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Lemma 16.1.5. There exists \( \mu \in K \) such that \( \iota \left( \beta \right) /\beta = {\mu }^{q} \). Furthermore, \( \mu \) is unique and satisfies \( \iota \left( \mu \right) = {\mu }^{-1} \). | Proof. By Lemma 16.1.1 we know that \( \beta \) is an algebraic integer coprime to its conjugates, not divisible by \( \mathfrak{p} \), and equal to a \( q \) th power of an ideal, so write \( \beta {\mathbb{Z}}_{K} = {\mathfrak{b}}^{q} \) for some integral ideal \( \mathfrak{b} \) of \( K \). Recall from Proposition 3... | Yes |
Lemma 16.1.6. Denote by \( r \) an inverse of \( q \) modulo \( p \) . Then \( \phi = \beta {\left( \mu + {\zeta }^{r}\right) }^{q} \) is a unit of \( K \) . | Proof. By the binomial theorem we have \( \beta {\left( \mu + {\zeta }^{r}\right) }^{q} = \mathop{\sum }\limits_{{0 \leq n \leq q}}{c}_{n}\beta {\mu }^{n} \) for some algebraic integers \( {c}_{n} \) . Since \( {\left( \beta {\mu }^{n}\right) }^{q} = {\beta }^{q - n}\iota {\left( \beta \right) }^{n} \) and \( \beta \) ... | Yes |
Lemma 16.1.7. We have\n\n\[ \mu = {\left( 1 + \lambda \right) }^{-1/q}{\left( 1 - \zeta \lambda \right) }^{1/q} \]\n\nwhere \( {\left( 1 + \lambda \right) }^{-1/q} \) (and similarly \( {\left( 1 - \zeta \lambda \right) }^{1/q} \) ) means the power series \( {\left( 1 + x\right) }^{-1/q} \) evaluated at \( x = \lambda \... | Proof. Denote by \( \nu \) the right-hand side. Since \( 1 + \lambda = \left( {x - \zeta }\right) /\left( {1 - \zeta }\right) = \beta \) and \( \iota \left( \lambda \right) = - {\zeta \lambda } \), we have \( {\nu }^{q} = \iota \left( \beta \right) /\beta = {\mu }^{q} \), so that \( \mu = {\eta \nu } \) for some \( q \... | Yes |
Lemma 16.1.8. (1) We have\n\n\[ \frac{\phi }{{\left( 1 + \zeta \right) }^{q}} = \left( {1 - \left( {x - 1}\right) \frac{{\zeta }^{r} - \zeta }{\left( {\zeta - 1}\right) \left( {{\zeta }^{r} + 1}\right) } + O\left( {\lambda }^{2}\right) }\right) . \] | Proof. (1). By definition we have \( \beta = \left( {x - \zeta }\right) /\left( {1 - \zeta }\right) = 1 + \lambda \) . On the other hand, by the preceding lemma we have \( \mu = {\left( 1 + \lambda \right) }^{-1/q}{\left( 1 - \zeta \lambda \right) }^{1/q} \) . Since \( 1/q \in {\mathbb{Z}}_{p} \) we have \( \left( \beg... | Yes |
Lemma 16.1.9. For \( \alpha \in {\mathbb{Z}}_{K} \) we have\n\n\[ \mathcal{N}\left( \alpha \right) \equiv 1 + \operatorname{Tr}\left( {\alpha - 1}\right) + O{\left( \alpha - 1\right) }^{2}. \] | Proof. Write \( \varepsilon = \alpha - 1 \) and \( k = \left( {p - 1}\right) {v}_{p}\left( {\alpha - 1}\right) = {v}_{\mathfrak{p}}\left( {\alpha - 1}\right) \) . For any \( \sigma \in G = \operatorname{Gal}\left( {K/\mathbb{Q}}\right) \) we have \( \sigma \left( \alpha \right) = 1 + \sigma \left( \varepsilon \right) \... | Yes |
Lemma 16.1.10. Denote by \( {\pi }_{K} \) the canonical reduction map from \( {\mathbb{Z}}_{K} \) to \( {\mathbb{Z}}_{K}/\mathfrak{p} = {\mathbb{F}}_{p} \), which is a ring homomorphism. For any \( \alpha \in {\mathbb{Z}}_{K} \) we have \( \operatorname{Tr}\left( \alpha \right) \equiv - {\pi }_{K}\left( \alpha \right) ... | Proof. We have \( \operatorname{Tr}\left( {\zeta }^{k}\right) = \operatorname{Tr}\left( \zeta \right) = - 1 \) if \( p \nmid k \), and \( \operatorname{Tr}\left( {\zeta }^{k}\right) = \operatorname{Tr}\left( 1\right) = \) \( p - 1 \equiv - 1\left( {\;\operatorname{mod}\;p}\right) \) if \( p \mid k \), so the result is ... | Yes |
Theorem 16.1.11 (Mihäilescu). Let \( p \) and \( q \) be distinct odd primes. If \( p \nmid {h}_{q}^{ - } \) or \( q \nmid {h}_{p}^{ - } \) the equation \( {x}^{p} - {y}^{q} = 1 \) does not have any nonzero solutions. | Proof. By Cassels’s theorem, we know that a solution to \( {x}^{p} - {y}^{q} = 1 \) implies \( \left( {{x}^{p} - 1}\right) /\left( {x - 1}\right) = p{v}^{q} \), which does not have any nontrivial solution for \( p \geq 5 \) by Dupuy’s theorem, and for \( p = 3 \) by Nagell’s Corollary 6.7.15. Symmetrically \( \left( {{... | Yes |
Corollary 16.1.12. If \( p \) and \( q \) are distinct odd primes and \( p \) or \( q \) is less than or equal to 43 then the equation \( {x}^{p} - {y}^{q} = 1 \) does not have any nonzero solutions. | Proof. Thanks to the above theorem it is sufficient to check that for any \( p \) and \( q \) with \( \min \left( {p, q}\right) \leq {43} \) we have \( p \nmid {h}_{q}^{ - } \) or \( q \nmid {h}_{p}^{ - } \) . For this we need to compute \( {h}_{p}^{ - } \) for small values of \( p \), which is easily done thanks to Co... | Yes |
Lemma 16.2.1. \( {\varepsilon }^{ \pm }R\left\lbrack G\right\rbrack = R{\left\lbrack G\right\rbrack }^{ \pm } \), and both are free \( R \) -modules of dimension \( \left( {p - 1}\right) /2 \) . | Proof. The left-hand side is always a submodule of the right-hand side. Thus let \( x = \mathop{\sum }\limits_{{1 \leq t \leq p - 1}}{a}_{t}{\sigma }_{t} \in R{\left\lbrack G\right\rbrack }^{ \pm } \) . Since \( \iota {\sigma }_{t} = {\sigma }_{p - t} \), this means that \( {a}_{p - t} = \pm {a}_{t} \) . Thus if we set... | No |
Proposition 16.2.5. (1) \( E \) is a \( \mathbb{Z}\left\lbrack G\right\rbrack \) -module and \( E = \mathbb{Z}{\left\lbrack {\zeta }_{p},1/p\right\rbrack }^{ * } \) . | Proof. Immediate consequences of the definition and left to the reader. | No |
Lemma 16.2.6. Keep all the above notation.\n\n(1) We have an exact sequence of \( {\mathbb{F}}_{q}\left\lbrack G\right\rbrack \) -modules\n\n\[ 0 \rightarrow E/{E}^{q} \rightarrow S \rightarrow {Cl}\left( K\right) \left\lbrack q\right\rbrack \rightarrow 0. \] | Proof. (1) is a general property of Selmer groups and is immediate to prove: if \( \bar{\alpha } \in S \) then \( \alpha {\mathbb{Z}}_{K} = {\pi }^{k}{\mathfrak{a}}^{q} \) for some ideal \( \mathfrak{a} \), and we send \( \bar{\alpha } \) to the ideal class of \( \mathfrak{a} \) . It is clear that this lands in \( {Cl}... | Yes |
Lemma 16.3.2. The map sending \( \theta \in X \) to \( \alpha \in {K}^{ * } \) such that \( {\left( x - {\zeta }_{p}\right) }^{\theta } = {\alpha }^{q} \) is a well-defined injective group homomorphism. | Proof. The map is well defined since \( K = \mathbb{Q}\left( {\zeta }_{p}\right) \) does not contain any other \( q \) th root of unity than 1 . It is clear that it is a group homomorphism from the additive group \( X \) to the multiplicative group \( {K}^{ * } \) . Let us show that it is injective: let \( \theta \in X... | Yes |
Lemma 16.3.4. The number of \( k \) -tuples of nonnegative integers \( {\lambda }_{i} \) such that \( \mathop{\sum }\limits_{{1 \leq i \leq k}}{\lambda }_{i} \leq s \) is equal to \( \left( \begin{matrix} s + k \\ s \end{matrix}\right) = \left( \begin{matrix} s + k \\ k \end{matrix}\right) \) . | Proof. The map that sends \( {\left( {\lambda }_{i}\right) }_{1 \leq i \leq k} \) to the set of \( \mathop{\sum }\limits_{{1 \leq i \leq j}}\left( {{\lambda }_{i} + 1}\right) \) for \( 1 \leq j \leq k \) is easily seen to be a bijection from the set of \( k \) -tuples with sum \( s \) to the set of subsets of cardinali... | Yes |
