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Lemma 5.1.2. Let \( {\Gamma }_{1} \) and \( {\Gamma }_{2} \) be congruence subgroups of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \), and let \( \alpha \) be an element of \( {\mathrm{{GL}}}_{2}^{ + }\left( \mathbb{Q}\right) \) . Set \( {\Gamma }_{3} = {\alpha }^{-1}{\Gamma }_{1}\alpha \cap {\Gamma }_{2} \), a sub... | Proof. The map \( {\Gamma }_{2} \rightarrow {\Gamma }_{1} \smallsetminus {\Gamma }_{1}\alpha {\Gamma }_{2} \) taking \( {\gamma }_{2} \) to \( {\Gamma }_{1}\alpha {\gamma }_{2} \) clearly surjects. It takes elements \( {\gamma }_{2},{\gamma }_{2}^{\prime } \) to the same orbit when \( {\Gamma }_{1}\alpha {\gamma }_{2} ... | Yes |
Proposition 5.2.1. Let \( N \in {\mathbb{Z}}^{ + } \), let \( {\Gamma }_{1} = {\Gamma }_{2} = {\Gamma }_{1}\left( N\right) \), and let \( \alpha = \left\lbrack \begin{array}{ll} 1 & 0 \\ 0 & p \end{array}\right\rbrack \) where \( p \) is prime. The operator \( {T}_{p} = {\left\lbrack {\Gamma }_{1}\alpha {\Gamma }_{2}\r... | \[ {T}_{p}f = \left\{ \begin{array}{ll} \mathop{\sum }\limits_{{j = 0}}^{{p - 1}}f{\left\lbrack \left\lbrack \begin{array}{ll} 1 & j \\ 0 & p \end{array}\right\rbrack \right\rbrack }_{k} & \text{ if }p \mid N, \\ \mathop{\sum }\limits_{{j = 0}}^{{p - 1}}f{\left\lbrack \left\lbrack \begin{array}{ll} 1 & j \\ 0 & p \end{... | Yes |
Let \( f \in {\mathcal{M}}_{k}\left( {{\Gamma }_{1}\left( N\right) }\right) \). Since \( \left\lbrack \begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right\rbrack \in {\Gamma }_{1}\left( N\right) \), \( f \) has period 1 and hence has a Fourier expansion\n\n\[ f\left( \tau \right) = \mathop{\sum }\limits_{{n = 0}}^{\infty... | Proof. For part (a), take \( 0 \leq j < p \) and compute\n\n\[ f{\left\lbrack \left\lbrack \begin{array}{ll} 1 & j \\ 0 & p \end{array}\right\rbrack \right\rbrack }_{k}\left( \tau \right) = {p}^{k - 1}{\left( 0\tau + p\right) }^{-k}f\left( \frac{\tau + j}{p}\right) = \frac{1}{p}\mathop{\sum }\limits_{{n = 0}}^{\infty }... | Yes |
Proposition 5.2.3. Let \( \chi \) modulo \( N,\psi ,\varphi \), and \( t \) be as above. Let \( p \) be prime. Excluding the case \( k = 2,\psi = \varphi = {\mathbf{1}}_{1} \) ,\n\n\[ \n{T}_{p}{E}_{k}^{\psi ,\varphi, t} = \left( {\psi \left( p\right) + \varphi \left( p\right) {p}^{k - 1}}\right) {E}_{k}^{\psi ,\varphi,... | This is a direct calculation (Exercise 5.2.5). | No |
Proposition 5.2.4. Let \( d \) and \( e \) be elements of \( {\left( \mathbb{Z}/N\mathbb{Z}\right) }^{ * } \), and let \( p \) and \( q \) be prime. Then\n\n(a) \( \langle d\rangle {T}_{p} = {T}_{p}\langle d\rangle \) ,\n\n(b) \( \langle d\rangle \langle e\rangle = \langle e\rangle \langle d\rangle = \langle {de}\rangl... | Proof. Part (a) has already been shown. Since the \( \langle d\rangle \) and \( {T}_{p} \) operators preserve the decomposition \( {\mathcal{M}}_{k}\left( {{\Gamma }_{1}\left( N\right) }\right) = \bigoplus {\mathcal{M}}_{k}\left( {N,\chi }\right) \), it suffices to check (b) and (c) on an arbitrary \( f \in {\mathcal{M... | No |
Proposition 5.3.1. Let \( f \in {\mathcal{M}}_{k}\left( {{\Gamma }_{1}\left( N\right) }\right) \) have Fourier expansion\n\n\[ f\left( \tau \right) = \mathop{\sum }\limits_{{m = 0}}^{\infty }{a}_{m}\left( f\right) {q}^{m}\;\text{ where }q = {e}^{2\pi i\tau }.\]\n\nThen for all \( n \in {\mathbb{Z}}^{ + },{T}_{n}f \) ha... | Proof. As usual, we may take \( f \in {\mathcal{M}}_{k}\left( {N,\chi }\right) \) and thus formula (5.13) reduces to (5.14). Formula (5.14) checks trivially when \( n = 1 \) and it reduces to (5.4) when \( n = p \) is prime (Exercise 5.3.3). Let \( r \geq 2 \) and assume (5.14) holds for \( n = 1, p,{p}^{2},\ldots ,{p}... | No |
Lemma 5.5.1. Let \( \Gamma \subset {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) be a congruence subgroup, and let \( \alpha \in \) \( {\mathrm{{GL}}}_{2}^{ + }\left( \mathbb{Q}\right) \) . (a) If \( \varphi : \mathcal{H} \rightarrow \mathbb{C} \) is continuous, bounded, and \( \Gamma \) -invariant, then \[ {\int }_{{... | Proof. Part (a) is immediate. | No |
Proposition 5.5.2. Let \( \Gamma \subset {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) be a congruence subgroup, and let \( \alpha \in \) \( {\mathrm{{GL}}}_{2}^{ + }\left( \mathbb{Q}\right) \) . Set \( {\alpha }^{\prime } = \det \left( \alpha \right) {\alpha }^{-1} \) . Then\n\n(a) If \( {\alpha }^{-1}{\Gamma \alpha ... | Proof. For part (a), expand the \( {\left\lbrack \alpha \right\rbrack }_{k} \) operator, note that \( {\alpha }^{\prime } \) acts as \( {\alpha }^{-1} \) on \( {\mathcal{H}}^{ * } \) , and apply Lemma 5.5.1(a) to get\n\n\[{\int }_{{\alpha }^{-1}{\Gamma \alpha } \smallsetminus {\mathcal{H}}^{ * }}\left( {f{\left\lbrack ... | Yes |
Proposition 5.6.2. The subspaces \( {\mathcal{S}}_{k}{\left( {\Gamma }_{1}\left( N\right) \right) }^{\text{old }} \) and \( {\mathcal{S}}_{k}{\left( {\Gamma }_{1}\left( N\right) \right) }^{\text{new }} \) are stable under the Hecke operators \( {T}_{n} \) and \( \langle n\rangle \) for all \( n \in {\mathbb{Z}}^{ + } \... | Proof. Let \( p \mid N \) . The argument breaks into cases. First, let \( T = \langle d\rangle \) with \( \left( {d, N}\right) = 1 \) or let \( T = {T}_{{p}^{\prime }} \) for \( {p}^{\prime } \) a prime other than \( p \) . Then the diagram\n\n. If \( f \in {\mathcal{S}}_{k}\left( {{\Gamma }_{1}\left( N\right) }\right) \) has Fourier expansion \( f\left( \tau \right) = \sum {a}_{n}\left( f\right) {q}^{n} \) with \( {a}_{n}\left( f\right) = 0 \) whenever \( \left( {n, N}\right) = 1 \), then \( f \) takes the form \( f = \mathop{\sum ... | The Main Lemma is due to Atkin and Lehner [AL70]. The elegant proof presented here is due to David Carlton [Car99, Car01], using some basic results about group representations and tensor products. Explaining these from scratch would take the exposition too far afield, so the reader should consult a text such as [FH91] ... | No |
