {\mathbb{P}}^{3}\;\text{ and }\;\phi = \left\lbrack {{F}_{\mathbf{a}},{F}_{\mathbf{b}}}\right\rbrack \in {\operatorname{Rat}}_{d} ... | No |
In principle it is possible to explicitly write down the action of \( {\mathrm{{PGL}}}_{2} \) on \( {\operatorname{Rat}}_{d} \), but in practice the expressions become hopelessly unwieldy for even moderate values of \( d \) . As illustration, we describe the action for \( d = 2 \) . Let\n\n\[ f = \left\lbrack {{\alpha ... | \[ {a}_{0}^{\prime } = {\alpha }^{2}\delta {a}_{0} + {\alpha \gamma \delta }{a}_{1} + {\gamma }^{2}\delta {a}_{2} - {\alpha }^{2}\beta {b}_{0} - {\alpha \beta \gamma }{b}_{1} - \beta {\gamma }^{2}{b}_{2}, \] \n\n\[ {a}_{1}^{\prime } = {2\alpha \beta \delta }{a}_{0} + \left( {{\alpha \delta } + {\beta \gamma }}\right) \... | Yes |
Theorem 4.36. There is an algebraic variety \( {\mathcal{M}}_{d} \) defined over \( \mathbb{Q} \) and a morphism\n\n\[ \langle \cdot \rangle : {\operatorname{Rat}}_{d} \rightarrow {\mathcal{M}}_{d} \]\n\n(4.26)\n\ndefined over \( \mathbb{Q} \) with the following properties:\n\n(a) The map (4.26) is \( {\mathrm{{PSL}}}_... | Proof Sketch. A full proof of Theorem 4.36 (see [416]) uses the machinery of geometric invariant theory [322] and is thus unfortunately beyond the scope of this book. Geometric invariant theory tells us that there is a certain subset of \( {\mathbb{P}}^{{2d} + 1} \), called the stable locus, on which the conjugation ac... | Yes |
Let \( \phi \in {\operatorname{Rat}}_{d}\left( \mathbb{C}\right) \) be a rational function of degree \( d \) defined over the complex numbers. Associated to each fixed point \( P \in \operatorname{Fix}\left( \phi \right) \) is its multiplier \( {\lambda }_{P}\left( \phi \right) \in {\mathbb{C}}^{ * } \) . A simple calc... | \[ \operatorname{Fix}\left( {\phi }^{f}\right) = {f}^{-1}\left( {\operatorname{Fix}\left( \phi \right) }\right) \;\text{ and }\;{\lambda }_{{f}^{-1}\left( P\right) }\left( {\phi }^{f}\right) = {\lambda }_{P}\left( \phi \right) \;\text{ for all }P \in \operatorname{Fix}\left( \phi \right) . \] | Yes |
We illustrate the construction of Example 4.44 for rational maps of degree 2. As usual, we write\n\n\[ \phi = \left\lbrack {{F}_{\mathbf{a}},{F}_{\mathbf{b}}}\right\rbrack = \left\lbrack {{a}_{0}{X}^{2} + {a}_{1}{XY} + {a}_{2}{Y}^{2},{b}_{0}{X}^{2} + {b}_{1}{XY} + {b}_{2}{Y}^{2}}\right\rbrack . \]\n\nThe map \( \phi \)... | Notice that the denominator of \( {\sigma }_{1}\left( \phi \right) \) is \( \operatorname{Res}\left( {{F}_{\mathbf{a}},{F}_{\mathbf{b}}}\right) \), so \( {\sigma }_{1}\left( \phi \right) \) is in \( \mathbb{Q}\left\lbrack {\operatorname{Rat}}_{d}\right\rbrack \) . It is far less obvious that this expression for \( {\si... | No |
Let \( \phi \left( z\right) = {z}^{d} \) with \( d \geq 2 \) . Then \( {\operatorname{Per}}_{n}\left( \phi \right) = \{ 0,\infty \} \cup {\mathbf{\mu }}_{{d}^{n} - 1} \) consists of the points \( 0,\infty \), and the \( {\left( {d}^{n} - 1\right) }^{\text{th }} \) roots of unity. | It is easy to check that \( {\lambda }_{0}\left( \phi \right) = {\lambda }_{\infty }\left( \phi \right) = 0 \), and for \( \zeta \in {\mathbf{\mu }}_{{d}^{n} - 1} \) we have\n\n\[ \n{\lambda }_{\zeta }\left( \phi \right) = {\left( {\phi }^{n}\right) }^{\prime }\left( \zeta \right) = {d}^{n}{\zeta }^{{d}^{n} - 1} = {d}^... | Yes |
Let \( \phi \left( z\right) = {z}^{2} + {bz} \). Then\n\n\[ \n{\operatorname{Per}}_{1}\left( \phi \right) = \{ 0,1 - b,\infty \} \;\text{ and }\;{\Lambda }_{1}\left( \phi \right) = \{ b,2 - b,0\} .\n\] | Next we compute\n\n\[ \n{\Phi }_{\phi ,2}^{ * } = \frac{{\phi }^{2}\left( z\right) - z}{\phi \left( z\right) - z} = {z}^{2} + \left( {b + 1}\right) z + b + 1.\n\]\n\nThe two points of formal period 2 are the roots of \( {\Phi }_{\phi ,2}^{ * } \),\n\n\[ \n{\operatorname{Per}}_{2}^{ * }\left( \phi \right) = \left\{ \fra... | Yes |
Continuing with Example 4.46, let \( \phi \left( z\right) = {z}^{d} \). Then | \[ \mathop{\prod }\limits_{{\lambda \in {\Lambda }_{n}\left( \phi \right) }}\left( {T + \lambda }\right) = {T}^{2}{\left( T + {d}^{n}\right) }^{{d}^{n} - 1}\;\text{ and }\;\mathop{\prod }\limits_{{\lambda \in {\Lambda }_{n}^{ * }\left( \phi \right) }}\left( {T + \lambda }\right) = {\left( T + {d}^{n}\right) }^{\varphi ... | Yes |
Continuing with Example 4.47, let \( \phi \left( z\right) = {z}^{2} + {bz} \) . We computed \( {\Lambda }_{1}\left( \phi \right) = \{ b,2 - b,0\} \) | \[ \mathop{\prod }\limits_{{\lambda \in {\Lambda }_{1}\left( \phi \right) }}\left( {T + \lambda }\right) = \left( {T + b}\right) \left( {T + 2 - b}\right) T = {T}^{3} + 2{T}^{2} + \left( {{2b} - {b}^{2}}\right) T \] which gives \[ {\sigma }_{1}^{\left( 1\right) } = 2,\;{\sigma }_{2}^{\left( 1\right) } = {2b} - {b}^{2},... | Yes |
For \( \phi \in {\operatorname{Rat}}_{d}, n \geq 1 \), and \( i \) in the appropriate range, let \( {\sigma }_{i}^{\left( n\right) }\left( \phi \right) \) and \( {\overset{ * }{\sigma }}_{i}^{\left( n\right) }\left( \phi \right) \) be the symmetric polynomials of the \( n \) -multiplier spectra of \( \phi \) . (a) The ... | Proof. We sketch the proof for \( {\sigma }_{i}^{\left( n\right) }\left( \phi \right) \) and leave \( {\overset{ * }{\sigma }}_{i}^{\left( n\right) }\left( \phi \right) \) as an exercise for the reader (Exercise 4.26).\n\n(a) We write\n\n\[ {\phi }^{n} = \left\lbrack {{F}_{\mathbf{a}, n}\left( {X, Y}\right) ,{F}_{\math... | No |
For each \( t \in {\mathbb{C}}^{ * } \) with \( t \neq - \frac{27}{4} \), consider the rational map \[ {\phi }_{t}\left( x\right) = \frac{{x}^{4} - {2t}{x}^{2} - {8tx} + {t}^{2}}{4{x}^{3} + {4tx} + {4t}}. \] It is the Lattès map associated to multiplication-by-2 on the elliptic curve \[ {E}_{t} : {y}^{2} = {x}^{3} + {t... | (For proofs of these statements, see Proposition 6.52 in Section 6.5.) | No |
Theorem 4.54. Define the degree of \( {\sigma }_{d, N} \) to be the number of points in \( {\sigma }_{d, N}^{-1}\left( P\right) \) for a generic point \( P \) in the image \( {\mathbf{\sigma }}_{d, N}\left( {\mathcal{M}}_{d}\right) \) . One can show that the degree of \( {\mathbf{\sigma }}_{d, N} \) stabilizes as \( N ... | Proof. We will prove this in Chapter 6 using Lattès maps associated to elliptic curves with complex multiplication; see Theorem 6.62. | No |
Theorem 4.61. Let \( {\overline{\mathcal{M}}}_{2} = {\mathcal{M}}_{2}^{s} = {\mathcal{M}}_{2}^{ss} \) be the completion of \( {\mathcal{M}}_{2} \) constructed using geometric invariant theory in Theorem 4.40. Then the isomorphism\n\n\[ \mathbf{\sigma } = \left( {{\sigma }_{1},{\sigma }_{2}}\right) : {\mathcal{M}}_{2} \... | Proof. See [302] and [416, Theorem 6.1 and Lemmas 6.2 and 6.3]. | No |