Lemma 16.3.5. Assume that \( \min \left( {p, q}\right) \geq {11} \) and that \( q > 4{p}^{2} \) . There exist at least \( q + 1 \) elements \( \theta \in I \) such that \( \parallel \theta \parallel \leq {3q}/\left( {2\left( {p - 1}\right) }\right) \) . | Proof. Recall from Lemma 16.2.3 that \( I \) has a basis of elements \( {e}_{i} \) for \( 1 \leq i \leq \left( {p - 1}\right) /2 \) that are such that \( \begin{Vmatrix}{e}_{i}\end{Vmatrix} = p - 1 \) . Consider the set of \( \theta = \mathop{\sum }\limits_{{1 \leq i \leq \left( {p - 1}\right) /2}}{\lambda }_{i}{e}_{i}... | Yes |
Theorem 16.3.7. Let \( p \) and \( q \) be odd primes such that \( \min \left( {p, q}\right) \geq {11} \), and let \( x \) and \( y \) be nonzero integers such that \( {x}^{p} - {y}^{q} = 1 \) . Then \( p < 4{q}^{2} \) and \( q < 4{p}^{2} \) . | Proof. By symmetry, it is enough to prove that \( q < 4{p}^{2} \) . Assume by contradiction that \( q > 4{p}^{2} \) . By Proposition 16.3.6 for all \( \tau \in G \) there exists a nonzero \( \theta \in I \) such that \( \parallel \theta \parallel \leq {3q}/\left( {p - 1}\right) \) and \( \left| {\operatorname{Arg}\left... | Yes |
Lemma 16.4.1. Let \( R \) be a commutative ring, \( \mathfrak{b} \) an ideal of \( R, M \) an \( R \) - module of finite type, and \( \phi \) an \( R \) -endomorphism of \( M \) such that \( \phi \left( M\right) \subset \mathfrak{b}M \) . There exists a nonzero monic polynomial \( P \in R\left\lbrack X\right\rbrack \) ... | Proof. Let \( {\left( {m}_{i}\right) }_{1 \leq i \leq n} \) be an \( R \) -generating set for \( M \), and let \( {b}_{i, j} \in \mathfrak{b} \) be such that \( \phi \left( {m}_{j}\right) = \mathop{\sum }\limits_{{1 \leq i \leq n}}{b}_{i, j}{m}_{i} \) for \( 1 \leq j \leq n \) . The module \( M \) can be considered as ... | Yes |
Lemma 16.4.2. Let \( R \) be a commutative ring, \( \mathfrak{b} \) an ideal of \( R, M \) an \( R \) - module of finite type, and denote by \( \psi \) the canonical surjection from \( R \) to \( R/\mathfrak{b} \) . If \( R/\left( {{\operatorname{Ann}}_{R}\left( M\right) + \mathfrak{b}}\right) \) has no nonzero nilpote... | \[ \psi \left( {{\operatorname{Ann}}_{R}\left( M\right) }\right) = {\operatorname{Ann}}_{R/\mathfrak{b}}\left( {M/\mathfrak{b}M}\right) . \] Proof. The inclusion \( \subset \) is trivial, so let us show the reverse inclusion. Thus, let \( \psi \left( \alpha \right) \in {\operatorname{Ann}}_{R/\mathfrak{b}}\left( {M/\ma... | Yes |
Lemma 16.4.3. Let \( H \) be a cyclic group of order \( n \), and assume that \( q \nmid n \) . Set \( s = \mathop{\sum }\limits_{{\sigma \in H}}\sigma \in {\mathbb{F}}_{q}\left\lbrack H\right\rbrack \) . The rings \( {\mathbb{F}}_{q}\left\lbrack H\right\rbrack \) and \( {\mathbb{F}}_{q}\left\lbrack H\right\rbrack /\le... | Proof. Since \( H \) is cyclic we have \( {\mathbb{F}}_{q}\left\lbrack H\right\rbrack \simeq {\mathbb{F}}_{q}\left\lbrack X\right\rbrack /\left( {\left( {{X}^{n} - 1}\right) {\mathbb{F}}_{q}\left\lbrack X\right\rbrack }\right) \), and \( {\mathbb{F}}_{q}\left\lbrack H\right\rbrack /\left( {s{\mathbb{F}}_{q}\left\lbrack... | Yes |
Lemma 16.4.5. Let \( H \) be a cyclic group of order \( n \), and assume that \( q \nmid n \) . Then \( {\mathbb{F}}_{q}\left\lbrack H\right\rbrack \) is a semisimple ring. | Proof. Let \( {X}^{n} - 1 = \mathop{\prod }\limits_{{1 \leq i \leq g}}{P}_{i}^{{e}_{i}}\left( X\right) \) be the decomposition of \( {X}^{n} - 1 \) as a power product of distinct monic irreducible polynomials in \( {\mathbb{F}}_{q}\left\lbrack X\right\rbrack \) . Since \( q \nmid n \) the polynomial \( {X}^{n} - 1 \) h... | Yes |
Lemma 16.4.7. We have \( {Cl}\left( {K}^{ + }\right) \left\lbrack q\right\rbrack = {Cl}\left( K\right) {\left\lbrack q\right\rbrack }^{ + } \) . | Proof. By Proposition 3.5.21 we can write by abuse of notation \( {Cl}\left( {K}^{ + }\right) \left\lbrack q\right\rbrack \subset \) \( {Cl}\left( K\right) \left\lbrack q\right\rbrack \), and since evidently \( {Cl}\left( {K}^{ + }\right) \) is invariant by \( \iota \) we have \( {Cl}\left( {K}^{ + }\right) \left\lbrac... | Yes |
Lemma 16.4.10. We have\n\n\[ \n{E}_{q} = \\left\\{ {u \\in E, u \\equiv {\\beta }^{q}\\left( {{\\;\\operatorname{mod}\\{q}^{2}}{R}_{p}}\\right) }\\right\\} .\n\] | Proof. If \( u \) belongs to the right-hand side then \( u \\in E, u \\equiv {\\beta }^{q}\\left( {{\\;\\operatorname{mod}\\{q}^{2}}{R}_{p}}\\right) \) , so \( {\\beta }^{q} \) modulo \( {q}^{2} \) is equal to \( u \) . Since elements of \( E \) are invertible in \( {R}_{p} \) , it follows that \( {\\beta }^{q} \) modu... | Yes |
Lemma 16.4.15. Let \( R \) be a commutative ring of characteristic 0, let \( f\left( T\right) = \) \( \mathop{\sum }\limits_{{k \geq 0}}\left( {{a}_{k}/k!}\right) {T}^{k} \) and \( g\left( T\right) = \mathop{\sum }\limits_{{k \geq 0}}\left( {{b}_{k}/k!}\right) {T}^{k} \), and let \( q \in R \) . Assume that there exist... | Proof. Immediate and left to the reader. | No |
Proposition 16.4.18. For simplicity, write \( F \) instead of \( {F}_{\theta } \) . (1) The coefficients of \( F\left( T\right) \) are integral outside \( q \), in other words have the form \( a/{q}^{k} \) for some \( a \in {\mathbb{Z}}_{K} \) and \( k \in {\mathbb{Z}}_{ \geq 0} \) . (2) More precisely, if \( \theta = ... | Proof. We have \[ {\left( 1 - \sigma \left( {\zeta }_{p}\right) T\right) }^{{n}_{\sigma }/q} = \mathop{\sum }\limits_{{k \geq 0}}\left( \begin{matrix} {n}_{\sigma }/q \\ k \end{matrix}\right) {\left( -\sigma \left( {\zeta }_{p}\right) \right) }^{k}{T}^{k}, \] hence (1) follows from Lemma 4.2.8. More precisely, we have ... | Yes |
Proposition 16.4.19. Keep the same assumptions and notation, but assume in addition that \( \theta \in \left( {1 + \iota }\right) \mathbb{Z}\left\lbrack G\right\rbrack \) . Then\n\n(1) \( {F}_{\theta } = F \in {K}^{ + }\left\lbrack \left\lbrack T\right\rbrack \right\rbrack \) .\n\n(2) Assume that \( t \in \mathbb{Q} \)... | Proof. Since \( \theta = \mathop{\sum }\limits_{{\sigma \in G}}{n}_{\sigma }\sigma \in \left( {1 + \iota }\right) \mathbb{Z}\left\lbrack G\right\rbrack \) we have \( {\iota \theta } = \theta \) hence \( {n}_{\iota \sigma } = {n}_{\sigma } \) for all \( \sigma \in G \) . Thus if as usual \( P \) is a set of representati... | Yes |
Theorem 16.4.21. Let \( p \) and \( q \) be odd primes such that \( \min \left( {p, q}\right) \geq {11} \), and let \( x \) and \( y \) be nonzero integers such that \( {x}^{p} - {y}^{q} = 1 \) . Then \( p \equiv 1\left( {\;\operatorname{mod}\;q}\right) \) or \( q \equiv 1\left( {\;\operatorname{mod}\;p}\right) \) . | Proof. By Theorem 16.4.20, \( {\operatorname{Ann}}_{R}\left( {\left\lbrack x - {\zeta }_{p}\right\rbrack }^{1 + \iota }\right) = 0 \) . By Mihäilescu’s first Theorem 16.1.3 we know that \( {q}^{2} \mid x \), and as usual \( \left( {-{\zeta }_{p}}\right) \) is a \( q \) th power since \( q \) and \( {2p} \) are coprime.... | Yes |