Lemma 5.7.2. \( {\alpha }_{M}{\Gamma }_{1}\left( M\right) {\alpha }_{M}^{-1} = {\Gamma }^{1}\left( M\right) \), and the same formula holds with \( M \) replaced by \( N \) . | The proof is Exercise 5.7.1. | No |
Lemma 5.7.4. A set of representatives for the quotient space \( \Gamma \left( N\right) \smallsetminus {\Gamma }_{d} \) is\n\n\[ \left\{ {\left\lbrack \begin{matrix} 1 & b & N/d \\ 0 & 1 & \end{matrix}\right\rbrack : 0 \leq b < d}\right\} \] | Proof. Exercise 5.7.2. | No |
Theorem 5.7.5 (Main Lemma, third version). \[ {\mathcal{S}}_{k}\left( {{\Gamma }^{1}\left( N\right) }\right) \cap \mathop{\sum }\limits_{{p \mid N}}{\mathcal{S}}_{k}\left( {{\Gamma }_{1}\left( N\right) \cap {\Gamma }^{0}\left( {N/p}\right) }\right) = \mathop{\sum }\limits_{{p \mid N}}{\mathcal{S}}_{k}\left( {{\Gamma }^... | This version of the Main Lemma reduces to group theory. The group \( G = \) \( {\mathrm{{SL}}}_{2}\left( {\mathbb{Z}/N\mathbb{Z}}\right) \) acts on the complex vector space \( {\mathcal{S}}_{k}\left( {\Gamma \left( N\right) }\right) \) from the right via the weight- \( k \) operator. (The full modular group \( {\mathrm... | Yes |
Lemma 5.7.6. For \( p \) prime and \( e \geq 1 \) ,\n\n\[ \left\langle {{\Gamma }^{1}\left( {p}^{e}\right) ,{\Gamma }_{1}\left( {p}^{e}\right) \cap {\Gamma }^{0}\left( {p}^{e - 1}\right) }\right\rangle = {\Gamma }^{1}\left( {p}^{e - 1}\right) . \]\n | Proof. To stay with the main argument, this will be done afterward. | No |
Theorem 5.8.2. Let \( f \in {\mathcal{S}}_{k}{\left( {\Gamma }_{1}\left( N\right) \right) }^{\text{new }} \) be a nonzero eigenform for the Hecke operators \( {T}_{n} \) and \( \langle n\rangle \) for all \( n \) with \( \left( {n, N}\right) = 1 \) . Then\n\n(a) \( f \) is a Hecke eigenform, i.e., an eigenform for \( {... | Proof. All that remains to prove is that the set of newforms in the space \( {\mathcal{S}}_{k}{\left( {\Gamma }_{1}\left( N\right) \right) }^{\text{new }} \) is linearly independent. To see this, suppose there is a nontrivial linear relation\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{n}{c}_{i}{f}_{i} = 0,\;{c}_{i} \in \mat... | Yes |
Theorem 5.8.3. The set\n\n\[ \n{\mathcal{B}}_{k}\left( N\right) = \{ f\left( {n\tau }\right) : f\text{is a newform of level}M\text{and}{nM} \mid N\} \]\n\nis a basis of \( {\mathcal{S}}_{k}\left( {{\Gamma }_{1}\left( N\right) }\right) \) . | Proof. (Partial.) Consider the decomposition\n\n\[ \n{\mathcal{S}}_{k}\left( {{\Gamma }_{1}\left( N\right) }\right) = {\mathcal{S}}_{k}{\left( {\Gamma }_{1}\left( N\right) \right) }^{\text{new }} \oplus \mathop{\sum }\limits_{{p \mid N}}{i}_{p}\left( {\left( {\mathcal{S}}_{k}\left( {\Gamma }_{1}\left( N/p\right) \right... | No |
Proposition 5.8.5. Let \( f \in {\mathcal{M}}_{k}\left( {N,\chi }\right) \) . Then \( f \) is a normalized eigenform if and only if its Fourier coefficients satisfy the conditions\n\n(1) \( {a}_{1}\left( f\right) = 1 \) ,\n\n(2) \( {a}_{{p}^{r}}\left( f\right) = {a}_{p}\left( f\right) {a}_{{p}^{r - 1}}\left( f\right) -... | Proof. The forward implication \( \left( \Rightarrow \right) \) follows from the definition of \( {T}_{n} \) in Section 5.3 (Exercise 5.8.7). For the reverse implication \( \left( \Leftarrow \right) \), suppose \( f \) satisfies the three conditions. Then \( f \) is normalized, and to be an eigenform for all the Hecke ... | No |
Proposition 5.9.1. If \( f \in {\mathcal{M}}_{k}\left( {{\Gamma }_{1}\left( N\right) }\right) \) is a cusp form then \( L\left( {s, f}\right) \) converges absolutely for all \( s \) with \( \operatorname{Re}\left( s\right) > k/2 + 1 \) . If \( f \) is not a cusp form then \( L\left( {s, f}\right) \) converges absolutel... | Proof. First assume \( f \) is a cusp form. Let \( g\left( q\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{q}^{n} \), a holomorphic function on the unit disk \( \{ q : \left| q\right| < 1\} \) . Then by Cauchy’s formula,\n\n\[ \n{a}_{n} = \frac{1}{2\pi i}{\int }_{\left| q\right| = r}g\left( q\right) {q}^{-n... | Yes |
Proposition 5.10.1. The Mellin transform of \( f \) is\n\n\[ g\left( s\right) = {\left( 2\pi \right) }^{-s}\Gamma \left( s\right) L\left( {s, f}\right) ,\;\operatorname{Re}\left( s\right) > k/2 + 1. \] | This is shown exactly as in Chapter 4 (Exercise 5.10.1). | No |
Theorem 5.10.2. Suppose \( f \in {\mathcal{S}}_{k}{\left( {\Gamma }_{1}\left( N\right) \right) }^{ \pm } \) . Then the Mellin transform \( {\Lambda }_{N}\left( s\right) \) extends to an entire function satisfying the functional equation\n\n\[ \n{\Lambda }_{N}\left( s\right) = \pm {\Lambda }_{N}\left( {k - s}\right) \n\... | Proof. Take \( f \in {\mathcal{S}}_{k}{\left( {\Gamma }_{1}\left( N\right) \right) }^{ \pm } \) and compute\n\n\[ \n{\Lambda }_{N}\left( s\right) = {N}^{s/2}{\int }_{t = 0}^{\infty }f\left( {it}\right) {t}^{s}\frac{dt}{t} = {\int }_{t = 0}^{\infty }f\left( {{it}/\sqrt{N}}\right) {t}^{s}\frac{dt}{t}. \n\]\n\nSince \( f\... | Yes |