Proposition 4.65. Let \( \\phi \\left( z\\right) \\in \\widetilde{K}\\left( z\\right) \) be a rational map of degree \( d \\geq 2 \) . Then \( \\operatorname{Aut}\\left( \\phi \\right) \) is a finite subgroup of \( {\\mathrm{{PGL}}}_{\\2}\\left( \\bar{K}\\right) \), and its order is bounded by a function of \( d \) . | Proof. Let \( f \\in \\operatorname{Aut}\\left( \\phi \\right) \) . Then for any point \( P \\in {\\mathbb{P}}^{1}\\left( \\bar{K}\\right) \) and any \( n \\geq 1 \) we have\n\n\[ \n{\\phi }^{n}\\left( P\\right) = {\\left( {\\phi }^{f}\\right) }^{n}\\left( P\\right) = \\left( {{f}^{-1}{\\phi }^{n}f}\\right) \\left( P\\... | Yes |
The map \( \phi \left( z\right) = \left( {{z}^{2} - {2z}}\right) /\left( {-{2z} + 1}\right) \) has an automorphism group \( \operatorname{Aut}\left( \phi \right) \) that is isomorphic to the symmetric group \( {\mathcal{S}}_{3} \) on three letters. | More precisely (see Exercise 4.36), the automorphism group of \( \phi \) consists of the following six linear fractional transformations:\n\n\[ \operatorname{Aut}\left( \phi \right) = \left\{ {z,\frac{1}{z},\frac{z - 1}{z},\frac{1}{1 - z},\frac{z}{z - 1},1 - z}\right\} \cong {\mathcal{S}}_{3}. \] | No |
We can use Remark 4.70 to prove that the map (4.43) is injective. (We assume that \( K \) does not have characteristic 2.) | A quick computation shows that \( {\Phi }_{{\phi }_{b},2}^{ * }\left( {X, Y}\right) = 2{X}^{2} + b{Y}^{2} \), so the primitive 2-periodic points of \( {\phi }_{b} \) are \( \pm \sqrt{-b/2} \) . Hence if \( {\phi }_{b} \) and \( {\phi }_{c} \) are \( K \) -isomorphic, then the fields \( K\left( \sqrt{-b/2}\right) \) and... | No |
Proposition 4.73. Let \( \phi \left( z\right) \in K\left( z\right) \) be a rational map of degree \( d \geq 2 \) and assume that its automorphism group \( \operatorname{Aut}\left( \phi \right) \) is trivial. Then \( \phi \) has no nontrivial twists, i.e., \( \operatorname{Twist}\left( {\phi /K}\right) \) has only one e... | Proof. Suppose that \( \psi \in K\left( z\right) \) is a twist of \( \phi \), so there is an \( f \in {\operatorname{PGL}}_{2}\left( \bar{K}\right) \) such that \( \psi = {\phi }^{f} \) . We let an element \( \sigma \in \operatorname{Gal}\left( {\bar{K}/K}\right) \) act on \( f\left( z\right) \) and \( \phi \left( z\ri... | Yes |
Let \( \phi \left( z\right) \in K\left( z\right) \) be an odd rational map, i.e., a rational map satisfying \( \phi \left( {-z}\right) = - \phi \left( z\right) \). Then for each \( b \in {K}^{ * } \) we can define a new rational map \( {\phi }_{b} \) by the formula\n\n\[ \n{\phi }_{b}\left( z\right) = \frac{1}{\sqrt{b}... | The odd parity of \( \phi \left( z\right) \) implies that \( \phi \) has the form \( \phi \left( z\right) = {z\psi }\left( {z}^{2}\right) \) for some rational map \( \psi \left( z\right) \in K\left( z\right) \), so \( {\phi }_{b}\left( z\right) = {z\psi }\left( {b{z}^{2}}\right) \) is in \( K\left( z\right) \). Further... | Yes |
Example 4.76. Our second example deals with twists of the variety \( {\mathbb{P}}^{1} \) . For any nonzero \( a \in {K}^{ * } \), let \( {C}_{a} \) be the plane curve \[ {C}_{a} : {x}^{2} + {y}^{2} = a. \] All of these curves are isomorphic over \( \bar{K} \) via the explicit isomorphism \[ i : {C}_{a} \rightarrow {C}_... | Notice that if two curves \( C \) and \( {C}^{\prime } \) are \( K \) -isomorphic, then the \( K \) -isomorphism \( i : C \rightarrow {C}^{\prime } \) identifies their \( K \) -rational points \( i : C\left( K\right) \rightarrow {C}^{\prime }\left( K\right) \) . This suffices to prove that the curves \( {C}_{1} \) and ... | Yes |
Proposition 4.77. Let \( X \) be an object defined over \( K \), let \( Y \) be a twist of \( X/K \) , choose a \( \bar{K} \) -isomorphism \( i : Y \rightarrow X \), and define a map\n\n\[ g : \operatorname{Gal}\left( {\bar{K}/K}\right) \rightarrow \operatorname{Aut}\left( X\right) ,\;{g}_{\sigma }\left( x\right) = \le... | Proof. (a) We have \( {g}_{\sigma \tau } = i \circ \left( {\sigma \tau }\right) \left( {i}^{-1}\right) = i \circ \sigma \left( {\tau {\left( i\right) }^{-1}}\right) \) . Consider the following commutative diagram of maps:\n\n\[ X\xrightarrow[]{\sigma \left( {\tau {\left( i\right) }^{-1}}\right) }Y\xrightarrow[]{i}X\n\]... | Yes |
Theorem 4.79. Let \( \phi \left( z\right) \in K\left( z\right) \) be a nonzero rational map and let\n\n\[ g : \operatorname{Gal}\left( {\bar{K}/K}\right) \rightarrow \operatorname{Aut}\left( \phi \right) \]\n\nbe a 1-cocycle with values in \( \operatorname{Aut}\left( \phi \right) \) . Then the following are equivalent:... | Proof. Suppose first that there is a twist \( {\phi }^{f} \) of \( \phi /K \) whose 1-cocycle is \( g \) . Then by the definition given in Proposition 4.77, the 1-cocycle \( g \) is given by \( {g}_{\sigma } = {f\sigma }\left( {f}^{-1}\right) \) . Hence \( g \) is the \( {\operatorname{PGL}}_{2}\left( \bar{K}\right) \)... | Yes |
Let \( K \) be a field of characteristic not dividing \( n \), and let \( \phi \left( z\right) \in K\left( z\right) \) be a rational map whose automorphism group is \[ \operatorname{Aut}\left( \phi \right) = \left\{ {{\zeta z} : \zeta \in {\mathbf{\mu }}_{n}}\right\} \] where we recall that \( {\mathbf{\mu }}_{n} \subs... | Note that by assumption we have \( \phi \left( z\right) = {\zeta }^{-1}\phi \left( {\zeta z}\right) \) for all \( \zeta \in {\mathbf{\mu }}_{n} \) . In particular, the function \( {z}^{-1}\phi \left( z\right) \) is invariant under the substitution \( z \rightarrow {\zeta z} \), so it has the form \( \phi \left( z\right... | No |
Let \( \phi \left( z\right) \in K\left( z\right) \) be a rational map with automorphism group \( \operatorname{Aut}\left( \phi \right) = \left\{ {z,{z}^{-1}}\right\} \). The Galois group \( \operatorname{Gal}\left( {\bar{K}/K}\right) \) acts trivially on \( \operatorname{Aut}\left( \phi \right) \), so we have\n\n\[ \n{... | The isomorphism is given explicitly by associating to any \( b \in {K}^{ * }/{K}^{*2} \) the cocycle\n\n\[ \n\sigma \mapsto \left\{ \begin{array}{ll} z & \text{ if }\sigma \left( \sqrt{b}\right) = \sqrt{b} \\ {z}^{-1} & \text{ if }\sigma \left( \sqrt{b}\right) = - \sqrt{b} \end{array}\right.\n\]\n\n(4.51)\n\nTo ease no... | Yes |