Proposition 0.3. Let \( G \) be an abelian group, let \( d \geq 2 \) be an integer, and let \( \phi : G \rightarrow G \) be the \( {d}^{\text{th }} \) power map \( \phi \left( \alpha \right) = {\alpha }^{d} \) . Then\n\n\[ \operatorname{PrePer}\left( {\phi, G}\right) = {G}_{\text{tors }} \] | Proof. The simple nature of the map \( \phi \) allows us to give an explicit formula for its iterates,\n\n\[ {\phi }^{n}\left( \alpha \right) = {\alpha }^{{d}^{n}} \]\n\nNow suppose that \( \alpha \in \operatorname{PrePer}\left( {{\phi }_{d}, G}\right) \) . This means that \( {\phi }^{m + n}\left( \alpha \right) = {\ph... | Yes |
Let \( S \) be a topological space and let \( \phi : S \rightarrow S \) be a continuous map. For a given \( \alpha \in S \), one might ask for a description of the accumulation points of \( {\mathcal{O}}_{\phi }\left( \alpha \right) \). | For example, a point \( \alpha \) is called recurrent if it is an accumulation point of \( {\mathcal{O}}_{\phi }\left( \alpha \right) \). In other words, \( \alpha \) is recurrent if there is a sequence of integers \( {n}_{1} < {n}_{2} < {n}_{3} < \cdots \) such that \( \mathop{\lim }\limits_{{i \rightarrow \infty }}{\... | No |
Theorem 1.1. (Riemann-Hurwitz Formula for \( {\mathbb{P}}^{1} \) )\n\n\[ \n{2d} - 2 = \mathop{\sum }\limits_{{\alpha \in {\mathbb{P}}^{1}}}\left( {{e}_{\alpha }\left( \phi \right) - 1}\right) \n\] | Proof. (Algebraic Proof) After a change of variables using a linear fractional transformation, we may assume that \( \infty \) is neither a ramification point nor the image of a ramification point, and also that \( \phi \left( \infty \right) = 0 \) . This means that \( \phi \left( z\right) \) has the form\n\n\[ \n\phi ... | Yes |
Corollary 1.3. Let \( \phi : {\mathbb{P}}^{1}\left( \mathbb{C}\right) \rightarrow {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) be a rational map of degree \( d \geq 1 \) . (a) Let \( \alpha \in {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) . Then \[ \mathop{\sum }\limits_{{\beta \in {\phi }^{-1}\left( \alpha \right) }}{e}... | Proof. Making a change of coordinates reduces us to the case that \( \alpha \neq \infty \) and \( \infty \notin {\phi }^{-1}\left( \alpha \right) \) . Writing \( \phi \left( z\right) = F\left( z\right) /G\left( z\right) \) and \( {\phi }^{-1}\left( \alpha \right) = \left\{ {{\beta }_{1},\ldots ,{\beta }_{r}}\right\} \)... | Yes |
Theorem 1.5. (Riemann-Hurwitz Formula) Let \( {C}_{1} \) and \( {C}_{2} \) be algebraic curves (Riemann surfaces) of genus \( {g}_{1} \) and \( {g}_{2} \), respectively, and let \( \phi : {C}_{1} \rightarrow {C}_{2} \) be a finite map of degree \( d \geq 1 \) . Then\n\n\[ 2{g}_{1} - 2 = d\left( {2{g}_{2} - 2}\right) + ... | Proof. See [198, IV §2]. | No |
Theorem 1.6. Let \( \phi : {\mathbb{P}}^{1}\left( \mathbb{C}\right) \rightarrow {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) be a rational map of degree \( d \geq 2 \), and let \( E \subset {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) be a finite set satisfying \( {\phi }^{-1}\left( E\right) = E \) . Then \( \# E \leq 2 ... | Proof. The assumption that \( E \) is finite and satisfies \( {\phi }^{-1}\left( E\right) = E \) combined with the fact that \( \phi : {\mathbb{P}}^{1}\left( \mathbb{C}\right) \rightarrow {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) is surjective implies that \( \phi \) acts as a permutation on \( E \) . Since \( E \) i... | Yes |
Theorem 1.7. Let \( \phi : {\mathbb{P}}^{1}\left( \mathbb{C}\right) \rightarrow {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) be a rational map of degree \( d \geq 2 \), and suppose that \( {\phi }^{n} \) is a polynomial map for some \( n \geq 1 \) . Then already \( {\phi }^{2} \) is a polynomial map. Further, if \( \phi... | Proof. Let \( \alpha \) be a totally ramified fixed point of \( {\phi }^{n} \), so \( {\left( {\phi }^{n}\right) }^{-1}\left( \alpha \right) = \{ \alpha \} \) . Consider the chain of maps\n\n\[ \n\{ \alpha \} \overset{\phi }{ \rightarrow }\{ {\phi \alpha }\} \overset{\phi }{ \rightarrow }\{ {\phi }^{2}\alpha \} \overse... | Yes |
Let \( \phi \in \mathbb{C}\left( z\right) \) be a rational map and let \( \alpha \) be a fixed point of \( \phi \). Let \( f \in {\mathrm{{PGL}}}_{2}\left( \mathbb{C}\right) \) be a change of coordinates and set \( \beta = {f}^{-1}\left( \alpha \right) \), so \( \beta \) is a fixed point of the conjugate map \( {\phi }... | \[ {\phi }^{\prime }\left( \alpha \right) = {\left( {\phi }^{f}\right) }^{\prime }\left( \beta \right) \] Proof. Two applications of the chain rule yield \[ {\left( {\phi }^{f}\right) }^{\prime }\left( w\right) = {\left( {f}^{-1} \circ \phi \circ f\right) }^{\prime }\left( w\right) = {\left( {f}^{-1}\right) }^{\prime }... | Yes |
Consider the map \( \phi : {\mathbb{P}}^{1}\left( \mathbb{C}\right) \rightarrow {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) given by \( \phi \left( z\right) = {z}^{d} \) . Then \( {\phi }^{n}\left( z\right) = {z}^{{d}^{n}} \), so | \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{\phi }^{n}\left( \alpha \right) = \left\{ \begin{array}{ll} 0 & \text{ if }\left| \alpha \right| < 1 \\ \infty & \text{ if }\left| \alpha \right| > 1 \end{array}\right. \] It is easy to see that if \( \left| \alpha \right| \neq 1 \), then \( \alpha \in \mathcal{F}\left... | Yes |
The Julia set of the polynomial \( \phi \left( z\right) = {z}^{2} - 2 \) consists of the closed interval on the real axis between -2 and 2. | See Exercise 1.28. | No |
Proposition 1.24. Let \( \phi : {\mathbb{P}}^{1}\left( \mathbb{C}\right) \rightarrow {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) be a rational map of degree \( d \geq 2 \), and let \( \mathcal{F} \) and \( \mathcal{J} \) be the Fatou and Julia sets of \( \phi \), respectively.\n\n(a) The Fatou set \( \mathcal{F} \) is ... | Proof Sketch. (a) Since \( \phi \) is surjective, it suffices to prove that \( {\phi }^{-1}\left( \mathcal{F}\right) = \mathcal{F} \) . Suppose first that \( \alpha \in \mathcal{F} \), and let \( \phi \left( \beta \right) = \alpha \) . For any point \( {\beta }^{\prime } \) that is close to \( \beta \), the Lipschitz p... | No |
Proposition 1.25. 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. Let \( \psi = {\phi }^{n} \) . It is clear from the definition that \( \mathcal{F}\left( \phi \right) \subset \mathcal{F}\left( \psi \right) \), since if we know that iteration of \( \phi \) maintains closeness of points, then the same is certainly true for \( {\phi }^{n} \) .\n\nTo prove the opposite inclusion,... | Yes |
Theorem 1.29. Let \( \phi \left( z\right) \in \mathbb{C}\left( z\right) \) be a rational function of degree \( d \geq 2 \) .\n\n(a) The Julia set \( \mathcal{J}\left( \phi \right) \) is nonempty. | Proof. (a) See [43, Theorem 4.2.1] or [95, Theorem III.1.2]. | No |
Theorem 1.30. Let \( \phi \left( z\right) \in \mathbb{C}\left( z\right) \) be a rational function of degree \( d \geq 2 \) . Then the following are equivalent:\n\n(a) The Julia set \( \mathcal{J}\left( \phi \right) \) is equal to \( {\mathbb{P}}^{1}\left( \mathbb{C}\right) \).\n\n(b) The Julia set \( \mathcal{J}\left( ... | Proof. See [43, Theorems 4.2.3, 4.3.2, 9.4.4] or [95, Theorems III.1.9, V.1.2]. | No |