Theorem 6.1.2 (Abel's Theorem). The map (6.1) descends to divisor classes, inducing an isomorphism\n\n\[ \n{\operatorname{Pic}}^{0}\left( X\right) \overset{ \sim }{ \rightarrow }\operatorname{Jac}\left( X\right) ,\;\left\lbrack {\mathop{\sum }\limits_{x}{n}_{x}x}\right\rbrack \mapsto \mathop{\sum }\limits_{x}{n}_{x}{\i... | Thus the degree- 0 divisors describing meromorphic functions on \( X \) are those that map to trivial integration on \( {\Omega }_{\text{hol }}^{1}\left( X\right) \) modulo integration over loops. If \( X \) has genus \( g > 0 \) then its embedding into its Picard group followed by the isomorphism of Abel's Theorem sho... | Yes |
Proposition 6.1.4. Let \( \varphi : {\mathbb{C}}^{g}/{\Lambda }_{g} \rightarrow {\mathbb{C}}^{h}/{\Lambda }_{h} \) be a holomorphic map of complex tori. Then\n\n\[ \varphi \left( {z + {\Lambda }_{g}}\right) = {Mz} + b + {\Lambda }_{h} \]\n\nwhere \( M \in {\mathrm{M}}_{h, g}\left( \mathbb{C}\right) \) is an h-by-g matr... | This is proved the same way as in Chapter 1. By topology the map \( \varphi \) lifts to a map \( \widetilde{\varphi } : {\mathbb{C}}^{g} \rightarrow {\mathbb{C}}^{h} \) of universal covers, and it follows that \( {\widetilde{\varphi }}^{\prime } \) is bounded. Applying Liouville’s Theorem in each direction makes \( {\w... | Yes |
Proposition 6.2.4. Let \( h : X \rightarrow Y \) be a nonconstant holomorphic map of compact Riemann surfaces. Let \( {\left. {h}_{ * } = {\left( {h}^{ * }\right) }^{ \land }\right| }_{{\mathrm{H}}_{1}\left( {X,\mathbb{Z}}\right) } \) be the induced forward homomorphism of homology groups,\n\n\[ \n{h}_{ * } : {\mathrm{... | Proof. Using (6.8) at the last step, compute that\n\n\[ \n{h}_{ * }\left( {{\mathrm{H}}_{1}\left( {X,\mathbb{Z}}\right) }\right) \supset \left( {{\left( {h}^{ * }\right) }^{ \land } \circ {\operatorname{tr}}_{h}^{ \land }}\right) \left( {{\mathrm{H}}_{1}\left( {Y,\mathbb{Z}}\right) }\right)\n\]\n\n\[ \n= {\left( {\oper... | Yes |
Proposition 6.3.2. The Hecke operators \( T = {T}_{p} \) and \( T = \langle d\rangle \) act by composition on the Jacobian associated to \( {\Gamma }_{1}\left( N\right) \) , | \[ T : {\mathrm{J}}_{1}\left( N\right) \rightarrow {\mathrm{J}}_{1}\left( N\right) ,\;\left\lbrack \varphi \right\rbrack \mapsto \left\lbrack {\varphi \circ T}\right\rbrack \text{ for }\varphi \in {\mathcal{S}}_{2}{\left( {\Gamma }_{1}\left( N\right) \right) }^{ \land }, \] and similarly for \( {T}_{p} \) on \( {\mathr... | Yes |
Theorem 6.4.1. Let \( \alpha \) be a complex number. The following conditions on \( \alpha \) are equivalent:\n\n(1) \( \alpha \) is an algebraic number, i.e., \( \alpha \in \overline{\mathbb{Q}} \) ,\n\n(2) The ring \( \mathbb{Q}\left\lbrack \alpha \right\rbrack \) is a finite-dimensional vector space over \( \mathbb{... | Proof. (1) \( \Rightarrow \) (2): Let \( \alpha \) satisfy the polynomial \( {x}^{n} + {c}_{1}{x}^{n - 1} + \cdots + {c}_{n} \) with \( {c}_{1},\ldots ,{c}_{n} \in \mathbb{Q} \) . Then \( {\alpha }^{n} = - \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{c}_{n - i}{\alpha }^{i} \), so the complex vector space generated by the... | Yes |
Corollary 6.4.3. The field \( \overline{\mathbb{Q}} \) of algebraic numbers is algebraically closed. | Proof. (Sketch.) Consider a monic polynomial \( {x}^{n} + {c}_{1}{x}^{n - 1} + \cdots + {c}_{n} \) with coefficients \( {c}_{i} \in \overline{\mathbb{Q}} \) and let \( \alpha \) be one of its roots. Since each ring \( \mathbb{Q}\left\lbrack {c}_{i}\right\rbrack \) is a finite-dimensional vector space over \( \mathbb{Q}... | No |
Proposition 6.5.5 (Nakayama's Lemma). Suppose that \( A \) is a commutative ring with unit and \( J \subset A \) is an ideal contained in every maximal ideal of \( A \), and suppose that \( M \) is a finitely generated \( A \) -module such that \( {JM} = M \) . Then \( M = \{ 0\} \) . | Proof. Suppose that \( M \neq \{ 0\} \) and let \( {m}_{1},\ldots ,{m}_{n} \) be a minimal set of generators for \( M \) over \( A \) . Since \( {JM} = M \), in particular \( {m}_{n} \in {JM} \), giving it the form \( {m}_{n} = {a}_{1}{m}_{1} + \cdots + {a}_{n}{m}_{n} \) with all \( {a}_{i} \in J \) . Thus\n\n\[ \left(... | Yes |
Corollary 6.5.6. The space \( {\mathcal{S}}_{2}\left( {{\Gamma }_{1}\left( N\right) }\right) \) has a basis of forms with rational integer coefficients. | Proof. Let \( f \) be any newform at level \( M \) where \( M \mid N \) . Let \( \mathbb{K} = {\mathbb{K}}_{f} \) be the number field of \( f \) . Let \( \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{d}}\right\} \) be a basis of \( {\mathcal{O}}_{\mathbb{K}} \) as a \( \mathbb{Z} \) -module and let \( \left\{ {{\sigma }_{1... | Yes |
Lemma 6.6.2. Let \( G \) be a finitely generated Abelian group and let \( \mathbf{k} \) be a field of characteristic 0 . Then\n\n(a) \( G \otimes \mathbf{k} \cong {\mathbf{k}}^{\operatorname{rank}\left( G\right) } \) .\n\n(b) \( \left( {G/K}\right) \otimes \mathbf{k} \cong \left( {G \otimes \mathbf{k}}\right) /\left( {... | Proof. Exercise 6.6.1. | No |
Theorem 6.6.6. The Jacobian associated to \( {\Gamma }_{1}\left( N\right) \) is isogenous to a direct sum of Abelian varieties associated to equivalence classes of newforms, \[ {\mathrm{J}}_{1}\left( N\right) \rightarrow {\bigoplus }_{f}{A}_{f}^{{m}_{f}} \] Here the sum is taken over a set of representatives \( f \in {... | Proof. By Theorem 5.8.3 and Theorem 6.5.4, \( {\mathcal{S}}_{2}\left( {{\Gamma }_{1}\left( N\right) }\right) \) has basis \[ {\mathcal{B}}_{2}\left( N\right) = \mathop{\bigcup }\limits_{f}\mathop{\bigcup }\limits_{n}\mathop{\bigcup }\limits_{\sigma }{f}^{\sigma }\left( {n\tau }\right) \] where the first union is taken ... | No |