Proposition 4.84. Let \( \phi \left( z\right) \in \bar{K}\left( z\right) \). (a) The set \( {G}_{\phi } \) is a subgroup of \( \operatorname{Gal}\left( {\bar{K}/K}\right) \). (b) Let \( {K}^{\prime } \) be a field of definition for \( \phi \). Then \( {K}_{\phi } \subseteq {K}^{\prime } \). Informally, we say that \( \... | Proof. The proof of this proposition is simply a matter of unsorting definitions. Thus let \( \sigma ,\tau \in {G}_{\phi } \). Then\n\n\[ \left( {\sigma \tau }\right) \left( \phi \right) = \sigma \left( {\tau \left( \phi \right) }\right) = \sigma \left( {\phi }^{{g}_{\tau }}\right) = \sigma {\left( \phi \right) }^{\sig... | Yes |
Let \[ \phi \left( z\right) = i{\left( \frac{z - 1}{z + 1}\right) }^{3}. \] Clearly \( \mathbb{Q}\left( i\right) \) is a field of definition for \( \phi \). Let \( \sigma \) be complex conjugation, so \( \operatorname{Gal}\left( {\mathbb{Q}\left( i\right) /\mathbb{Q}}\right) = \{ 1,\sigma \} \), and let \( g\left( z\ri... | \[ = i{\left( \frac{-1/z + 1}{-1/z - 1}\right) }^{3} = - i{\left( \frac{z - 1}{z + 1}\right) }^{3} = \sigma \left( \phi \right) \left( z\right) . \] This shows that \( \sigma \in {G}_{\phi } \), so \( {G}_{\phi } = \{ 1,\sigma \} \) and \( {K}_{\phi } = \mathbb{Q} \). In other words, \( \mathbb{Q} \) is the field of mo... | Yes |
Proposition 4.86. Let \( \phi \in \bar{K}\left( z\right) \) be a rational map of degree \( d \geq 2 \) with field of moduli \( K \) and satisfying \( \operatorname{Aut}\left( \phi \right) = 1 \), and for each \( \sigma \in \operatorname{Gal}\left( {\bar{K}/K}\right) \) write \( \sigma \left( \phi \right) = \) \( {\phi ... | Proof. (a) Let \( \sigma ,\tau \in \operatorname{Gal}\left( {\bar{K}/K}\right) \) . Then\n\n\[ {\phi }^{{g}_{\sigma \tau }} = \sigma \left( {\tau \left( \phi \right) }\right) = \sigma \left( {\phi }^{{g}_{\tau }}\right) = \sigma {\left( \phi \right) }^{\sigma \left( {g}_{\tau }\right) } = {\phi }^{{g}_{\sigma }\sigma \... | Yes |
Proposition 4.87. Let\n\n\\[ \n g : \\operatorname{Gal}\\left( {\\bar{K}/K}\\right) \\rightarrow {\\operatorname{PGL}}_{2}\\left( \\bar{K}\\right) \n\\]\n\nbe a 1-cocycle, and assume that \\( g \\) has the property that there is a finite extension \\( L/K \\) such that \\( {g}_{\\sigma } = 1 \\) for all \\( \\sigma \\i... | Proof. We construct the curve \\( C \\) by describing its field of rational functions. Note that the field of rational functions for the curve \\( {\\mathbb{P}}^{1} \\) is the field \\( \\bar{K}\\left( z\\right) \\) and that the Galois group \\( \\operatorname{Gal}\\left( \\bar{K}/K\\right) \\) acts naturally on \\( \\... | No |
Define a map \( g \) by the rule\n\n\[ g : \operatorname{Gal}\left( {\overline{\mathbb{Q}}/\mathbb{Q}}\right) \rightarrow {\operatorname{PGL}}_{2}\left( \overline{\mathbb{Q}}\right) ,\;{g}_{\sigma }\left( z\right) = \left\{ \begin{array}{ll} z & \text{ if }\sigma \left( i\right) = i, \\ - 1/z & \text{ if }\sigma \left(... | It is easy to check that \( g \) is a 1-cocycle, and indeed it is the 1-cocycle described in Example 4.85. Fix an embedding of \( \mathbb{Q} \) into \( \mathbb{C} \) and let \( \rho \in \operatorname{Gal}\left( {\mathbb{Q}/\mathbb{Q}}\right) \) denote complex conjugation. The twisted action on \( \overline{\mathbb{Q}}\... | No |
Corollary 4.90. (Riemann-Roch in Genus 0) Let \( C/K \) be a smooth projective curve of genus 0 defined over \( K \) .\n\n(a) There is a \( K \) -rational divisor of degree 2 on \( C \) .\n\n(b) Let \( D \in \operatorname{Div}\left( C\right) \) be a divisor defined over \( K \) and satisfying \( \deg \left( D\right) \g... | Proof. The proof of the Riemann-Roch theorem over an algebraically closed field is given in most introductory texts on algebraic geometry, such as [198, IV.1.3], or see [255, Chapter I] for an elementary proof due to Weil and [410, II, §5] for an overview. The divisor of degree \( {2g} - 2 \) in (a) is a canonical divi... | No |
Proposition 4.91. Let \( C \) be a twist of \( {\mathbb{P}}^{1}/K \) . The following are equivalent:\n\n(a) \( C \) is the trivial twist of \( {\mathbb{P}}^{1}/K \), i.e., \( C \) is \( K \) -isomorphic to \( {\mathbb{P}}^{1} \).\n\n(b) \( C\left( K\right) \) is nonempty, i.e., \( C \) has a point with coordinates in \... | Proof. If \( C \) is the trivial twist of \( {\mathbb{P}}^{1}/K \), then there is a \( K \) -isomorphism \( j : C \rightarrow {\mathbb{P}}^{1} \) . In particular, \( j : C\left( K\right) \rightarrow {\mathbb{P}}^{1}\left( K\right) \) is a bijection, so \( C\left( K\right) \) is certainly nonempty. This proves that (a) ... | Yes |
Even if \( \phi \) has bad reduction at \( \mathfrak{p} \), it may be possible to change coordinates and achieve good reduction. In other words, there may be some \( f \in {\operatorname{PGL}}_{2}\left( K\right) \) such that \( \operatorname{Res}\left( {\phi }^{f}\right) \) is a p-unit. | For example, the map \( \phi \left( z\right) = z + {p}^{2}{z}^{-1} \) has bad reduction at \( p \), since\n\n\[ \phi = \left\lbrack {{X}^{2} + {p}^{2}{Y}^{2},{XY}}\right\rbrack ,\;\text{ so }\;{\operatorname{Res}}_{p}\left( \phi \right) = \operatorname{Res}\left( {{X}^{2} + {p}^{2}{Y}^{2},{XY}}\right) = {p}^{2}. \]\n\n... | Yes |
Proposition 4.95. Let \( \phi = \left\lbrack {F, G}\right\rbrack \) be a rational map of degree \( d \) described by homogeneous polynomials \( F, G \in K\left\lbrack {X, Y}\right\rbrack \) and let \( \mathfrak{p} \) be a prime ideal. (a) The valuation of the minimal resultant of \( \phi \) is given by the formula\n\n\... | Proof. (a) Choose a constant \( c \in {K}^{ * } \) satisfying\n\n\[ \n{\operatorname{ord}}_{\mathfrak{p}}\left( c\right) = \min \left\{ {{\operatorname{ord}}_{\mathfrak{p}}\left( F\right) ,{\operatorname{ord}}_{\mathfrak{p}}\left( G\right) }\right\} \n\]\n\nThen\n\n\[ \n{\operatorname{Res}}_{\mathfrak{p}}\left( \phi \r... | Yes |
Proposition 4.100. Let \( K \) be a number field and \( \phi \left( z\right) \in K\left( z\right) \) a rational map of degree \( d \geq 2 \) . If \( \phi \) has a global minimal model over \( K \), then its Weierstrass class \( {\overline{\mathfrak{a}}}_{\phi /K} \) is trivial. | Proof. Replacing \( \phi \) by \( {\phi }^{f} \) for an appropriate choice of \( f \in {\operatorname{PGL}}_{2}\left( K\right) \), we may assume that \( \phi = \left\lbrack {F, G}\right\rbrack \) with polynomials \( F \) and \( G \) having coefficients in the ring of integers of \( K \) and satisfying\n\n\[ \n{\operato... | Yes |