Theorem 1.33. Let \( \phi \left( z\right) \in \mathbb{C}\left\lbrack z\right\rbrack \) be a polynomial of degree \( d \geq 2 \) .\n\n(a) The Julia set \( \mathcal{J}\left( \phi \right) \) is connected if and only if for every critical point \( \alpha \neq \infty \) , the orbit \( {\mathcal{O}}_{\phi }\left( \alpha \rig... | Proof. See [95, Theorems III.4.1, III.4.2]. | No |
Theorem 1.35. Let \( \phi \left( z\right) \in \mathbb{C}\left( z\right) \) be a rational function of degree \( d \geq 2 \) .\n\n(a) The map \( \phi \) has at most \( {2d} - 2 \) nonrepelling periodic cycles in \( {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) . If \( \phi \) is a polynomial map, then it has at most \( d -... | Proof. (a) The sharp bound of \( {2d} - 2 \) for rational maps is due to Shishikura, see [43, Theorem 9.6]. Much earlier, Fatou gave a weaker bound that is sufficient for many applications, see [95, Theorem III.2.7]. The bound for polynomial maps is due to Douady, see [95, Theorem VI.1.2].\n\n(b) This important result ... | Yes |
Theorem 1.36. (Sullivan’s No Wandering Domains Theorem) A rational map \( \phi \in \) \( \mathbb{C}\left( z\right) \) has no wandering domains. | Proof. See [43, Chapter 8], [95, Theorem IV.1.3] or [426]. | No |
Theorem 1.37. Let \( U \) be a periodic connected component of the Fatou set of a rational map \( \phi \in \mathbb{C}\left( z\right) \) . Then \( U \) fits into exactly one of the following categories:\n\n(a) \( U \) contains an attracting periodic point.\n\n(b) \( U \) is parabolic.\n\n(c) \( U \) is a Siegel disk.\n\... | Proof. See [43, Theorem 7.1] or [95, Theorem IV.2.1]. We mention that it is nontrivial to prove that Siegel disks and Herman rings exist. In particular, the Fatou set of a polynomial map cannot contain a Herman ring. | No |
Let \( E : {y}^{2} = {x}^{3} + {ax} + b \) be an elliptic curve as above. Then the duplication map \( \left\lbrack 2\right\rbrack : E\left( \mathbb{C}\right) \rightarrow E\left( \mathbb{C}\right) \) leads to the rational function | \[ {\phi }_{E,2}\left( x\right) = \frac{{x}^{4} - {2a}{x}^{2} - {8bx} + {a}^{2}}{4{x}^{3} + {4ax} + {4b}}. \] | Yes |
Proposition 1.42. Let \( E : {y}^{2} = {x}^{3} + {ax} + b \) be an elliptic curve, let \( d \geq 2 \) be an integer, and let \( {\phi }_{E, d} : {\mathbb{P}}^{1}\left( \mathbb{C}\right) \rightarrow {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) be the associated rational map as above. Then\n\n\[ \operatorname{PrePer}\left... | Proof. We leave the proof as an exercise; see Exercise 1.32. | No |
The field \( \mathbb{Q} \) has the usual real absolute value | \[ {\left| \alpha \right| }_{\infty } = \max \{ \alpha , - \alpha \} \] | No |
Lemma 2.3. Let \( K \) be a field with a nonarchimedean absolute value \( {\left| \cdot \right| }_{v} \) and let \( \alpha ,\beta \in K \) . Then\n\n\[ \n{\left| \alpha \right| }_{v} \neq {\left| \beta \right| }_{v} \Rightarrow {\left| \alpha + \beta \right| }_{v} = \max \left\{ {{\left| \alpha \right| }_{v},{\left| \b... | Proof. We suppose that \( {\left| \alpha \right| }_{v} > {\left| \beta \right| }_{v} \) . The strict inequality\n\n\[ \n{\left| \beta \right| }_{v} < {\left| \alpha \right| }_{v} = {\left| \left( \alpha + \beta \right) - \beta \right| }_{v} \leq \max \left\{ {{\left| \alpha + \beta \right| }_{v},{\left| \beta \right| }... | Yes |
Proposition 2.4. The \( v \) -adic chordal metric has the following properties.\n\n(a) \( 0 \leq {\rho }_{v}\left( {{P}_{1},{P}_{2}}\right) \leq 1 \) .\n\n(b) \( {\rho }_{v}\left( {{P}_{1},{P}_{2}}\right) = 0 \) if and only if \( {P}_{1} = {P}_{2} \) .\n\n(c) \( {\rho }_{v}\left( {{P}_{1},{P}_{2}}\right) = {\rho }_{v}\... | Proof. The lower bound in (a) and parts (b) and (c) of the proposition are obvious from the definition. For the upper bound in (a), we use the nonarchimedean nature of \( v \) to compute\n\n\[ \n{\left| {X}_{1}{Y}_{2} - {X}_{2}{Y}_{1}\right| }_{v} \leq \max \left\{ {{\left| {X}_{1}{Y}_{2}\right| }_{v},{\left| {X}_{2}{Y... | No |
Lemma 2.5. Let\n\n\\[ \nR = \\left\\{ \\alpha \\in K : {\\left| \\alpha \\right| }_{v} \\leq 1\\right\\} \n\\]\n\nbe the ring of integers of \\( K \\), and let \\( f : {\\mathbb{P}}^{1} \\rightarrow {\\mathbb{P}}^{1} \\) be a linear fractional transformation of the form\n\n\\[ \nf\\left( \\left\\lbrack {X, Y}\\right\\r... | Proof. Write each point as \\( {P}_{i} = \\left\\lbrack {{X}_{i},{Y}_{i}}\\right\\rbrack \\) with \\( {X}_{i},{Y}_{i} \\in R \\) and at least one of \\( {X}_{i} \\) or \\( {Y}_{i} \\) in \\( {R}^{ * } \\) . Then \\( \\max \\left\\{ {{\\left| {X}_{i}\\right| }_{v},{\\left| {Y}_{i}\\right| }_{v}}\\right\\} = 1 \\), so\n\... | Yes |
Consider the point \( P = \left\lbrack {\frac{3}{14},\frac{9}{35},\frac{24}{49},\frac{27}{245}}\right\rbrack \in {\mathbb{P}}^{1}\left( \mathbb{Q}\right) \) . We can reduce \( P \) modulo 11 without any modification, since every coordinate is an 11-adic integer and not all coordinates vanish modulo 11. Thus \( \widetil... | \[ P = \left\lbrack {\frac{3}{14},\frac{9}{35},\frac{24}{49},\frac{27}{245}}\right\rbrack = \left\lbrack {\frac{1}{14},\frac{3}{35},\frac{8}{49},\frac{9}{245}}\right\rbrack , \] and then \( \widetilde{P} = \left\lbrack {2,0,2,0}\right\rbrack \left( {\;\operatorname{mod}\;3}\right) \) . Similarly, in order to compute \(... | Yes |
Proposition 2.7. Let \( P = \left\lbrack {{x}_{0},\ldots ,{x}_{N}}\right\rbrack \in {\mathbb{P}}^{N}\left( K\right) \) . Then the reduction \( \widetilde{P} \) is independent of the choice of \( \alpha \) satisfying (2.2). | Proof. Suppose that \( \alpha \) and \( \beta \) both satisfy (2.2). Then \( \alpha \) and \( \beta \) have the same valuation, so \( \alpha {\beta }^{-1} \in {R}^{ * } \) . This allows us to compute\n\n\[ \left\lbrack {\widetilde{{\alpha }^{-1}{x}_{0}},\widetilde{{\alpha }^{-1}{x}_{1}},\ldots ,\widetilde{{\alpha }^{-1... | Yes |
Lemma 2.8. Let \( {P}_{1} \) and \( {P}_{2} \) be points in \( {\mathbb{P}}^{1}\left( K\right) \). Then\n\n\[ \n{\widetilde{P}}_{1} = {\widetilde{P}}_{2}\;\text{ if and only if }\;{\rho }_{v}\left( {{P}_{1},{P}_{2}}\right) < 1.\n\] | Proof. Write \( {P}_{1} = \left\lbrack {{X}_{1},{Y}_{1}}\right\rbrack \) and \( {P}_{2} = \left\lbrack {{X}_{2},{Y}_{2}}\right\rbrack \) using normalized coordinates, so in particular \( {\rho }_{v}\left( {{P}_{1},{P}_{2}}\right) = {\left| {X}_{1}{Y}_{2} - {X}_{2}{Y}_{1}\right| }_{v} \). Suppose first that \( {\widetil... | Yes |
Proposition 2.9. Let \( P, Q \in {\mathbb{P}}^{1}\left( K\right) \) and \( f \in {\operatorname{PGL}}_{2}\left( R\right) \) . Then\n\n\[ \n\widetilde{P} = \widetilde{Q}\;\text{ if and only if }\;\widetilde{f\left( P\right) } = \widetilde{f\left( Q\right) }.\n\] | Proof. We combine Lemmas 2.5 and 2.8. Thus\n\n\[ \n\widetilde{P} = \widetilde{Q}\; \Leftrightarrow \;{\rho }_{v}\left( {P, Q}\right) < 1 \n\]\nfrom Lemma 2.8,\n\n\[ \n\Leftrightarrow \;{\rho }_{v}\left( {f\left( P\right), f\left( Q\right) }\right) < 1 \n\]\nfrom Lemma 2.5,\n\n\[ \n\Leftrightarrow \;\widetilde{f\left( P... | Yes |