Theorem 7.1.3. Let \( \mathcal{E} \) be an elliptic curve over a field \( \mathbf{k} \) of characteristic 0 and let \( N \) be a positive integer. Then \( \mathcal{E}\left\lbrack N\right\rbrack \cong {\left( \mathbb{Z}/N\mathbb{Z}\right) }^{2} \) . | Let \( \mathbb{K} \) be a Galois extension field of \( \mathbf{k} \) containing the \( x \) - and \( y \) -coordinates of \( \mathcal{E}\left\lbrack N\right\rbrack - \left\{ {0\varepsilon }\right\} \) . The relations \( E\left( {{x}^{\sigma },{y}^{\sigma }}\right) = E{\left( x, y\right) }^{\sigma } \) and \( {\widetild... | Yes |
Proposition 7.2.3. For any nonsingular point \( P \in C \) the ideal \( {M}_{P} \) is principal. | Proof. By the lemma and the isomorphism \( i \), some function \( t \in \overline{\mathbf{k}}\left\lbrack C\right\rbrack \) generates \( {M}_{P}/{M}_{P}^{2} \) as a vector space over \( \overline{\mathbf{k}} \) . This function will also generate \( {M}_{P} \) as an ideal of \( \overline{\mathbf{k}}{\left\lbrack C\right... | Yes |
Proposition 7.2.4. Let \( C \) be a nonsingular algebraic curve. For any point \( P \in C \) the valuation \( {\nu }_{P} : \overline{\mathbf{k}}\left( C\right) \rightarrow \mathbb{Z} \cup \{ + \infty \} \) surjects, and for all nonzero \( F, G \in \overline{\mathbf{k}}\left( C\right) \) it satisfies the conditions\n\n(... | Now we can describe the mapping from \( C \) to \( {\mathbb{P}}^{1}\left( \overline{\mathbf{k}}\right) \) defined by a rational function \( F \in \overline{\mathbf{k}}\left( C\right) \) . The zero function defines the zero mapping. Any nonzero function takes the form\n\n\[ F = {t}^{{\nu }_{P}\left( F\right) }\frac{f}{g... | No |
Theorem 7.2.5 (Curves-Fields Correspondence, Part 1). The map\n\n\[ \nC \mapsto \mathbf{k}\left( C\right) \n\]\n\ninduces a bijection from the set of isomorphism classes over \( \mathbf{k} \) of nonsingular projective algebraic curves over \( \mathbf{k} \) to the set of conjugacy classes over \( \mathbf{k} \) of functi... | The theorem gives the map from curve-classes to field-classes explicitly. To describe the map from field-classes to curve-classes, let \( \mathbb{K} \) be a function field over \( \mathbf{k} \) . Since the extension \( \mathbb{K}/\mathbf{k}\left( t\right) \) is finite and we are in characteristic 0 the Primitive Elemen... | No |
Lemma 7.3.2. Let \( \mathcal{E} \) be an elliptic curve over \( \mathbf{k} \). (a) For any points \( P, Q \in \mathcal{E} \), \[ \text{the divisor}\left( P\right) - \left( Q\right) \text{is principal} \Leftrightarrow P = Q\text{.} \] | Proof. (a) The implication ( \( \Rightarrow \) ) was shown in the calculation above. The other implication is immediate. | No |
Theorem 7.3.3. Let \( \mathcal{E} \) be an elliptic curve. Then the map\n\n\[ \operatorname{Div}\left( \mathcal{E}\right) \rightarrow \mathcal{E},\;\sum {n}_{P}\left( P\right) \mapsto \sum \left\lbrack {n}_{P}\right\rbrack P \]\n\ninduces an isomorphism\n\n\[ {\operatorname{Pic}}^{0}\left( \mathcal{E}\right) \overset{ ... | Proof. The map \( \operatorname{Div}\left( \mathcal{E}\right) \rightarrow \mathcal{E} \) of the theorem is clearly a homomorphism. Its restriction to \( {\operatorname{Div}}^{0}\left( \mathcal{E}\right) \) is surjective since \( \left( P\right) - \left( {0}_{\mathcal{E}}\right) \mapsto P \) for any \( P \in \mathcal{E}... | Yes |
Proposition 7.4.1 (Properties of the Weil pairing).\n\n(a) The Weil pairing is bilinear,\n\n\[ \n{e}_{N}\left( {{aP} + b{P}^{\prime },{cQ} + d{Q}^{\prime }}\right) = {e}_{N}{\left( P, Q\right) }^{ac}{e}_{N}{\left( P,{Q}^{\prime }\right) }^{ad}{e}_{N}{\left( {P}^{\prime }, Q\right) }^{bc}{e}_{N}{\left( {P}^{\prime },{Q}... | Proof. (a) For linearity in the first argument let \( g = {g}_{Q} \) and compute\n\n\[ \n{e}_{N}\left( {{P}_{1} + {P}_{2}, Q}\right) = \frac{g\left( {X + {P}_{1} + {P}_{2}}\right) }{g\left( X\right) } = \frac{g\left( {X + {P}_{1} + {P}_{2}}\right) }{g\left( {X + {P}_{2}}\right) }\frac{g\left( {X + {P}_{2}}\right) }{g\l... | Yes |
Proposition 7.5.1. The fields of meromorphic functions on \( X\left( N\right) ,{X}_{1}\left( N\right) \) , and \( {X}_{0}\left( N\right) \) are\n\n\[ \mathbb{C}\left( {X\left( N\right) }\right) = \mathbb{C}\left( {j,\left\{ {{f}_{0}^{\pm \bar{v}} : \pm \bar{v} \in \left( {{\left( \mathbb{Z}/N\mathbb{Z}\right) }^{2} - \... | Proof. Since \( {\wp }_{\tau }\left( z\right) = {\wp }_{\tau }\left( {z}^{\prime }\right) \) if and only if \( z \equiv \pm {z}^{\prime }\left( {\;\operatorname{mod}\;{\Lambda }_{\tau }}\right) \), each pair of functions \( {f}_{0}^{\bar{v}} \) and \( {f}_{0}^{-\bar{v}} \) are equal but otherwise the \( {f}_{0}^{\bar{v... | Yes |
Lemma 7.6.1. The function \( \det \rho \) describes how \( {H}_{\mathbb{Q}} \) permutes \( {\mathbf{\mu }}_{N} \), \[ {\mu }^{\sigma } = {\mu }^{\det \rho \left( \sigma \right) },\;\mu \in {\mathbf{\mu }}_{N},\sigma \in {H}_{\mathbb{Q}}. \] (Here \( {\mu }^{\sigma } \) is \( \mu \) acted on by \( \sigma \) while \( {\m... | This is shown with the Weil pairing as in the proof of Corollary 7.5.3 (Exercise 7.6.1). | No |