Theorem 5.2. Let \( \\left( {K,{\\left. \\mid \\cdot \\right| }_{K}}\\right) \) be a valued field. Then there exists a valued field \( \\left( {\\widehat{K},{\\left. \\cdot \\right| }_{\\widehat{K}}}\\right) \), unique up to isomorphism of valued fields, with the following properties:\n\n(a) There is an inclusion \( K ... | Proof. (Sketch) The field \( \\widehat{K} \) may be constructed as follows. Let \( C \) be the set of all Cauchy sequences in \( K \) and make \( C \) into a ring by setting\n\n\[ \n\\left\\{ {\\alpha }_{n}\\right\\} + \\left\\{ {\\beta }_{n}\\right\\} = \\left\\{ {{\\alpha }_{n} + {\\beta }_{n}}\\right\\} \\;\\text{ a... | Yes |
Theorem 5.3. Let \( \left( {K,{\left| \cdot \right| }_{K}}\right) \) be a complete field and let \( L/K \) be a finite extension. Then there is a unique absolute value \( {\left. \mid \cdot \mid \right| }_{L} \) on \( L \) extending the absolute value on \( K \) . The field \( L \) is complete with respect to \( {\left... | Proof. (Sketch) One checks that \( {\left. \mid \cdot \right| }_{L} \) is given by\n\n\[{\left| \alpha \right| }_{L} = {\left| {\mathrm{N}}_{L/K}\left( \alpha \right) \right| }_{K}^{1/\left\lbrack {L : K}\right\rbrack }.\]\n\nSee [259, Proposition XII.2.6] or [382, II §4]. | No |
Theorem 5.6. Let\n\n\\[ \n{\\mathbb{C}}_{p} = {\\widehat{\\mathbb{Q}}}_{p} = \\text{the completion of the algebraic closure of}{\\mathbb{Q}}_{p}\\text{.}\n\\]\n\nThen \\( {\\mathbb{C}}_{p} \\) is both complete and algebraically closed. It is the smallest complete algebraically closed field containing \\( {\\mathbb{Q}}_... | Proof. More generally, the completion of an algebraically closed nonarchimedean field is algebraically closed. We briefly sketch the proof. Let \\( \\alpha \\) be algebraic over \\( {\\mathbb{C}}_{p} \\) , say a root of \\( f\\left( X\\right) = \\sum {a}_{i}{X}^{i} \\in {\\mathbb{C}}_{p}\\left\\lbrack X\\right\\rbrack ... | Yes |
Proposition 5.8. (a) Let \( \phi \left( z\right) \) be a holomorphic function on the closed disk \( \bar{D}\left( {a, r}\right) \) and let \( b \in \bar{D}\left( {a, r}\right) \) . Then \( \phi \left( z\right) \) is a holomorphic function on \( \bar{D}\left( {b, r}\right) \), i.e., \( \phi \left( z\right) \) is given b... | Proof. (a) Let \( \phi \left( z\right) = \sum {a}_{i}{\left( z - a\right) }^{i} \), and for \( k \geq 0 \), define coefficients \( {b}_{k} \) by the formula\n\n\[ {b}_{k} = \mathop{\sum }\limits_{{i = k}}^{\infty }\left( \begin{array}{l} i \\ k \end{array}\right) {a}_{i}{\left( b - a\right) }^{i - k}. \]\n\n(These valu... | Yes |
Proposition 5.10. Let \( \phi \left( z\right) \in K\llbracket z\rrbracket \) be a power series converging on \( \bar{D}\left( {a, r}\right) \). Then \[ \left| {\phi \left( z\right) - \phi \left( w\right) }\right| \leq \frac{\parallel \phi \parallel }{r}\left| {z - w}\right| \;\text{ for all }z, w \in \bar{D}\left( {a, ... | Proof. We take \( z, w \in \bar{D}\left( {a, r}\right) \) and compute \[ \left| {\phi \left( z\right) - \phi \left( w\right) }\right| = \left| {\mathop{\sum }\limits_{{i = 0}}^{\infty }{a}_{i}\left( {{\left( z - a\right) }^{i} - {\left( w - a\right) }^{i}}\right) }\right| \] \[ = \left| {z - w}\right| \left| {\mathop{\... | Yes |
Theorem 5.11. Let \( \phi \left( z\right) \in {\mathbb{C}}_{p}\llbracket z\rrbracket \) be a power series. Suppose that the Newton polygon of \( \phi \) includes a line segment of slope \( m \) whose horizontal length is \( N \), i.e., the Newton polygon has a line segment running from\n\n\[ \left( {n, v\left( {a}_{n}\... | Proof. See [249, IV.4, Corollary to Theorem 14]. We observe that the proof of this result for polynomials or rational functions is quite easy. For power series, one first proves a \( p \) -adic version of the Weierstrass preparation theorem saying, roughly, that \( \phi \left( z\right) \) factors into the product of a ... | Yes |
The Newton polygon of the power series\n\n\[ \phi \left( z\right) = {p}^{5} + {p}^{4}z + p{z}^{2} + p{z}^{3} + {p}^{-1}{z}^{4} + {p}^{-1}{z}^{5} + {p}^{3}{z}^{6} + {p}^{-1}{z}^{7} + {p}^{2}{z}^{8} + {p}^{3}{z}^{9} + \cdots \] | is illustrated in Figure 5.1(a). The leftmost line segment has slope -2 and width 2, so \( \phi \left( z\right) \) has exactly 2 roots \( \alpha \) satisfying \( \left| \alpha \right| = {p}^{-2} \) (assuming that \( \phi \left( z\right) \) converges on the appropriate disk). Similarly, \( \phi \left( z\right) \) has ex... | Yes |
Theorem 5.13. (Maximum Modulus Principle) Let \( \phi \left( z\right) \in {\mathbb{C}}_{p}\llbracket z\rrbracket \) be a power series that converges on a disk \( \bar{D}\left( {a, r}\right) \) of rational radius.\n\n(a) There is a point \( \beta \in \bar{D}\left( {a, r}\right) \) satisfying\n\n\[ \left| {\phi \left( \b... | Proof of Theorem 5.13. Write \( \phi \left( z\right) = \sum {a}_{i}{\left( z - a\right) }^{i} \) and choose constants \( b, c \in {K}^{ * } \) with \( \left| c\right| = r \) and \( \left| b\right| = \parallel \phi \parallel \) (cf. Remark 5.9). Consider the series\n\n\[ \psi \left( z\right) = {b}^{-1}\phi \left( {{cz} ... | Yes |
Corollary 5.17. In each of the following situations, the indicated map \( \phi \) is both open and continuous: | (a) The continuity is a consequence of Proposition 5.10, which gives the stronger assertion that \( \phi \) is Lipschitz. The openness of \( \phi \) follows easily from Proposition 5.16, since the collection of \ | No |
Proposition 5.18. For every integer \( n \geq 1 \) ,\n\n\[ \mathcal{F}\left( {\phi }^{n}\right) = \mathcal{F}\left( \phi \right) \;\text{ and }\;\mathcal{J}\left( {\phi }^{n}\right) = \mathcal{J}\left( \phi \right) . \] | Proof. We proved this over \( \mathbb{C} \) in Proposition 1.25, and the same proof works for nonarchimedean fields using the nonarchimedean Lipschitz property (Theorem 2.14) in place of the archimedean version cited in Chapter 1. | Yes |
Corollary 5.19. Let \( K \) be an algebraically closed field of characteristic 0 that is complete with respect to a nonarchimedean absolute value. Let \( \phi \left( z\right) \in K\left( z\right) \) be a rational function of degree \( d \geq 2 \) . Then \( \phi \) has a nonrepelling fixed point. | Proof. If some fixed point \( P \) has multiplier \( {\lambda }_{P}\left( \phi \right) = 1 \), then \( P \) is nonrepelling and we are done. Otherwise, we can use Theorem 1.14 to estimate\n\n\[ 1 = \left| {\mathop{\sum }\limits_{{P \in \operatorname{Fix}\left( \phi \right) }}\frac{1}{1 - {\lambda }_{P}\left( \phi \righ... | Yes |