Example 2.10. Let \( f = \left( \begin{array}{ll} 5 & 2 \\ 5 & 8 \end{array}\right) \), so \( f \notin {\operatorname{PGL}}_{2}\left( {\mathbb{Z}}_{3}\right) \) . Consider the points \( P = \left\lbrack {7,5}\right\rbrack \) and \( Q = \left\lbrack {4,2}\right\rbrack \) in \( {\mathbb{P}}^{1}\left( {\mathbb{Q}}_{3}\rig... | \[ f\left( P\right) = \left\lbrack {{45},{75}}\right\rbrack = \left\lbrack {3,5}\right\rbrack \equiv \left\lbrack {0,1}\right\rbrack \;\left( {\;\operatorname{mod}\;3}\right) ,\] \[ f\left( Q\right) = \left\lbrack {{24},{36}}\right\rbrack = \left\lbrack {2,3}\right\rbrack \equiv \left\lbrack {1,0}\right\rbrack \;\left(... | Yes |
Proposition 2.11. Let \( {P}_{1},{P}_{2},{P}_{3} \in {\mathbb{P}}^{1}\left( K\right) \) be points whose reductions \( {\widetilde{P}}_{1},{\widetilde{P}}_{2},{\widetilde{P}}_{3} \) are distinct. Then there is a linear fractional transformation \( f \in {\mathrm{{PGL}}}_{2}\left( R\right) \) such that\n\n\[ f\left( {P}_... | Proof. Write \( {P}_{i} = \left\lbrack {{X}_{i},{Y}_{i}}\right\rbrack \) with normalized coordinates. If \( v\left( {X}_{1}\right) > v\left( {Y}_{1}\right) \), we begin by applying the map \( f = Y/X \in {\operatorname{PGL}}_{2}\left( R\right) \) to each of the three points, so we may assume that \( v\left( {X}_{1}\rig... | Yes |
Let \( a \in {K}^{ * } \) and consider the rational map\n\n\[ \n{\phi }_{a}\left( {X, Y}\right) = \left\lbrack {a{X}^{d},{Y}^{d}}\right\rbrack \n\]\n\nIf \( a \in {R}^{ * } \), then \( \widetilde{a} \neq 0 \) and the reduced map \( {\widetilde{\phi }}_{a}\left( {X, Y}\right) = \left\lbrack {\widetilde{a}{X}^{d},{Y}^{d}... | Keep in mind that our goal is to use the dynamics of \( \widetilde{\phi } \) to help us understand the dynamics of \( \phi \) . In the above example, if \( v\left( a\right) = 0 \), then it is easy to see that for any \( P \in {\mathbb{P}}^{1}\left( K\right) \)\n\n\[ \n\widetilde{\phi \left( P\right) } = \widetilde{\phi... | Yes |
Proposition 2.13. Let\n\n\[ A\left( {X, Y}\right) = {a}_{0}{X}^{n} + {a}_{1}{X}^{n - 1}Y + \cdots + {a}_{n - 1}X{Y}^{n - 1} + {a}_{n}{Y}^{n}, \]\n\n\[ B\left( {X, Y}\right) = {b}_{0}{X}^{m} + {b}_{1}{X}^{m - 1}Y + \cdots + {b}_{m - 1}X{Y}^{m - 1} + {b}_{m}{Y}^{m} \]\n\nbe homogeneous polynomials of degrees \( n \) and ... | Proof. We begin by showing that the following three conditions are equivalent.\n\n(i) \( A\left( {X, Y}\right) \) and \( B\left( {X, Y}\right) \) have a common zero in \( {\mathbb{P}}^{1}\left( \bar{K}\right) \).\n\n(ii) \( A\left( {X, Y}\right) \) and \( B\left( {X, Y}\right) \) have a common (nonconstant) factor in t... | Yes |
Theorem 2.15. Let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a rational map defined over \( K \) and write \( \phi = \left\lbrack {F, G}\right\rbrack \) in normalized form. The following are equivalent:\n\n(a) \( \deg \left( \phi \right) = \deg \left( \widetilde{\phi }\right) \) .\n\n(b) The equation... | Proof. The equivalence of (b), (c), and (d) is immediate from the basic properties of the resultant given in Proposition 2.13, once we observe that\n\n\[ \operatorname{Res}\left( {\widetilde{F},\widetilde{G}}\right) = \widetilde{\operatorname{Res}\left( {F, G}\right) }.\]\n\nThis equality follows from the fact that the... | Yes |
Theorem 2.17. Let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a rational map that has good reduction.\n\n(a) The map \( \phi \) is everywhere nonexpanding,\n\n\[{\rho }_{v}\left( {\phi \left( {P}_{1}\right) ,\phi \left( {P}_{2}\right) }\right) \leq {\rho }_{v}\left( {{P}_{1},{P}_{2}}\right) \;\text{ f... | Proof. (a) This is immediate from Theorem 2.14 and the fact that good reduction is equivalent to \( \operatorname{Res}\left( \phi \right) \in {R}^{ * } \) .\n\n(b) It is clear from the definition of equicontinuity that a nonexpanding map is equicontinuous. Indeed, the iterates of a nonexpanding map are uniformly contin... | No |
Theorem 2.18. Let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a rational map that has good reduction.\n\n(a) \( \widetilde{\phi }\left( \widetilde{P}\right) = \widetilde{\phi \left( P\right) }\; \) for all \( P \in {\mathbb{P}}^{1}\left( K\right) \) . | Proof. (a) Write \( \phi = \left\lbrack {F\left( {X, Y}\right), G\left( {X, Y}\right) }\right\rbrack \) in normalized form with homogeneous polynomials \( F, G \in R\left\lbrack {X, Y}\right\rbrack \), and write \( P = \left\lbrack {\alpha ,\beta }\right\rbrack \) in normalized form with \( \alpha ,\beta \in R \) . The... | Yes |
Corollary 2.20. Let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a rational map with good reduction. Then the reduction map sends periodic points to periodic points and preperiodic points to preperiodic points:\n\n\[ \operatorname{Per}\left( \phi \right) \rightarrow \operatorname{Per}\left( \widetilde{... | Proof. Suppose first that \( P \) is periodic of exact period \( n \), so \( P = {\phi }^{n}\left( P\right) \) . Reducing both sides modulo \( \mathfrak{p} \) and using Theorem 2.18 yields\n\n\[ \widetilde{P} = \widetilde{{\phi }^{n}\left( P\right) } = {\widetilde{\phi }}^{n}\left( \widetilde{P}\right) \]\n\nwhich show... | Yes |
Theorem 2.21. Let \( \phi : {\mathbb{P}}^{1}\left( K\right) \rightarrow {\mathbb{P}}^{1}\left( K\right) \) be a rational function of degree \( d \geq 2 \) defined over a local field with a nonarchimedean absolute value \( {\left| \cdot \right| }_{v} \) . Assume that \( \phi \) has good reduction, let \( P \in {\mathbb{... | Proof. We make frequent use of Theorem 2.18, which tells us that\n\n\[ \widetilde{{\phi }^{i}\left( Q\right) } = {\widetilde{\phi }}^{i}\left( \widetilde{Q}\right) \;\text{ for all }Q \in {\mathbb{P}}^{1}\left( K\right) \text{ and all }i \geq 0. \]\n\nRecall that we used this relation in Corollary 2.20 to prove that th... | Yes |
Corollary 2.26. Let \( K \) be a number field, let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a rational map defined over \( K \), and let \( \mathfrak{p} \) and \( \mathfrak{q} \) be primes of \( K \) such that \( \phi \) has good reduction at both \( \mathfrak{p} \) and \( \mathfrak{q} \) and such ... | Proof. Using the obvious notation, we have\n\n\[ {m}_{\mathfrak{p}} = \left( {\operatorname{period}\text{ of }\widetilde{\phi }\left( \widetilde{P}\right) {\;\operatorname{mod}\;\mathfrak{p}}}\right) \leq \# {\mathbb{P}}^{1}\left( {\mathbb{F}}_{\mathfrak{p}}\right) = \mathrm{N}\mathfrak{p} + 1, \]\n\n\[ {r}_{\mathfrak{... | Yes |
Let \( q \) be a power of an odd prime \( p \), let \( \zeta \in {\overline{\mathbb{Q}}}_{p} \) be a primitive \( {q}^{\text{th }} \) root of unity, let \( K = {\mathbb{Q}}_{p}\left( \zeta \right) \), and let \( \phi \left( z\right) = 1 + {\zeta z} - {z}^{q} \). The maximal ideal of \( K \) is \( \mathfrak{p} = \left( ... | \[ v\left( p\right) = \left\lbrack {K : {\mathbb{Q}}_{p}}\right\rbrack = q\left( {1 - 1/p}\right) = {p}^{e - 1}\left( {p - 1}\right) . \] Thus the inequality (2.9) in Theorem 2.28 becomes \( {p}^{e - 1} \leq 2{p}^{e - 1} \), which shows that the power of \( p \) cannot be improved. | Yes |