Lemma 7.6.2 (Restriction Lemma). Let \( \mathbf{k} \) and \( \mathbb{F} \) be extension fields of \( \mathbf{f} \) inside \( \mathbb{K} \) with \( \mathbb{F}/\mathbf{f} \) Galois. Suppose \( \mathbb{K} = \mathbf{k}\mathbb{F} \) . Then \( \mathbb{K}/\mathbf{k} \) is Galois, there is a natural injection\n\n\[ \operatorna... | Proof. The situation is shown in Figure 7.4. Any map \( \sigma : \mathbb{K} \rightarrow \overline{\mathbb{K}} \) fixing \( \mathbf{k} \) restricts to a map \( \mathbb{F} \rightarrow \overline{\mathbb{F}} \) fixing \( \mathbf{k} \cap \mathbb{F} \) . Since the extension \( \mathbb{F}/\left( {\mathbf{k} \cap \mathbb{F}}\r... | Yes |
Theorem 7.6.3. Let \( {H}_{\mathbb{Q}} \) denote the Galois group of the field extension \( \mathbb{Q}\left( {j,{E}_{j}\left\lbrack N\right\rbrack }\right) /\mathbb{Q}\left( j\right) \) . There is an isomorphism\n\n\[ \rho : {H}_{\mathbb{Q}}\overset{ \sim }{ \rightarrow }{\mathrm{{GL}}}_{2}\left( {\mathbb{Z}/N\mathbb{Z... | The last statement in the theorem follows from Theorem 7.2.5. | No |
Theorem 7.7.2 (Modularity Theorem, Version \( {X}_{\mathbb{Q}} \) ). Let \( E \) be an elliptic curve over \( \mathbb{Q} \) . Then for some positive integer \( N \) there exists a surjective morphism over \( \mathbb{Q} \) of curves over \( \mathbb{Q} \) from the modular curve \( {X}_{0}{\left( N\right) }_{\text{alg }} ... | \[ {X}_{0}{\left( N\right) }_{\text{alg }} \rightarrow E \] | No |
Theorem 7.8.1. Let \( \mathbf{k} \) be a field. Isogeny over \( \mathbf{k} \) between elliptic curves over \( \mathbf{k} \) is an equivalence relation. That is, if \( \varphi : E \rightarrow {E}^{\prime } \) is an isogeny over \( \mathbf{k} \) of elliptic curves over \( \mathbf{k} \) then there exists a dual isogeny \(... | Proof. Given \( \varphi : E \rightarrow {E}^{\prime } \), let \( C = \ker \left( \varphi \right) \), let \( N = \left| C\right| \), and consider also \( \left\lbrack N\right\rbrack : E \rightarrow E \), again an isogeny over \( \mathbf{k} \) . Similarly to before, identify each point \( P \in E\left\lbrack N\right\rbra... | Yes |
Proposition 8.1.3. Let \( E \) be a Weierstrass equation over \( \mathbf{k} \). Then\n\n- \( E \) describes an elliptic curve \( \Leftrightarrow \Delta \neq 0 \),\n\n- \( E \) describes a curve with a node \( \Leftrightarrow \Delta = 0 \) and \( {c}_{4} \neq 0 \),\n\n- \( E \) describes a curve with a cusp \( \Leftrigh... | In the case of a node the set of projective solutions of \( E \) other than the singular point forms a multiplicative group isomorphic to \( {\overline{\mathbf{k}}}^{ * } \), and in the case of a cusp the set forms an additive group isomorphic to \( \overline{\mathbf{k}} \). (See [Sil86] for the proof of this, and also... | No |
Proposition 8.3.2. Let \( E \) be an elliptic curve over \( \mathbb{Q} \) and let \( p \) be a prime such that \( E \) has good reduction modulo \( p \) . Let \( {\sigma }_{p, * } \) and \( {\sigma }_{p}^{ * } \) be the forward and reverse maps of \( {\operatorname{Pic}}^{0}\left( \widetilde{E}\right) \) induced by \( ... | Proof. An element \( x \in {\overline{\mathbb{F}}}_{p} \) satisfies \( {x}^{p} = x \) if and only if \( x \in {\mathbb{F}}_{p} \) . Thus\n\n\[ \n\widetilde{E}\left( {\mathbb{F}}_{p}\right) = \left\{ {P \in \widetilde{E} : {P}^{{\sigma }_{p}} = P}\right\} = \ker \left( {{\sigma }_{p} - 1}\right) ,\n\]\n\nand so since \(... | No |
Proposition 8.3.3. Let \( E \) be an elliptic curve over \( \mathbb{Q} \), and let \( p \) be a prime such that \( E \) has good reduction at \( p \) . Then the reduction is\n\n\[ \n\\left\\{ \\begin{array}{ll} \\text{ ordinary } & \\text{ if }{a}_{p}\\left( E\\right) ≢ 0\\left( {\\;\\operatorname{mod}\\;p}\\right) , \... | The estimate \( \\left| {{a}_{p}\\left( E\\right) }\\right| \\leq 2\\sqrt{p} \) due to Hasse (which we do not address in this book) combines with the previous proposition to give the result stated earlier in the section that \( {a}_{p}\\left( E\\right) = 0 \) if \( E \) has supersingular reduction at \( p \) and \( p \... | Yes |
Lemma 8.4.1. Let \( \mathfrak{p} \) be a maximal ideal of \( \overline{\mathbb{Z}} \) . Let \( \alpha \) be a nonzero element of \( \overline{\mathbb{Q}} \) . Then at least one of \( \alpha \) or \( 1/\alpha \) belongs to \( {\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) } \) . | Proof. Consider the number field \( \mathbb{K} = \mathbb{Q}\left( \alpha \right) \) . Let \( {\mathcal{O}}_{\mathbb{K}} \) be its ring of algebraic integers and let \( {\mathfrak{p}}_{\mathbb{K}} = \mathfrak{p} \cap {\mathcal{O}}_{\mathbb{K}} \) . The localization of \( {\mathcal{O}}_{\mathbb{K}} \) at \( {\mathfrak{p}... | Yes |
Proposition 8.4.2. Ordinary reduction, supersingular reduction, and multiplicative reduction are well defined on equivalence classes of \( \mathfrak{p} \) -minimal Weierstrass equations. If \( E \) and \( {E}^{\prime } \) are equivalent \( \mathfrak{p} \) -minimal Weierstrass equations with good reduction at \( \mathfr... | Proof. If \( E \) and \( {E}^{\prime } \) are equivalent then by Exercise 8.1.1(b)\n\n\[ \n{u}^{12}{\Delta }^{\prime } = \Delta ,\;{u}^{4}{c}_{4}^{\prime } = {c}_{4}, \n\]\n\nwhere \( u \) comes from the admissible change of variable taking \( E \) to \( {E}^{\prime } \) . Recall the disjoint union mentioned early in t... | No |