Proposition 5.20. Let \( K \) be an algebraically closed field of characteristic 0 that is complete with respect to a nonarchimedean absolute value. Let \( \phi \left( z\right) \in K\left( z\right) \) be a rational function of degree \( d \geq 2 \) .\n\n(a) Let \( P \in {\mathbb{P}}^{1}\left( K\right) \) be a nonrepell... | Proof. Making a change of variables, we can move \( P \) to 0, and then Proposition 5.18 lets us replace \( \phi \) by \( {\phi }^{n} \), so we may assume that 0 is a fixed point. This puts \( \phi \) into\nthe form\n\[ \phi \left( z\right) = {\lambda z} + \frac{{z}^{2}F\left( z\right) }{G\left( z\right) } \]\nwith \( ... | Yes |
(a) \( \;\left| z\right| > \frac{1}{p}\; \) and \( \;\left| {z - 1}\right| > \frac{1}{p}\; \Rightarrow \;\mathop{\lim }\limits_{{n \rightarrow \infty }}\left| {{\phi }^{n}\left( z\right) }\right| = \infty \) . | Proof. (a) We consider two cases. First, if \( \left| z\right| > 1 \), then \( \left| z\right| = \left| {z - 1}\right| \), so we find that\n\n\[ \left| {\phi \left( z\right) }\right| = \left| \frac{z\left( {z - 1}\right) }{p}\right| = p \cdot \left| z\right| \cdot \left| {z - 1}\right| = p \cdot {\left| z\right| }^{2}.... | Yes |
Proposition 5.23. Let \( {S}^{\mathbb{N}} \) be the space of \( S \) -sequences with associated metric as above and let \( L : {S}^{\mathbb{N}} \rightarrow {S}^{\mathbb{N}} \) be the left shift map.\n\n(a) If \( \rho \left( {\alpha ,\beta }\right) < 1 \), then \( \rho \left( {L\left( \alpha \right), L\left( \beta \righ... | Proof. (a) The condition \( \rho \left( {\alpha ,\beta }\right) < 1 \) is equivalent to \( {\alpha }_{0} = {\beta }_{0} \) . It is then clear from the definition that \( \rho \left( {L\left( \alpha \right), L\left( \beta \right) }\right) = p \cdot \rho \left( {\alpha ,\beta }\right) \), since \( L{\left( \alpha \right)... | Yes |
Proposition 5.24. With notation as above, the itinerary map \( \beta : \Lambda \rightarrow \{ 0,1{\} }^{\mathbb{N}} \) has the following properties:\n\n(a) \( \beta \) is injective.\n\n(b) \( \beta \left( {\Lambda \cap {\mathbb{Q}}_{p}}\right) = \{ 0,1{\} }^{\mathbb{N}} \), i.e., \( \beta \) restricted to \( \Lambda \c... | Proof. (a) We begin with the following observation. Let \( u \) be 0 or 1 .\n\n\[ \text{If}z, w \in {I}_{u}\text{, then}\left| {\phi \left( z\right) - \phi \left( w\right) }\right| = p \cdot \left| {z - w}\right| \text{.} \]\n\n(5.6)\n\nTo verify (5.6), we use the assumption that \( z, w \in {I}_{u} \) to write \( z = ... | Yes |
Corollary 5.25. Let \( p \geq 3 \) be a prime and let\n\n\[ \phi \left( z\right) = \frac{{z}^{2} - z}{p}\;\text{ and }\;\Lambda = \left\{ {z \in {\mathbb{C}}_{p} : {\phi }^{n}\left( z\right) \text{ is bounded for all }n \geq 0}\right\} .\n\]\n\n(a) \( \mathcal{J}\left( \phi \right) = \Lambda \subset {\mathbb{Q}}_{p} \)... | Proof. Let \( \beta : \Lambda \rightarrow \{ 0,1{\} }^{\mathbb{N}} \) be the itinerary map. Proposition 5.24 tells us that \( \beta \) is injective, and further that it is surjective even when restricted to \( \Lambda \cap {\mathbb{Q}}_{p} \) . It follows that \( \Lambda \subset {\mathbb{Q}}_{p} \), which proves one pa... | Yes |
Theorem 5.27. (Nonarchimedean Montel Theorem, Hsia [208]) Let \( \Phi \) be a collection of rational, or more generally meromorphic, functions \( \bar{D}\left( {a, r}\right) \rightarrow {\mathbb{P}}^{1}\left( K\right) \), and suppose that the union\n\n\[ \mathop{\bigcup }\limits_{{\phi \in \Phi }}\phi \left( {\bar{D}\l... | Proof. Let \( \alpha = \left\lbrack {{\alpha }_{1},{\alpha }_{2}}\right\rbrack \) and \( \beta = \left\lbrack {{\beta }_{1},{\beta }_{2}}\right\rbrack \) be two points of \( {\mathbb{P}}^{1}\left( K\right) \) that are not in the union (5.13). Consider the family of rational (or meromorphic) functions\n\n\[ \Psi = \left... | Yes |
Proposition 5.29. Let \( \phi : {\mathbb{P}}^{1}\left( K\right) \rightarrow {\mathbb{P}}^{1}\left( K\right) \) be a rational map of degree \( d \geq 2 \) and let \( U \subset {\mathbb{P}}^{1}\left( K\right) \) be an open set such that \( U \cap \mathcal{J}\left( \phi \right) \neq \varnothing \) . In particular, we are ... | Proof. The set \( U \) is covered by disks \( \bar{D}\left( {a, r}\right) \), which are both open and closed, so it suffices to prove the proposition under the assumption that \( U = \widetilde{D}\left( {a, r}\right) \) . If the union omits two or more points of \( {\mathbb{P}}^{1}\left( K\right) \), then Montel’s theo... | Yes |
Proposition 5.30. Let \( \phi : {\mathbb{P}}^{1}\left( K\right) \rightarrow {\mathbb{P}}^{1}\left( K\right) \) be a rational map of degree \( d \geq 2 \) , and let \( E \subseteq {\mathbb{P}}^{1}\left( K\right) \) be a closed completely invariant subset for \( \phi \) containing at least three points. Then \( E \) is a... | Proof. Theorem 1.6 tells us that a finite completely invariant subset contains at most two points, so our assumption that \( \# E \geq 3 \) implies that \( E \) is infinite. Notice that the complete invariance of the closed set \( E \) implies the complete invariance of its complement \( U \), which is an open set. It ... | Yes |
Corollary 5.32. Let \( \phi : {\mathbb{P}}^{1}\left( K\right) \rightarrow {\mathbb{P}}^{1}\left( K\right) \) be a rational map of degree \( d \geq 2 \), and assume that \( \mathcal{J}\left( \phi \right) \neq \varnothing \) . (a) \( \mathcal{J}\left( \phi \right) \) has empty interior. | Proof. (a) Let \( \partial \mathcal{J}\left( \phi \right) \) denote the boundary of the Julia set \( \mathcal{J}\left( \phi \right) \) . Theorem 1.24 tells us that \( \mathcal{F}\left( \phi \right) \) and \( \partial \mathcal{J}\left( \phi \right) \) are completely invariant, so the same is true of their union \( \part... | Yes |
Lemma 5.33. Let \( {\phi }_{1}\left( z\right) \) and \( {\phi }_{2}\left( z\right) \) be power series that converge on \( \bar{D}\left( {a, r}\right) \), and suppose that \( {\phi }_{1}\left( {\bar{D}\left( {a, r}\right) }\right) \cap {\phi }_{2}\left( {\bar{D}\left( {a, r}\right) }\right) = \varnothing \) . Then\n\n\[... | Proof. Let\n\n\[ {M}_{1} = \mathop{\sup }\limits_{{z \in \bar{D}\left( {a, r}\right) }}\left| {{\phi }_{1}\left( z\right) }\right| \;\text{ and }\;{M}_{2} = \mathop{\sup }\limits_{{z \in \bar{D}\left( {a, r}\right) }}\left| {{\phi }_{2}\left( z\right) }\right| .\n\nThe maximum modulus principle (Theorem 5.13(a)) says t... | Yes |