Proposition 2.32. Let \( \phi \left( z\right) \in K\left( z\right) \) be a rational function of degree \( d \geq 2 \) with good reduction.\n\n(a) Let \( P \in {\mathbb{P}}^{1}\left( K\right) \) be a point of period \( n \) for \( \phi \) . Then\n\n\[{\rho }_{v}\left( {{\phi }^{i}P,{\phi }^{j}P}\right) = {\rho }_{v}\lef... | Proof. (a) Proposition 2.14 and the good-reduction assumption imply that\n\n\[{\rho }_{v}\left( {Q, R}\right) \geq {\rho }_{v}\left( {{\phi Q},{\phi R}}\right) \;\text{ for all }Q, R \in {\mathbb{P}}^{1}\left( K\right) .\n\nApplying this repeatedly yields\n\n\[{\rho }_{v}\left( {Q, R}\right) \geq {\rho }_{v}\left( {{\p... | Yes |
Theorem 2.33. (Narkiewicz [325]) Let \( \\phi \\left( z\\right) \\in R\\left\\lbrack z\\right\\rbrack \) be a polynomial of degree \( d \\geq 2 \) whose leading coefficient is a unit in \( R \) . Let \( \\alpha \\in K \) be a periodic point of \( \\phi \) of exact period \( n \) (with \( n \\geq 2 \) ), and let \( i, j... | Proof. The assumption that \( \\phi \\left( z\\right) \) is a polynomial with unit leading coefficient implies that \( \\phi \) has good reduction, since\n\n\[ \n\\operatorname{Res}\\left( {{a}_{0}{X}^{d} + {a}_{1}{X}^{d - 1}Y + \\cdots + {a}_{d}{Y}^{d},{Y}^{d}}\\right) = {a}_{0}^{d}.\n\]\n\nWe next observe that every ... | Yes |
Theorem 2.34. (Morton-Silverman [313, Theorem 6.4(a)]) Let \( \phi \in K\left( z\right) \) be a rational map of degree \( d \geq 2 \) with good reduction. Let \( P \in {\mathbb{P}}^{1}\left( K\right) \) be a periodic point for \( \phi \) of exact period \( n \), and let \( i \) and \( j \) be integers satisfying\n\n\[ ... | Proof. Comparing the definition of the cross-ratio to the definition of the chordal metric, we see that\n\n\[ {\left| \kappa \left( {P}_{1},{P}_{2},{P}_{3},{P}_{4}\right) \right| }_{v} = \frac{{\rho }_{v}\left( {{P}_{1},{P}_{3}}\right) {\rho }_{v}\left( {{P}_{2},{P}_{4}}\right) }{{\rho }_{v}\left( {{P}_{1},{P}_{2}}\rig... | Yes |
Theorem 2.35. Let \( \phi \in K\left( z\right) \) be a rational map of degree \( d \geq 2 \) with good reduction. Let \( {n}_{1},{n}_{2} \in \mathbb{Z} \) be integers with \( {n}_{1} \nmid {n}_{2} \) and \( {n}_{2} \nmid {n}_{1} \), let \( {P}_{1},{P}_{2} \in {\mathbb{P}}^{1}\left( K\right) \) be periodic points of exa... | Proof. Since we have taken normalized homogeneous coordinates, the chordal metric is given by\n\n\[ \n{\rho }_{v}\left( {{P}_{1},{P}_{2}}\right) = {\left| {x}_{1}{y}_{2} - {x}_{2}{y}_{1}\right| }_{v} \n\]\n\nThe assumptions on \( {n}_{1} \) and \( {n}_{2} \) and Proposition 2.32(c) tell us that \( {\rho }_{v}\left( {{P... | Yes |
Example 3.1. Let \( P \in {\mathbb{P}}^{N}\left( \mathbb{Q}\right) \) and write \( P \) using homogeneous coordinates as\n\n\[ P = \left\lbrack {{x}_{0},{x}_{1},\ldots ,{x}_{N}}\right\rbrack \]\n\nSince the coordinates are homogeneous, we can multiply through by an integer to clear the denominators, and we can also can... | Indeed, this set clearly has fewer than \( {\left( 2B + 1\right) }^{N + 1} \) elements, since each coordinate \( {x}_{i} \) of \( P \) is an integer satisfying \( \left| {x}_{i}\right| \leq B \), so has at most \( {2B} + 1 \) possible values. (See Exercise 3.2 for an asymptotic estimate for the size of the set (3.1).) | No |
Proposition 3.3. (Product Formula) Let \( K/\mathbb{Q} \) be a number field. Then\n\n\[ \mathop{\prod }\limits_{{v \in {M}_{K}}}{\left| \alpha \right| }_{v}^{{n}_{v}} = 1\;\text{ for all }\alpha \in {K}^{ * }. \] | For proofs of these two formulas, see for example [258, Section II.1 and Section V.1]. | No |
Proposition 3.4. Let \( K/\mathbb{Q} \) be a number field and \( P \in {\mathbb{P}}^{N}\left( K\right) \) a point.\n\n(a) The height \( {H}_{K}\left( P\right) \) is independent of the choice of homogeneous coordinates for \( P \) .\n\n(b) \( {H}_{K}\left( P\right) \geq 1 \) .\n\n(c) Let \( L/K \) be a finite extension.... | Proof. (a) Any other choice of homogeneous coordinates for \( P = \left\lbrack {{x}_{0},\ldots ,{x}_{N}}\right\rbrack \) has the form \( P = \left\lbrack {\alpha {x}_{0},\ldots ,\alpha {x}_{N}}\right\rbrack \) for some \( \alpha \in {K}^{ * } \) . Then the product formula (Proposition 3.3) yields\n\n\[ \n\mathop{\prod ... | Yes |
Theorem 3.6. Let \( K/\mathbb{Q} \) be a number field, let \( P \in {\mathbb{P}}^{N}\left( \bar{K}\right) \) , and let \( \sigma \in \operatorname{Gal}\left( {\bar{K}/K}\right) \) . Then \[ H\left( {\sigma \left( P\right) }\right) = H\left( P\right) \] In other words, the height is invariant under the action of the Gal... | Proof. Let \( L/K \) be a finite Galois extension such that \( P \in {\mathbb{P}}^{N}\left( L\right) \) . For any absolute value \( v \in {M}_{L} \) and any \( \sigma \in \operatorname{Gal}\left( {L/K}\right) \), we can define a new absolute value \( \sigma \left( v\right) \) on \( L \) by the formula \[ {\left| \alpha... | Yes |
Theorem 3.8. Let \( \alpha \in \overline{\mathbb{Q}} \) be a nonzero algebraic number. Then\n\n\[ H\left( \alpha \right) = 1\;\text{ if and only if }\alpha \text{ is a root of unity. } \] | Proof. If \( \alpha \) is a root of unity, then \( {\left| \alpha \right| }_{v} = 1 \) for every absolute value \( v \), so we clearly have \( H\left( \alpha \right) = 1 \) . Now suppose that \( H\left( \alpha \right) = 1 \) . Directly from the definition of the height we see that\n\n\[ H\left( {\beta }^{n}\right) = H{... | Yes |
Theorem 3.10. (Hilbert’s Nullstellensatz) Let \( I \) and \( J \) be homogeneous ideals properly contained in \( \bar{K}\left\lbrack {{X}_{0},\ldots ,{X}_{N}}\right\rbrack \) . Then\n\n\[ V\left( I\right) = V\left( J\right) \;\text{ if and only if }\;\sqrt{I} = \sqrt{J}. \] | Proof. Suppose that \( \sqrt{I} = \sqrt{J} \) . Let \( P \in V\left( I\right) \) and \( f \in J \) . Then \( {f}^{n} \in I \) for some \( n \geq 1 \), so \( {f}^{n}\left( P\right) = 0 \), so \( f\left( P\right) = 0 \) . This is true for every \( f \in J \), so \( P \in V\left( J\right) \) . This proves that \( V\left( ... | No |
Theorem 3.12. (Northcott [343]) Let \( \phi : {\mathbb{P}}^{N} \rightarrow {\mathbb{P}}^{N} \) be a morphism of degree \( d \geq 2 \) defined over a number field \( K \) . Then the set of preperiodic points \( \operatorname{PrePer}\left( \phi \right) \subset {\mathbb{P}}^{N}\left( \bar{K}\right) \) is a set of bounded ... | Proof. Theorem 3.11 tells us that there is a constant \( C = C\left( \phi \right) \) such that\n\n\[ h\left( {\phi \left( Q\right) }\right) \geq {dh}\left( Q\right) - C\;\text{ for all }Q \in {\mathbb{P}}^{N}\left( \bar{K}\right) .\n\nApplying this with \( Q = R,\phi \left( R\right) ,{\phi }^{2}\left( R\right) ,\ldots ... | Yes |
Theorem 3.22. Let \( \phi : {\mathbb{P}}^{N} \rightarrow {\mathbb{P}}^{N} \) be a morphism of degree \( d \geq 2 \) defined over \( \overline{\mathbb{Q}} \) and let \( P \in {\mathbb{P}}^{N}\left( \overline{\mathbb{Q}}\right) \) . Then \[ P \in \operatorname{PrePer}\left( \phi \right) \;\text{ if and only if }\;{\wideh... | Proof. If \( P \) is preperiodic, then the quantity \( h\left( {{\phi }^{n}\left( P\right) }\right) \) takes on only finitely many values, so it is clear that \( {d}^{-n}h\left( {{\phi }^{n}\left( P\right) }\right) \rightarrow 0 \) as \( n \rightarrow \infty \) . Now suppose that \( {\widehat{h}}_{\phi }\left( P\right)... | Yes |