Proposition 8.4.3. Let \( E \) be an elliptic curve over \( \overline{\mathbb{Q}} \) and let \( \mathfrak{p} \) be a maximal ideal of \( \overline{\mathbb{Z}} \) . Then \( E \) has good reduction at \( \mathfrak{p} \) if and only if \( j\left( E\right) \in {\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) } \) . | Proof. Suppose \( E \) has good reduction at \( \mathfrak{p} \) . We may assume the Weierstrass equation for \( E \) is \( \mathfrak{p} \) -minimal. Thus \( \Delta \in {\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) }^{ * } \) and \( {c}_{4} \in {\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) } \), so that \( j... | Yes |
Proposition 8.4.4. Let \( E \) be an elliptic curve over \( \overline{\mathbb{Q}} \) with good reduction at \( \mathfrak{p} \). Then:\n\n(a) The reduction map on \( N \) -torsion,\n\n\[ E\left\lbrack N\right\rbrack \rightarrow \widetilde{E}\left\lbrack N\right\rbrack \]\n\nis surjective for all \( N \).\n\n(b) Any isog... | We have shown part (a) when \( p \nmid {6N} \), but a complete proof is beyond the scope of this book. The substance of part (b) is that good reduction remains good under the isogeny, also not proved here. Granting this, it follows that ordinary reduction remains ordinary and supersingular reduction remains supersingul... | No |
Lemma 8.5.2. Let \( \mathbf{k} \) be a field. Let \( I \subset \mathbf{k}\left\lbrack x\right\rbrack \) be the homogenization of a prime ideal \( {I}_{\left( 0\right) } \subset \mathbf{k}\left\lbrack {x}_{\left( 0\right) }\right\rbrack \) that defines an affine algebraic curve. Then \( I \) is prime. For \( i = 0,\ldot... | Proof. If \( {\varphi \psi } \in I \), where \( \varphi ,\psi \in \mathbf{k}\left\lbrack x\right\rbrack \), then \( {\varphi }_{\left( 0\right) }{\psi }_{\left( 0\right) } = {\left( \varphi \psi \right) }_{\left( 0\right) } \in {I}_{\left( 0\right) } \) and so (up to symmetry) \( {\varphi }_{\left( 0\right) } \in {I}_{... | Yes |
Theorem 8.5.4. Let \( C \) be a nonsingular projective algebraic curve over \( \mathbb{Q} \) with good reduction at \( p \) . Then the reduction map \( C \rightarrow \widetilde{C} \) is surjective. | Proof. Let \( {Q}^{\prime } \) be a point of \( \widetilde{C} \) . Then \( {Q}^{\prime } \) lies in some affine piece \( {\widetilde{C}}_{i} \) of \( \widetilde{C} \) . Since \( i \) is fixed we will simply call the affine piece \( C \) throughout the proof. Thus the affine coordinate ring is \( {\mathbb{F}}_{p}\left\l... | Yes |
Lemma 8.5.6 (Krull Intersection Theorem, special case). Let \( R \) be a Noetherian domain and let \( \alpha \in R \) be a nonunit. Then \( \mathop{\bigcap }\limits_{{e = 0}}^{\infty }{\alpha }^{e}R = \{ 0\} \) . | Proof. Let \( J = \mathop{\bigcap }\limits_{{e = 0}}^{\infty }{\alpha }^{e}R \), let \( {m}_{1},\ldots ,{m}_{k} \) be a set of generators for \( J \), and let \( m \) be the column vector that they form. Since \( {\alpha J} = J \) (Exercise 8.5.8), also \( {\alpha m} \) is a column vector of generators, and so there ex... | Yes |
Theorem 8.5.7. Let \( C \) and \( {C}^{\prime } \) be nonsingular projective algebraic curves over \( \mathbb{Q} \) with good reduction at \( p \), and let \( {C}^{\prime } \) have positive genus. For any morphism \( h : C \rightarrow {C}^{\prime } \) the following diagram commutes:\n\n![bea1ef69-414f-4cdb-8846-ec2fa99... | Since \( C \rightarrow \widetilde{C} \) surjects, \( \widetilde{h} \) is the unique morphism from \( \widetilde{C} \) to \( \widetilde{{C}^{\prime }} \) that makes the diagram commute. Such a diagram can fail to commute when \( {C}^{\prime } \) has genus 0, for example the map \( h : {\mathbb{P}}^{1} \rightarrow {\math... | Yes |
Theorem 8.5.9. Let \( C \) be a nonsingular projective algebraic curve over \( \mathbb{Q} \) with good reduction at \( p \) . The map on degree-0 divisors induced by reduction,\n\n\[ \n{\operatorname{Div}}^{0}\left( C\right) \rightarrow {\operatorname{Div}}^{0}\left( \widetilde{C}\right) ,\;\sum {n}_{P}\left( P\right) ... | The first statement of Theorem 8.5.9 is not obvious because the induced map on divisors does not send the principal divisor \( \operatorname{div}\left( f\right) \) to \( \operatorname{div}\left( \widetilde{f}\right) \) . | No |
Theorem 8.5.10. Let\n\n\\[\\varphi : E \\rightarrow {E}^{\\prime }\n\\]\n\nbe an isogeny over \\( \\overline{\\mathbb{Q}} \\) of elliptic curves over \\( \\overline{\\mathbb{Q}} \\) . Then there is a reduction\n\n\\[\\widetilde{\\varphi } : \\widetilde{E} \\rightarrow \\widetilde{{E}^{\\prime }}\n\\]\n\nwith the proper... | Part (a) of Theorem 8.5.10 follows from the general results of this section and from the fact that \\( \\widetilde{\\varphi } \\) is a nonconstant map taking 0 to 0 . The statement in Section 8.2 that the map \\( \\left\\lbrack p\\right\\rbrack \\) is an isogeny of degree \\( {p}^{2} \\) on elliptic curves over \\( {\\... | Yes |
Theorem 8.6.1 (Igusa). Let \( N \) be a positive integer and let \( p \) be a prime with \( p \nmid N \) . The modular curve \( {X}_{1}\left( N\right) \) has good reduction at \( p \) . There is an isomorphism of function fields | \[ {\mathbb{F}}_{p}\left( {{\widetilde{X}}_{1}\left( N\right) }\right) \overset{ \sim }{ \rightarrow }{\mathbb{K}}_{1}\left( N\right) \] Moreover, reducing the modular curve is compatible with reducing the moduli space in that the following diagram commutes: . Let \( p \nmid N \) . The following diagram commutes: | In particular since \( \widetilde{\langle p\rangle } \) acts trivially on \( {\widetilde{X}}_{0}\left( N\right) \), the following diagram commutes as well:\n\n(8.38)\n\nThe Hecke operator \( {\widetilde{T}}_{p} \) in characteristic \( p \) is now described in terms of the Frobenius map \( {\sigma }_{p} \), as desired. | No |