Lemma 5.34. Let \( A, B \subset {\mathbb{C}}_{p} \) be bounded sets that are at a positive distance from one another. In other words, there are constants \( \Delta ,\delta > 0 \) such that\n\n\[ \mathop{\sup }\limits_{{\alpha \in A}}\left| \alpha \right| \leq \Delta ,\;\mathop{\sup }\limits_{{\beta \in B}}\left| \beta ... | Proof of Lemma 5.34. To ease notation, for \( x, y \in {\mathbb{C}}_{p} \) we write\n\n\[ \left| {x, y}\right| = \max \{ \left| x\right| ,\left| y\right| \} \]\n\nWe also assume (without loss of generality) that \( \Delta \geq 1 \) and \( \delta \leq 1 \) . Then for any \( \alpha ,{\alpha }^{\prime } \in A \) and \( \b... | Yes |
Theorem 5.37. (Hsia [208]) Let \( \phi \left( z\right) \in K\left( z\right) \) be a rational function of degree \( d \) with \( d \geq 2 \) . Then\n\n\[ \mathcal{J}\left( \phi \right) \subset \overline{\operatorname{Per}\left( \phi \right) } \]\n\ni.e., the closure of the periodic points of \( \phi \) contains the Juli... | Proof. We may clearly assume that \( \mathcal{J}\left( \phi \right) \) is not empty. Take any open set \( U \) having nontrivial intersection with \( \mathcal{J}\left( \phi \right) \) . We must show that \( U \) contains a periodic point.\n\nThe Julia set is a perfect set (Corollary 5.32), so the open set \( U \) actua... | Yes |
Theorem 5.40. (Bézivin) If a rational function \( \phi \left( z\right) \in {\mathbb{C}}_{p}\left( z\right) \) has at least one repelling periodic point, then \( \mathcal{J}\left( \phi \right) \) is the closure of the repelling periodic points of \( \phi \) . In particular, one repelling periodic point implies infinitel... | Proof. See [71] for the first assertion. The second then follows immediately from Corollary 5.32(d), since an uncountable set cannot be the closure of a finite set. | No |
Consider the polynomial map \[ \phi \left( z\right) = \frac{{z}^{p} - z}{p} \]. It is clear that the Julia set of \( \phi \) is contained in \( \bar{D}\left( {0,1}\right) \), since if \( \left| \alpha \right| > 1 \), then \( \left| {\alpha }^{p}\right| > \) \( \left| \alpha \right| \), so \[ \left| {\phi \left( \alpha ... | We also observe that if \( \alpha \in \bar{D}\left( {0,1}\right) \cap {\mathbb{Q}}_{p} = {\mathbb{Z}}_{p} \), then Fermat’s little theorem tells us that \( {\alpha }^{p} \equiv \alpha \left( {\;\operatorname{mod}\;p}\right) \), so \( \phi \left( \alpha \right) \in {\mathbb{Z}}_{p} \) . Thus \( {\mathbb{Z}}_{p} \) is a ... | Yes |
Let \( p \geq 5 \) be a prime, and let \( \phi \left( z\right) = p{z}^{3} + a{z}^{2} + b \) with \( a, b \in {\mathbb{Z}}_{p}^{ * } \) . We first consider the fixed points of \( \phi \), which are the roots of the equation\n\n\[ p{z}^{3} + a{z}^{2} - z + b = 0. \] | The assumption that \( a, b \in {\mathbb{Z}}_{p}^{ * } \) implies that the roots satisfy \( \left| {\alpha }_{1}\right| = p \) and \( \left| {\alpha }_{2}\right| = \) \( \left| {\alpha }_{3}\right| = 1 \) . (Look at the Newton polygon!) We also observe that \( p{\alpha }_{1}^{3} \) and \( a{\alpha }_{1}^{2} \) have nor... | Yes |
Let \( X = \mathbb{C} \) and let \( \mathcal{D} \) be the usual collection of open disks in \( \mathbb{C} \). Then the disk components of an open set \( U \subset X \) are the same as the usual path-connected components. | This is clear, since if \( \Gamma \) is a path from \( P \) to \( Q \), then \( \Gamma \) can be covered by open disks contained in \( U \), and the compactness of \( \Gamma \) shows that it suffices to take a finite subcover. Thus the definition of disk components and the related notion of disk connectivity (see Exerc... | No |
Proposition 5.45. Let \( {\mathcal{D}}_{\text{open }} \) and \( {\mathcal{D}}_{\text{closed }} \) be, respectively, the collections of standard open and closed disks in \( {\mathbb{P}}^{1}\left( {\mathbb{C}}_{p}\right) \) as defined above.\n\n(a) Let \( {D}_{1},{D}_{2} \in {\mathcal{D}}_{\text{closed }} \) . Then one o... | Proof. (a) If \( {D}_{1} \cup {D}_{2} = {\mathbb{P}}^{1}\left( {\mathbb{C}}_{p}\right) \), we are done. Otherwise, choose any point in the complement of \( {D}_{1} \cup {D}_{2} \) and use a linear fractional transformation to move that point to \( \infty \) . This reduces us to the case that neither \( {D}_{1} \) nor \... | No |
Theorem 5.46. (Benedetto [56]) Let \( K/{\mathbb{Q}}_{p} \) be a finite extension of \( p \) -adic fields and let \( \phi \left( z\right) \in K\left( z\right) \) be a rational function of degree \( d \geq 2 \) . Proposition 5.20(c) tells us that \( \mathcal{F}\left( \phi \right) \neq \varnothing \), so changing variabl... | Proof of Theorem 5.46. The implication (b) \( \Rightarrow \) (a) is clear, since if \( \alpha \) is a critical point in \( \mathcal{J}\left( \phi \right) \), we can take \( L = K\left( \alpha \right) \) and observe that\n\n\[ {\left( {\phi }^{m}\right) }^{\prime }\left( \alpha \right) = \mathop{\prod }\limits_{{i = 0}}... | Yes |
Example 5.49. Let \( p \) be an odd prime and let\n\n\[ \phi \left( z\right) = \frac{{z}^{2} - z}{p} \]\n\nbe the function that we studied in Section 5.5. We proved (Proposition 5.22) that the Julia set of \( \phi \) is contained in the union of two open disks,\n\n\[ \mathcal{J}\left( \phi \right) \subset D\left( {0,1}... | To see this, we observe that \( B\left( {-1}\right) \) cannot contain any larger disk, since it does not contain 0 . On the other hand, \( D\left( {-1,1}\right) \) is in \( \mathcal{F}\left( \phi \right) \) , since it is disjoint from \( D\left( {0,1}\right) \cup D\left( {1,1}\right) \) . Hence \( B\left( {-1}\right) =... | Yes |
Theorem 5.56. (Benedetto [59]) For \( c \in {\mathbb{C}}_{p} \), let \( {\phi }_{c}\left( z\right) \) be the polynomial\n\n\[ \n{\phi }_{c}\left( z\right) = \left( {1 - c}\right) {z}^{p + 1} + c{z}^{p}.\n\]\n\nThen there exists a value \( a \in {\mathbb{C}}_{p} \) such that:\n\n(1) \( \mathcal{J}\left( {\phi }_{a}\righ... | Proof. See [59] for a proof of this specific theorem, and see [63, 62, 373, 378, 380] for generalizations and related results. | No |
Proposition 5.57. Let \( K \) be a field with an absolute value \( v \) . Let\n\n\[ \Phi = \left( {F, G}\right) : {\mathbb{A}}^{2} \rightarrow {\mathbb{A}}^{2} \]\n\nbe given by homogeneous polynomials \( F, G \in K\left\lbrack {x, y}\right\rbrack \) of degree \( d \geq 1 \), and assume that \( F \) and \( G \) have no... | Proof. (a) We proved inequality (5.33) for general morphisms \( {\mathbb{P}}^{N} \rightarrow {\mathbb{P}}^{M} \) during the course of proving Theorem 3.11. More precisely, see (3.6) on page 92 for the upper bound with an explicit value for \( {c}_{2}\left( {\Phi, v}\right) \), and see (3.7) on page 93 for the lower bou... | Yes |