Theorem 3.27. Let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a rational function of degree \( d \geq 2 \) defined over \( K \), write \( \phi \left( z\right) = F\left( z\right) /G\left( z\right) \) using polynomials \( F, G \in K\left\lbrack z\right\rbrack \) having no common factors, and let \( v \)... | Proof. We defer the proof until Section 5.9; see Theorem 5.60. | No |
Theorem 3.29. Let \( K \) be a number field, and for each \( v \in {M}_{K} \), let \( {\widehat{\lambda }}_{\phi, v} \) be the local canonical height function constructed in Theorem 3.27. Then\n\n\[ \n{\widehat{h}}_{\phi }\left( \alpha \right) = \mathop{\sum }\limits_{{v \in {M}_{K}}}{n}_{v}{\widehat{\lambda }}_{\phi, ... | Proof. We defer the proof until Section 5.9; see Theorem 5.61. | No |
Theorem 3.34. (Roth) Fix \( \epsilon > 0 \) and let \( \alpha \in \overline{\mathbb{Q}} \) be an algebraic number with \( \alpha \notin \mathbb{Q} \) . Then there exists a constant \( \kappa = \kappa \left( {\epsilon ,\alpha }\right) > 0 \) such that\n\n\[ \left| {\frac{x}{y} - \alpha }\right| \geq \frac{\kappa }{{\lef... | Proof. The proof of Roth's theorem is beyond the scope of this book. A nice exposition may be found in [393]. For more general versions, see [205, Part D] and [256, Chapter 7]. | No |
Theorem 3.36. (Siegel) Let \( \phi \left( z\right) \in \mathbb{Q}\left( z\right) \) be a rational function with at least three distinct poles in \( {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) . Then\n\n\[ \n\{ \alpha \in \mathbb{Q} : \phi \left( \alpha \right) \in \mathbb{Z}\}\n\]\n\nis a finite set. | Proof. Write\n\n\[ \n\phi = \left\lbrack {F\left( {X, Y}\right), G\left( {X, Y}\right) }\right\rbrack \n\]\n\nusing homogeneous polynomials \( F\left( {X, Y}\right), G\left( {X, Y}\right) \in \mathbb{Z}\left\lbrack {X, Y}\right\rbrack \) of degree \( d \) having no common factors. For any fraction \( \alpha = a/b \in \... | Yes |
Theorem 3.39. (Siegel [403, 404]) Let \( E/\mathbb{Q} \) be an elliptic curve, let \( \phi \in \mathbb{Q}\left( E\right) \) be a nonconstant rational function on \( E \), and for each rational point \( P \in E\left( \mathbb{Q}\right) \), write\n\n\[ \phi \left( P\right) = \left\lbrack {a\left( P\right), b\left( P\right... | Proof. See [410, IX.3.3]. For a general version on curves of arbitrary genus \( g \geq 1 \) , see for example [205, D.9.4], although as noted above, if \( g \geq 2 \), then Faltings [164, 165] proves that \( C\left( \mathbb{Q}\right) \) is finite. | No |
Proposition 3.44. Let \( \phi \in \mathbb{C}\left( z\right) \) be a rational function of degree \( d \geq 2 \) satisfying \( {\phi }^{2}\left( z\right) \notin \mathbb{C}\left\lbrack z\right\rbrack \), so Theorem 1.7 implies that no iterate of \( \phi \) is a polynomial map. Then\n\n\[ \n\# {\phi }^{-4}\left( \infty \ri... | Proof. We give a pictorial proof using the Riemann-Hurwitz formula. Suppose that \( \phi \) is a rational map with \( \# {\phi }^{-3}\left( \infty \right) \leq 2 \) . There are four possible pictures for the backward orbit of \( \infty \), as illustrated in Figure 3.1.\n\n![aad50936-3f2e-45c3-8a73-b814eb18acbf_117_0.jp... | Yes |
In order to create a rational function whose orbit contains a large number of integral points, we simply take a finite (reasonably random) sequence of integers \( {z}_{0},{z}_{2},\ldots ,{z}_{k - 1} \) and treat the equations\n\n\[ \n{\phi }^{n + 1}\left( {z}_{n}\right) = {z}_{n + 1}\;\text{ for }n = 0,1,\ldots, k - 1 ... | We carry out this procedure with \( d = 2 \) to find the rational function\n\n\[ \n\phi \left( z\right) = \frac{{899}{x}^{2} - {2002x} + {275}}{{33}{x}^{2} - {584x} + {275}} \n\]\n\nsuch that the orbit of 0 contains quite a few integer points: | Yes |
Proposition 3.46. For all integers \( N \geq 0 \) and \( d \geq 2 \) there exists a rational map \( \phi \left( z\right) \in \mathbb{Q}\left( z\right) \) with the following properties:\n\n- \( {\phi }^{2}\left( z\right) \notin \mathbb{C}\left\lbrack z\right\rbrack \) .\n\n- 0 is a wandering point for \( \phi \) .\n\n- ... | Proof. Let \( \psi \left( z\right) \in \mathbb{Q}\left( z\right) \) be any rational map of degree \( d \) for which 0 is not a pre-periodic point. For each \( 0 \leq n \leq N \), write\n\n\[ \n{\psi }^{n}\left( 0\right) = \frac{{a}_{n}}{{b}_{n}} \in \mathbb{Q} \n\]\n\nas a fraction in lowest terms, and let \( B = {b}_{... | Yes |
To illustrate Theorem 3.48, we take \( \phi \left( z\right) = z + 1/z \) and \( \alpha = 1 \) and list the first few values of \( {\phi }^{n}\left( 1\right) = {a}_{n}/{b}_{n} \) in Table 3.1. | Proof. The idea underlying the proof of Theorem 3.48 is fairly simple. Choose some \( \epsilon > 0 \), and suppose that\n\n\[ \left| {a}_{n}\right| \geq {\left| {b}_{n}\right| }^{1 + \epsilon }\;\text{ for infinitely many }n \geq 0. \]\n\n(3.26)\n\nThis means that \( {\phi }^{n}\left( \alpha \right) = {a}_{n}/{b}_{n} \... | No |
Lemma 3.51. Let \( \phi : {\mathbb{P}}^{1}\left( \mathbb{C}\right) \rightarrow {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) be a rational map of degree \( d \geq 2 \), let \( \rho : {\mathbb{P}}^{1}\left( \mathbb{C}\right) \times {\mathbb{P}}^{1}\left( \mathbb{C}\right) \rightarrow \mathbb{R} \) be the chordal metric as... | Proof. We dehomogenize using a parameter \( z \) such that \( Q \neq \infty \) and \( \infty \notin {\phi }^{-1}\left( Q\right) \) . This means that we can write \( Q = \beta \) and \( P = \alpha \) and that we are looking for the \( {\beta }^{\prime } \in {\phi }^{-1}\left( \beta \right) \) that is closest to \( \alph... | Yes |
Lemma 3.53. Let \( \rho \) be the chordal metric on \( {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) . Then for all \( x, y \in \mathbb{C} \subset {\mathbb{P}}^{1}\left( \mathbb{C}\right) \) , \[ \rho \left( {x, y}\right) \leq \frac{1}{2}\rho \left( {y,\infty }\right) \; \Rightarrow \;\rho \left( {x, y}\right) \geq \left... | Proof. Note that \( \rho \left( {z,\infty }\right) = 1/\sqrt{{\left| z\right| }^{2} + 1} \), so directly from the definition of \( \rho \) we have \[ \rho \left( {x, y}\right) = \frac{\left| x - y\right| }{\sqrt{{\left| x\right| }^{2} + 1}\sqrt{{\left| y\right| }^{2} + 1}} = \left| {x - y}\right| \rho \left( {x,\infty ... | Yes |
Proposition 3.54. Let \( \phi \left( z\right) \in K\left( z\right) \) be a rational function of degree \( d \) . The set of \( n \) -periodic points \( {\operatorname{Per}}_{n}\left( \phi \right) \) and the set of primitive \( n \) -periodic points \( {\operatorname{Per}}_{n}^{* * }\left( \phi \right) \) are Galois-inv... | Proof. Let \( \sigma \in \operatorname{Gal}\left( {\bar{K}/K}\right) \) . Then\n\n\[ \phi \left( {\sigma \left( P\right) }\right) = \sigma \left( {\phi \left( P\right) }\right) \;\text{ for all points }P \in {\mathbb{P}}^{1}\left( \bar{K}\right) ,\]\n\nsince \( \phi \) is a rational function with coefficients in \( K \... | Yes |