Theorem 8.8.2. Let \( E \) be an elliptic curve over \( \mathbb{Q} \) with conductor \( {N}_{E} \), let \( N \) be a positive integer, and let\n\n\[ \n\alpha : {X}_{0}\left( N\right) \rightarrow E \n\] \n\nbe a nonzero morphism over \( \mathbb{Q} \) of curves over \( \mathbb{Q} \) . Then for some newform \( f \in {\mat... | Proof. For any \( p \nmid {N}_{E}N \) the route from \( {a}_{p}\left( f\right) \) for some \( f \) to \( {a}_{p}\left( E\right) \) is that\n\n- \( {a}_{p}\left( f\right) \) on \( {A}_{f}^{\prime } \) is \( {T}_{p} \) for each \( f \), by a variant of diagram (6.15), and a sum of factors \( {A}_{f}^{\prime } \) over all... | Yes |
Theorem 8.8.4 (Modularity Theorem, strong Version \( {A}_{\mathbb{Q}} \) ). Let \( E \) be an elliptic curve over \( \mathbb{Q} \) with conductor \( {N}_{E} \) . Then for some newform \( f \in \) \( {\mathcal{S}}_{2}\left( {{\Gamma }_{0}\left( {N}_{E}\right) }\right) \) the Abelian variety \( {A}_{f}^{\prime } \) is al... | To see this, suppose that by Version \( L \) we have \( L\left( {s, f}\right) = L\left( {s, E}\right) \) . Then \( f \) has rational coefficients, making \( {A}_{f}^{\prime } \) an elliptic curve. Equation (8.42) shows that \( L\left( {s,{A}_{f}^{\prime }}\right) \) and \( L\left( {s, f}\right) \) have the same Euler p... | No |
Theorem 9.3.1. For each maximal ideal \( \mathfrak{p} \) of \( \overline{\mathbb{Z}} \) lying over any but a finite set of rational primes \( p \), choose an absolute Frobenius element \( {\operatorname{Frob}}_{\mathfrak{p}} \) . The set of such elements forms a dense subset of \( {G}_{\mathbb{Q}} \) . | Let \( \chi : {\left( \mathbb{Z}/N\mathbb{Z}\right) }^{ * } \rightarrow {\mathbb{C}}^{ * } \) be a primitive Dirichlet character. Consider the following diagram, in which \( {\pi }_{N} \) is restriction to \( \operatorname{Gal}\left( {\mathbb{Q}\left( {\mu }_{N}\right) /\mathbb{Q}}\right) \), the horizontal arrow is th... | Yes |
Proposition 9.3.5. Let \( \rho : {G}_{\mathbb{Q}} \rightarrow {\mathrm{{GL}}}_{d}\left( \mathbb{L}\right) \) be a Galois representation. Then \( \rho \) is similar to a Galois representation \( {\rho }^{\prime } : {G}_{\mathbb{Q}} \rightarrow {\mathrm{{GL}}}_{d}\left( {\mathcal{O}}_{\mathbb{L}}\right) \) . | Proof. Let \( V = {\mathbb{L}}^{d} \) and let \( \Lambda = {\mathcal{O}}_{\mathbb{L}}^{d} \) . Then \( \Lambda \) is a lattice of \( V \), hence a finitely generated \( {\mathbb{Z}}_{\ell } \) -module, hence compact as noted at the end of Section 9.2. Since \( {G}_{\mathbb{Q}} \) is compact as well, so is the image \( ... | Yes |
Theorem 9.4.1. Let \( \ell \) be prime and let \( E \) be an elliptic curve over \( \mathbb{Q} \) with conductor \( N \) . The Galois representation \( {\rho }_{E,\ell } \) is unramified at every prime \( p \nmid \ell N \) . For any such \( p \) let \( \mathfrak{p} \subset \overline{\mathbb{Z}} \) be any maximal ideal ... | Proof. Let \( p \nmid \ell N \) and let \( \mathfrak{p} \) lie over \( p \) . Recall the absolute Galois group of \( {\mathbb{F}}_{p} \) , \( {G}_{{\mathbb{F}}_{p}} = \operatorname{Aut}\left( {\overline{\mathbb{F}}}_{p}\right) \) . For each \( n \in {\mathbb{Z}}^{ + } \) there is a commutative diagram\n\n\n\nThe map down the right sid... | Yes |
Lemma 9.5.2. The map \( {\operatorname{Pic}}^{0}\left( {{X}_{1}\left( N\right) }\right) \left\lbrack {\ell }^{n}\right\rbrack \rightarrow {A}_{f}\left\lbrack {\ell }^{n}\right\rbrack \) is a surjection. Its kernel is stable under \( {G}_{\mathbb{Q}} \) . | Proof. Multiplication by \( {\ell }^{n} \) is surjective on \( {I}_{f}{\mathrm{\;J}}_{1}\left( N\right) \) . Indeed, it is surjective on the complex torus \( {\mathrm{J}}_{1}\left( N\right) \), and the commutative Hecke algebra \( {\mathbb{T}}_{\mathbb{Z}} \) contains both \( {I}_{f} \) and \( {\ell }^{n} \), so that \... | Yes |
Theorem 9.6.2 (Modularity Theorem, Version \( R \) ). Let \( E \) be an elliptic curve over \( \mathbb{Q} \) . Then \( {\rho }_{E,\ell } \) is modular for some \( \ell \) . | This is the version that was proved, for semistable curves in [Wil95] and [TW95] and then for all curves in [BCDT01]. | Yes |
Theorem 9.6.3 (Modularity Theorem, strong Version \( R \) ). Let \( E \) be an elliptic curve over \( \mathbb{Q} \) with conductor \( N \) . Then for some newform \( f \in \) \( {\mathcal{S}}_{2}\left( {{\Gamma }_{0}\left( N\right) }\right) \) with number field \( {\mathbb{K}}_{f} = \mathbb{Q} \) , \[ {\rho }_{f,\ell }... | Given Version \( R \) of Modularity, let \( E \) be an elliptic curve over \( \mathbb{Q} \) with conductor \( N \) . Then there exists a newform \( f \in {\mathcal{S}}_{2}\left( {{\Gamma }_{0}\left( {M}_{f}\right) }\right) \) as in Definition 9.6.1, so \( {\rho }_{f,\lambda } \sim {\rho }_{E,\ell } \) for some suitable... | Yes |
Theorem 9.6.6. Let \( f = \frac{1}{2}{E}_{k}^{\psi ,\varphi, t} \in {\mathcal{E}}_{k}\left( {N,\chi }\right) \) where \( {E}_{k}^{\psi ,\varphi, t} \) is an eigenform as just described. Let \( \ell \) be prime. For each maximal ideal \( \lambda \) of \( {\mathcal{O}}_{{\mathbb{K}}_{f}} \) lying over \( \ell \) , the re... | To prove this, again recall from Section 9.3 that any primitive Dirichlet character \( \phi \) modulo \( N \) acts as a Galois representation unramified at the primes \( p \) not dividing its conductor, and it takes \( {\operatorname{Frob}}_{\mathfrak{p}} \) to \( \phi \left( p\right) \) for such \( p \) . Thus if also... | Yes |