Proposition 5.58. Let \( K \) be a field with an absolute value \( v \), let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a morphism of degree \( d \geq 2 \), and let \( \Phi = \left( {F, G}\right) : {\mathbb{A}}^{2} \rightarrow {\mathbb{A}}^{2} \) be a lift of \( \phi \) . (a) For all \( \left( {x, y}... | Proof. We consider the two functions\n\n\[ \n\Phi : {\mathbb{A}}_{ * }^{2}\left( K\right) \rightarrow {\mathbb{A}}_{ * }^{2}\left( K\right) \;\text{ and }\;\log \parallel \cdot \parallel : {\mathbb{A}}_{ * }^{2}\left( K\right) \rightarrow \mathbb{R}.\n\]\n\nProposition 5.57(a) tel | No |
Proposition 5.57(a) tells us that they satisfy\n\n\\[ \log \\parallel \\Phi \\left( {x, y}\\right) {\\parallel }_{v} = d\\log \\parallel \\left( {x, y}\\right) {\\parallel }_{v} + O\\left( 1\\right) \\;\\text{ for all }\\left( {x, y}\\right) \\in {\\mathbb{A}}_{ * }^{2}\\left( K\\right) .\n\\] | This is exactly the situation needed to apply Theorem 3.20, from which we conclude that the limit (5.35) exists and satisfies (5.36) and (5.37). Further, Theorem 3.20 says that \\( {\\mathcal{G}}_{\\Phi } \\) is the unique function satisfying (5.36) and (5.37). This completes the proof of (a) and (b). | Yes |
Theorem 5.59. Let \( K \) be a number field, let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a rational function of degree \( d \geq 2 \) defined over \( K \), and let \( \Phi \) be a fixed lift of \( \phi \) . For each absolute value \( v \in {M}_{K} \), let \( {\mathcal{G}}_{\Phi, v} \) be the assoc... | Proof. Let\n\n\[ \n\eta \left( {x, y}\right) = \mathop{\sum }\limits_{{v \in {M}_{K}}}{n}_{v}{\mathcal{G}}_{\Phi, v}\left( {x, y}\right) \;\text{ for }\left( {x, y}\right) \in {\mathbb{A}}_{ * }^{2}\left( K\right) ,\n\]\n\nso a priori the function \( \eta \) is a function on \( {\mathbb{A}}_{ * }^{2}\left( K\right) \) ... | Yes |
Theorem 5.60. Let \( K \) be a field with an absolute value \( v \) and let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a rational function of degree \( d \geq 2 \) defined over \( K \) . Fix a lift \( \Phi = \left( {F, G}\right) \) of \( \phi \) and let \( {\mathcal{G}}_{\Phi } \) be the associated G... | Proof. The Green function satisfies \( {\mathcal{G}}_{\Phi }\left( {{cx},{cy}}\right) = {\mathcal{G}}_{\Phi }\left( {x, y}\right) + \log {\left| c\right| }_{v} \), while the polynomial \( E \) satisfies \( E\left( {{cx},{cy}}\right) = {c}^{e}E\left( {x, y}\right) \), so the difference\n\n\[ \ne{\mathcal{G}}_{\Phi }\lef... | Yes |
Theorem 5.61. Let \( K \) be a number field, let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a rational function of degree \( d \geq 2 \) defined over \( K \), and fix a lift \( \Phi = \left( {F, G}\right) \) of \( \phi \) . Choose a homogeneous polynomial \( E\left( {x, y}\right) \in K\left\lbrack {x... | Proof. We use the definition of \( {\widehat{\lambda }}_{\phi, E, v} \) in terms of the associated Green function \( {\mathcal{G}}_{\Phi, v} \) from Theorem 5.60(a) to compute\n\n\[ \n\frac{1}{\deg E}\mathop{\sum }\limits_{{v \in {M}_{K}}}{n}_{v}{\widehat{\lambda }}_{\phi, E, v}\left( P\right) = \frac{1}{\deg E}\mathop... | Yes |
Theorem 5.68. (Berkovich) The Berkovich disk \( {\bar{D}}^{\mathcal{B}} \) with the Gel’fond topology is a compact path-connected Hausdorff space. | Proof. See [64, Theorem 1.2.1] or [29, Appendix D] for the proof that \( {\bar{D}}^{\mathcal{B}} \) is compact and Hausdorff and [64, Corollary 3.2.3] for the proof that it is path connected. | No |
Theorem 5.72. (a) The Berkovich disks \( {\bar{D}}_{R}^{\mathcal{B}} \) are compact, Hausdorff, and uniquely path connected. | Proof. See [26] and [64]. | No |
Theorem 5.78. Let \( \phi \left( z\right) \in {\mathbb{C}}_{p}\left( z\right) \) be a rational map of degree at least 2 . The support of the canonical measure \( {\mu }_{\phi } \) is equal to the Julia set \( {\mathcal{J}}^{\mathcal{B}}\left( \phi \right) \) . In particular, the Berkovich Julia set \( {\mathcal{J}}^{\m... | Proof. This theorem is an amalgamation of results due to Baker, Rumely, and Rivera-Letelier. We refer the reader to [26, Section 7.5] for the construction of the canonical measure and to [26, Theorem 7.18], [27, Theorems 8.9 and A.7], and [381] for the proof that \( {\mu }_{\phi } \) is supported exactly on \( {\mathca... | No |
Let \( \phi \left( z\right) \in {\mathbb{C}}_{p}\left( z\right) \) be a rational map of degree at least 2 and suppose that \( \phi \) has good reduction. We know (Theorem 2.17) that the classical Julia set \( \mathcal{J}\left( \phi \right) \subset {\mathbb{P}}^{1}\left( {\overline{\mathbb{C}}}_{p}\right) \) is empty. U... | \[ {\mu }_{\phi }\left( U\right) = 1\;\text{ if }\;{\xi }_{0,1} \in U\;\text{ and }\;{\mu }_{\phi }\left( U\right) = 0\;\text{ if }\;{\xi }_{0,1} \notin U. \] Thus \( {\mathcal{J}}^{\mathcal{B}}\left( \phi \right) = \left\{ {\xi }_{0,1}\right\} \), so the nonempty Julia set guaranteed by Theorem 5.78 is not very intere... | Yes |
Theorem 5.80. (Strong Montel Theorem on \( {\mathbb{P}}^{\mathcal{B}} \) ) Let \( \phi \in {\mathbb{C}}_{p}\left( z\right) \) be a rational map of degree at least 2, let \( \xi \in {\mathbb{P}}^{\mathcal{B}} \), let \( U \subset {\mathbb{P}}^{\mathcal{B}} \) be an open neighborhood of \( \xi \), and let \( V \) be the ... | Proof. This theorem is due to Baker and Rumely [27, Theorem 7.1] for maps \( \phi \) defined over a finite extension of \( {\mathbb{Q}}_{p} \), and to Rivera-Letelier in the general case; see [27, Theorem A.1] and [381]. | No |
Theorem 5.81. (Rivera-Letelier) Let \( {\mathcal{U}}_{\phi } \) be the set of points \( \xi \in {\mathbb{P}}^{\mathcal{B}} \) with the property that there is a neighborhood \( U \) of \( \xi \) such that\n\n\[ \n{\mathbb{P}}^{1}\left( {\mathbb{C}}_{p}\right) \smallsetminus \mathop{\bigcup }\limits_{{n \geq 1}}{\phi }^{... | Proof. The proof of (a) is given in [27, Theorem A.2] and the proof of (b) is in [27, Corollary A.5]. | Yes |
Theorem 5.82. Let \( \phi \in {\mathbb{C}}_{p}\left( z\right) \) be a rational map of degree at least 2 . (a) If there is some \( f \in {\mathrm{{PGL}}}_{2}\left( {\mathbb{C}}_{p}\right) \) such that the conjugate \( {\phi }^{f} \) has good reduction, then the Julia set \( {\mathcal{J}}^{\mathcal{B}}\left( \phi \right)... | Proof. (a) See [27, Lemma 8.1]. | No |