Consider the quadratic polynomial \( \phi \left( z\right) = {z}^{2} + 1 \) . The sets of primitive \( {2}^{\text{nd }},{3}^{\text{rd }},{4}^{\text{th }} \), and \( {6}^{\text{th }} \) periodic points are given, respectively, by the roots of the polynomials | \[ {\phi }_{2}^{ * }\left( z\right) = \frac{{\phi }^{2}\left( z\right) - z}{\phi \left( z\right) - z} = {z}^{2} + z + 2 \] \[ {\phi }_{3}^{ * }\left( z\right) = \frac{{\phi }^{3}\left( z\right) - z}{\phi \left( z\right) - z} = {z}^{6} + {z}^{5} + 4{z}^{4} + 3{z}^{3} + 7{z}^{2} + {4z} + 5, \] \[ {\phi }_{4}^{ * }\left( ... | Yes |
Theorem 3.56. Let \( \phi \in K\left( z\right) \) be a rational function of degree \( d \) . Let \( {\mathcal{O}}_{1},\ldots ,{\mathcal{O}}_{r} \) be the distinct \( \phi \) orbits in \( {\operatorname{Per}}_{n}^{* * }\left( \phi \right) \) and choose a point \( {P}_{j} \in {\mathcal{O}}_{j} \) in each orbit. For each ... | Proof. In order to show that \( W \) is a homomorphism, we need to verify that\n\n\[ W\left( {\sigma \tau }\right) = W\left( \sigma \right) W\left( \tau \right) \]\n\nWriting this out in terms of the (twisted) definition of the group law on the wreath product, we need to prove that\n\n\[ \left( {{i}_{\sigma \tau },{\pi... | Yes |
Proposition 3.59. Let \( \phi : {\mathbb{P}}^{N} \rightarrow {\mathbb{P}}^{N} \) be a morphism of degree \( d \) defined over \( \mathbb{C} \) . There is a unique probability measure \( {\mu }_{\phi } \) on \( {\mathbb{P}}^{N}\left( \mathbb{C}\right) \) satisfying\n\n\[ \n{\phi }_{ * }{\mu }_{\phi } = {\mu }_{\phi }\;\... | Proof. For a general construction that covers both archimedean and nonarchimedean base fields, see [453]. We also mention a standard result in dynamics (the Krylov-Bogolubov theorem [226, Theorem 4.1.1]), which says that any continuous map \( \phi : X \rightarrow X \) on a metrizable compact topological space \( X \) a... | Yes |
Theorem 3.61. (Yuan [450]) Let \( \phi : {\mathbb{P}}^{N} \rightarrow {\mathbb{P}}^{N} \) be a morphism of degree \( d \geq 2 \) defined over \( K \) and let \( {P}_{1},{P}_{2},{P}_{3},\ldots \in {\mathbb{P}}^{N}\left( \widetilde{K}\right) \) be a sequence of points satisfying the following two conditions:\n\n(a) Every... | Proof. The proof is beyond the scope of this book. See Yuan [450] for a general version over archimedean and nonarchimedean base fields and algebraic dynamical systems on arbitrary projective varieties. Earlier results and generalizations are given by Autissier [15, 16], Baker-Ih [24], Baker-Rumely [28], Chambert-Loir ... | No |
Proposition 3.63. Let \( K \) be a number field, let \( \phi \in K\left( z\right) \) be a rational map of degree \( d \geq 2 \), let \( P \in {\operatorname{Per}}_{n}^{* * }\left( {\phi, K}\right) \) be a point in \( {\mathbb{P}}^{1}\left( K\right) \) of exact period \( n \), and let \( \mathfrak{p} \) be a prime of \(... | Proof. Let \( \mathfrak{p} \) be a prime of \( K \) satisfying (3.45),(3.46), and (3.47). Let \( m \) be the exact period of \( \widetilde{P} \) and let \( r \) be the order of \( {\lambda }_{\widetilde{\phi }}\left( \widetilde{P}\right) \) in \( {\mathbb{F}}_{\mathfrak{p}}^{ * } \) . Theorem 2.21 tells us that either ... | Yes |
Corollary 3.64. Let \( K \) be a number field, let \( \phi \left( z\right) \in K\left( z\right) \) be a rational map of degree \( d \geq 2 \), and let \( {K}_{n,\phi } \) be the \( {n}^{\text{th }} \) dynatomic field for \( \phi \) . Let \( S \) be the set of primes \( \mathfrak{p} \) of \( K \) such that either \( \ph... | Proof. The field extension \( {K}_{n,\phi }/K \) is generated by the points of \( {\operatorname{Per}}_{n}^{* * }\left( \phi \right) \) . Proposition 3.63 tells us that if \( \mathfrak{p} \notin S \), then those points remain distinct when reduced modulo primes lying over \( \mathfrak{p} \) . Hence the extension \( {K}... | Yes |
We continue studying the quadratic polynomial \( \phi \left( z\right) = {z}^{2} + 1 \) from Example 3.55 (see also Example 2.37). The polynomial \( \phi \left( z\right) \) has everywhere good reduction, so if we assume for the moment that none of its periodic points have multiplier equal to 1, then the quantity (3.48) ... | \[ \mathop{\prod }\limits_{{{\phi }_{n}^{ * }\left( \alpha \right) = 0}}\left( {{\lambda }_{\phi }\left( \alpha \right) - 1}\right) = \mathop{\prod }\limits_{{{\phi }_{n}^{ * }\left( \alpha \right) = 0}}\left( {{\left( {\phi }^{n}\right) }^{\prime }\left( \alpha \right) - 1}\right) = \operatorname{Res}\left( {{\phi }_{... | Yes |
Consider the rational map\n\n\\[ \n\\phi \\left( z\\right) = {z}^{2} - 4 \n\\] | After some algebra, we find that\n\n\\[ \n\\phi \\left( z\\right) - z = {z}^{2} - z - 4 \n\\]\n\n\\[ \n\\frac{{\\phi }^{2}\\left( z\\right) - z}{\\phi \\left( z\\right) - z} = {z}^{2} + z - 3 \n\\]\n\n\\[ \n\\frac{{\\phi }^{3}\\left( z\\right) - z}{\\phi \\left( z\\right) - z} = \\left( {{z}^{3} - {z}^{2} - {6z} + 7}\\... | Yes |
The \( {n}^{\text{th }} \) cyclotomic polynomial is defined using an inclusion-exclusion product, \[ {n}^{\text{th }}\text{ cyclotomic polynomial } = \mathop{\prod }\limits_{{k \mid n}}{\left( {z}^{k} - 1\right) }^{\mu \left( {n/k}\right) }.\] | It is the polynomial whose roots are the primitive \( {n}^{\text{th }} \) roots of unity. Here \( \mu \) is the Möbius function \( \mu \) defined by \( \mu \left( 1\right) = 1 \) and \[ \mu \left( {{p}_{1}^{{e}_{1}}\cdots {p}_{r}^{{e}_{r}}}\right) = \left\{ \begin{array}{ll} {\left( -1\right) }^{r} & \text{ if }{e}_{1}... | No |
Let \( \phi \left( z\right) \) be the polynomial\n\n\[ \phi \left( z\right) = {z}^{2} - \frac{3}{4} \] | Then\n\n\[ \phi \left( z\right) - z = {z}^{2} - z - \frac{3}{4} = \left( {z - \frac{3}{2}}\right) \left( {z + \frac{1}{2}}\right) ,\]\n\n\[ {\phi }^{2}\left( z\right) - z = {z}^{4} - \frac{3}{2}{z}^{2} - z - \frac{3}{16} = \left( {z - \frac{3}{2}}\right) {\left( z + \frac{1}{2}\right) }^{3},\]\n\n\[ \frac{{\phi }^{2}\l... | Yes |
Theorem 4.5. Let \( \phi \left( z\right) \in K\left( z\right) \) be a rational function of degree \( d \geq 2 \) . For each \( P \in \) \( {\mathbb{P}}^{1}\left( \bar{K}\right) \), let\n\n\[ \n{a}_{P}\left( n\right) = {\operatorname{ord}}_{P}\left( {{\Phi }_{\phi, n}\left( {X, Y}\right) }\right) \;\text{ and }\;{a}_{P}... | Proof. By the definition of \( {\Phi }_{n}^{ * } \), we have the relation\n\n\[ \n{a}_{P}^{ * }\left( n\right) = \mathop{\sum }\limits_{{m \mid n}}\mu \left( {n/m}\right) {a}_{P}\left( m\right)\n\] | Yes |
Lemma 4.6. Let \( \psi \left( z\right) \in K\left( z\right) \) be a rational function of degree \( d \geq 2 \), let \( P \in \) \( {\mathbb{P}}^{1}\left( \bar{K}\right) \) be a fixed point of \( \psi \), let \( \lambda = {\lambda }_{P}\left( \psi \right) = {\psi }^{\prime }\left( P\right) \) be the multiplier of \( \ps... | Proof. Making a change of variables, we may assume that \( P = 0 \), and then the assumption that \( P \) is a fixed point of \( \psi \) means that \( \psi \left( z\right) \) has the form\n\n\[ \n\psi \left( z\right) = {\lambda z} + \alpha {z}^{e} + O\left( {z}^{e + 1}\right) \n\]\n\n(4.9)\n\nwhere \( \alpha \neq 0, e ... | Yes |
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