Proposition 6.1. Let \( d \in \mathbb{Z} \) with \( \left| d\right| \geq 2 \) and let \( {M}_{d} : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be the power map \( {M}_{d}\left( z\right) = {z}^{d} \) . Then\n\n\[ \operatorname{PrePer}\left( {M}_{d}\right) = {\left( {\mathbb{G}}_{m}\right) }_{\text{tors }} = \left\{... | Proof. We proved this long ago for any abelian group \( G \) and homomorphism \( z \mapsto {z}^{d} \) with \( d \geq 2 \) ; see Proposition 0.3 . The proof for \( d \leq - 2 \) is similar and left to the reader. | No |
Proposition 6.2. Let \( \\left| d\\right| \\geq 2 \) and let \( \\zeta \\in {\\operatorname{Per}}_{n}^{* * }\\left( {M}_{d}\\right) \) be a point of exact period \( n \\geq 2 \) . Then the multiplier of \( {M}_{d} \) at \( \\zeta \) is given by\n\n\[ \n{\\lambda }_{\\zeta }\\left( {M}_{d}\\right) = {d}^{n} \n\] | Proof. Using \( {M}_{d}^{n}\\left( z\\right) = {z}^{{d}^{n}} \), we can directly compute\n\n\[ \n{\\lambda }_{\\zeta }\\left( {M}_{d}\\right) = {\\left. \\frac{d{M}_{d}^{n}\\left( z\\right) }{dz}\\right| }_{z = \\zeta } = {\\left. \\frac{d{z}^{{d}^{n}}}{dz}\\right| }_{z = \\zeta } = {d}^{n}{\\zeta }^{{d}^{n} - 1} = {d}... | Yes |
Let \( K \) be a field and let \( {M}_{d}\left( z\right) = {z}^{d} \) be the \( {d}^{\text{th }} \) -power map for some \( \left| d\right| \geq 2 \) . Further, if \( K \) has finite characteristic \( p \), assume that \( p \nmid d \) . (a) The set of rational maps that commute with \( {M}_{d}\left( z\right) \) is given... | It is clear by a direct computation that the indicated maps \( c{z}^{e} \) commute with \( {M}_{d}\left( z\right) \), so it suffices to prove that they are the only commuting maps. Suppose that \( \phi \left( z\right) \in K\left( z\right) \) commutes with \( {M}_{d}\left( z\right) \), so\n\n\[ \phi \left( {z}^{d}\right... | Yes |
We can use the map \( {M}_{d}\left( z\right) = {z}^{d} \) to illustrate the construction of dynamical units in Section 3.11. First we use Theorem 3.66, which says that if \( \alpha \) has exact order \( n \) and \( \gcd \left( {i - j, n}\right) = 1 \), then\n\n\[ \frac{{\alpha }^{{d}^{i}} - {\alpha }^{{d}^{j}}}{{\alpha... | Taking \( j = 0 \) and \( \gcd \left( {i, n}\right) = 1 \), this implies that\n\n\[ \frac{{\alpha }^{{d}^{i} - 1} - 1}{{\alpha }^{d - 1} - 1}\text{is a unit for all primitive}{\left( {d}^{n} - 1\right) }^{\text{st }}\text{roots of unity}\alpha \text{.} \] | Yes |
Proposition 6.6. For each integer \( d \geq 0 \) there exists a unique polynomial \( {T}_{d}\left( w\right) \in \) \( \mathbb{Q}\left\lbrack w\right\rbrack \) satisfying\n\n\[ \n{T}_{d}\left( {z + {z}^{-1}}\right) = {z}^{d} + {z}^{-d}\;\text{ in the field }\mathbb{Q}\left( z\right) .\n\] | Proof. Suppose first that there do exist polynomials \( {T}_{d}\left( w\right) \) satisfying (6.3). Then\n\n\[ \n{T}_{0}\left( {z + {z}^{-1}}\right) = {z}^{0} + {z}^{-0} = 2, \n\]\n\n\[ \n{T}_{1}\left( {z + {z}^{-1}}\right) = z + {z}^{-1} \n\]\n\n\[ \n{T}_{2}\left( {z + {z}^{-1}}\right) = {z}^{2} + {z}^{-2} = {\left( z... | Yes |
Proposition 6.8. Let \( {T}_{d}\left( w\right) \) be the \( {d}^{\text{th }} \) Chebyshev polynomial for some \( d \geq 2 \) . (a) The fixed points of \( {T}_{d} \) in \( {\mathbb{A}}^{1}\left( \mathbb{C}\right) \) are\n\n\[ \left\{ {2\cos \left( \frac{2\pi j}{d + 1}\right) : 0 \leq j \leq \frac{d + 1}{2}}\right\} \cup... | Proof. See Exercise 6.5. | No |
Theorem 6.9. Let \( K \) be a field and let \( {T}_{d}\left( w\right) \) be the \( {d}^{\text{th }} \) Chebyshev polynomial for some \( d \geq 2 \) . Further, if \( K \) has finite characteristic \( p \), assume that \( p \nmid d \) .\n\n(a) The automorphism group of \( {T}_{d} \) is given by\n\n\[ \n\operatorname{Aut}... | Proof. (a) The assertion that \( \operatorname{Aut}\left( {T}_{d}\right) \subset {\mathbf{\mu }}_{2} \) is an immediate consequence of (b), since (b) implies that any \( f \in \operatorname{Aut}\left( {T}_{d}\right) \) satisfies \( f\left( w\right) = \pm {T}_{1}\left( w\right) = \pm w \) . However, since the proof of (... | Yes |
Proposition 6.6(c) says that \( {T}_{d}\left( w\right) \) satisfies \( {T}_{d}\left( {-w}\right) = {\left( -1\right) }^{d}{T}_{d}\left( w\right) \), so in particular,\n\n\[ \n{T}_{d}\left( w\right) = {w}^{d} + \left( {\text{ terms of degree at most }d - 2}\right) .\n\] | The identity \( {T}_{d}^{f}\left( w\right) = {T}_{d}\left( w\right) \) with \( f\left( w\right) = {aw} + b \) can be written as\n\n\[ \n{T}_{d}\left( {{aw} + b}\right) = a{T}_{d}\left( w\right) + b.\n\]\n\nWe evaluate both sides using (6.7) and look at the top degree terms. This gives\n\n\[ \n{a}^{d}{w}^{d} + d{a}^{d -... | Yes |
Lemma 6.10. Assume that \( K \) does not have characteristic 2. Let \( d \geq 1 \) and let \( F\left( w\right) \) be a polynomial solution to the differential equation\n\n\[ \left( {4 - {w}^{2}}\right) {F}^{\prime }{\left( w\right) }^{2} = {d}^{2}\left( {4 - F{\left( w\right) }^{2}}\right) . \]\n\n(6.8)\n\nThen \( F\le... | Proof. We first check that \( \pm {T}_{d}\left( w\right) \) are solutions. We differentiate the functional equation (6.2) defining the Chebyshev polynomials to obtain the identity\n\n\[ {T}_{d}^{\prime }\left( {z + {z}^{-1}}\right) \left( {1 - {z}^{-2}}\right) = d{z}^{d - 1} - d{z}^{-d - 1}, \]\n\nand then solve for \(... | Yes |
Suppose further that \( G\left( w\right) \) is a polynomial of degree \( e \geq 0 \) that commutes with \( F \) , i.e., \( F\left( {G\left( w\right) }\right) = G\left( {F\left( w\right) }\right) \) . Then\n\n\[ A\left( w\right) {G}^{\prime }{\left( w\right) }^{r} = {e}^{r}A\left( {G\left( w\right) }\right) . \] | Proof. Consider the polynomial\n\n\[ B\left( w\right) = A\left( w\right) {G}^{\prime }{\left( w\right) }^{r} - {e}^{r}A\left( {G\left( w\right) }\right) . \]\n\nWe assume that \( B\left( w\right) \neq 0 \) and derive a contradiction, which will prove the desired result. First we observe that the leading coefficients of... | Yes |
Corollary 6.12. Continuing with the notation and assumptions from Theorem 6.9, if \( d \) is even, then \( {T}_{d} \) has no nontrivial \( \bar{K}/K \) -twists, and if \( d \) is odd, then each \( a \in {K}^{ * } \) yields a twist\n\n\[ \n{T}_{d, a}\left( w\right) = \frac{1}{\sqrt{a}}{T}_{d}\left( {\sqrt{a}w}\right) \n... | Proof. We use the description of \( \operatorname{Aut}\left( \phi \right) \) from (a). If \( d \) is even, then the automorphism group \( \operatorname{Aut}\left( \phi \right) \) is trivial, so Proposition 4.73 says that \( \phi \) has no nontrivial twists. For odd \( d \) we have \( \operatorname{Aut}\left( \phi \righ... | Yes |
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