Theorem 9.6.7. Let the Galois representation \( \rho : {G}_{\mathbb{Q}} \rightarrow {\mathrm{{GL}}}_{2}\left( \mathbb{L}\right) \) be odd, reducible, and semisimple, i.e. \( \rho \sim \left\lbrack \begin{matrix} \psi & 0 \\ 0 & \phi \end{matrix}\right\rbrack \) . If \( \psi \) has finite image and \( \det \rho = \) \( ... | \[ {\rho }_{f,\lambda } \sim \rho \] Here we are using the result from Section 9.3 that 1-dimensional Galois representations with finite image can be viewed as Dirichlet characters. | Yes |
Proposition 2.5. Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be a monotone continuous function. Then the following statements are equivalent:\n\n2.5.1. The inverse image \( {f}^{-1}\left( A\right) \) is a Lebesgue measurable subset of \( \left\lbrack {a, b}\right\rbrack \) for every \( A \subs... | Proof. \( {2.5.2} \Rightarrow {2.5.1} \) . Evidently, \( {L}_{f} \) is open (in the relative topology of \( \left\lbrack {a, b}\right\rbrack \) ) and \( f \) is constant on each component of \( {L}_{f} \) . Let \( E \) be a set of endpoints of components of \( {L}_{f} \) . Let us denote by \( {f}_{0} \) and \( {f}_{1} ... | Yes |
Proposition 3.2. The Cantor function \( G \) is a first modulus of continuity of itself, i.e., \n\n\[ \n\mathop{\sup }\limits_{\substack{{\left| {x - y}\right| \leq \delta } \\ {x, y \in \left\lbrack {0,1}\right\rbrack } }}\left| {G\left( x\right) - G\left( y\right) }\right| = G\left( \delta \right) \n\] \n\nfor every ... | The proof of Propositions 3.1 and 3.2 can be found in Timan's book [54, Section 3.2.4] or in the paper of Doboš [18]. | No |
Lemma 3.6. Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be a continuous function. Then \( \left\lbrack {a, b}\right\rbrack \smallsetminus {L}_{f} \) is a compact perfect set. | Proof. By definition \( {L}_{f} \) is relatively open in \( \left\lbrack {a, b}\right\rbrack \) . Hence \( \left\lbrack {a, b}\right\rbrack \smallsetminus {L}_{f} \) is a compact subset of \( \mathbb{R} \) . If \( p \) is an isolated point of \( \left\lbrack {a, b}\right\rbrack \smallsetminus {L}_{f} \), then either it... | Yes |
Lemma 3.9. Let \( f : \left( {a, b}\right) \rightarrow \mathbb{R} \) be continuous increasing function. If equality (3.8) holds, then \( {V}_{f} \) is a dense subset of \( \left( {a, b}\right) \smallsetminus {L}_{f} \) . | Proof of Theorem 3.7. It is obvious that \( \varphi \cdot f \) is locally monotone at \( {x}_{0} \) for every \( {x}_{0} \in {L}_{f} \) . Suppose that \( \varphi \cdot f \) is monotone on an open interval \( J \subset \left( {a, b}\right) \) and \( {x}_{0} \in \left( {\left( {a, b}\right) \smallsetminus {L}_{f}}\right)... | No |
Proposition 4.2. The Cantor function \( G \) is the unique element of \( \mathcal{M}\left\lbrack {0,1}\right\rbrack \) for which\n\n\[ G\left( x\right) = \left\{ \begin{array}{ll} \frac{1}{2}G\left( {3x}\right) & \text{ if }0 \leq x \leq \frac{1}{3}, \\ \frac{1}{2} & \text{ if }\frac{1}{3} < x < \frac{2}{3}, \\ \frac{1... | Proof. Define a map \( F : \mathcal{M}\left\lbrack {0,1}\right\rbrack \rightarrow \mathcal{M}\left\lbrack {0,1}\right\rbrack \) as\n\n\[ F\left( f\right) \left( x\right) = \left\{ \begin{array}{ll} \frac{1}{2}f\left( {3x}\right) , & 0 \leq x \leq \frac{1}{3}, \\ \frac{1}{2}, & \frac{1}{3} < x < \frac{2}{3}, \\ \frac{1}... | Yes |
Lemma 4.19. Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be a continuous function and let \( p \in \left( {1,\infty }\right) \) . Suppose that the graph of \( f \) is symmetric with respect to the point \( \left( {a + b/2;f\left( {a + b/2}\right) }\right) \) . Then the inequality\n\n\[ \frac{1}... | Proof. We may assume without loss of generality that \( a = - 1, b = 1 \) . Now for \( f\left( 0\right) = 0 \) inequality (4.20) is trivial. Hence replacing \( f \) with \( - f \), if necessary, we may assume that \( f\left( 0\right) < 0 \) . Write\n\n\[ \Psi \left( x\right) = f\left( x\right) - f\left( 0\right) . \]\n... | Yes |
Lemma 4.25. If \( F : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R} \) is a continuous function satisfying (4.9) and (4.13), then the graph of the restriction \( {\left. F\right| }_{J} \) is symmetric with respect to the point \( \left( {{x}_{J},{G}_{J}}\right) \) for each \( J \in \mathcal{I} \) . | Proof. It follows from (4.13) that\n\n\[ F\left( \frac{1}{2}\right) = G\left( \frac{1}{2}\right) = \frac{1}{2} \]\n\nthus we can rewrite equation (4.13) as\n\n\[ F\left( \frac{1}{2}\right) = \frac{1}{2}\left( {F\left( {\frac{1}{2} + x}\right) + F\left( {\frac{1}{2} - x}\right) }\right) . \]\n\nConsequently a graph of \... | Yes |
Proposition 5.2. There is the unique Borel regular probability measure \( \mu \) such that\n\n\[ \mu \left( A\right) = \frac{1}{2}\mu \left( {{\varphi }_{0}^{-1}\left( A\right) }\right) + \frac{1}{2}\mu \left( {{\varphi }_{1}^{-1}\left( A\right) }\right) \]\n\n(5.3)\n\nfor every Borel set \( A \subseteq \mathbb{R} \) .... | For the proof see [24, Theorem 2.8, Lemma 6.4]. | No |
Corollary 5.9. The extended Cantor function \( \widehat{G} \) is the cumulative distribution function of the restriction of the Hausdorff measure \( {\mathcal{H}}^{{s}_{c}} \) to the Cantor set \( C \) . | This description of the Cantor function enables us to suggest a method for the proof of the following proposition. Write\n\n\[ \n{\widehat{G}}_{h}\left( x\right) \mathrel{\text{:=}} \widehat{G}\left( {x + h}\right) - \widehat{G}\left( x\right) \]\n\nfor each \( h \in \mathbb{R} \) . The function \( {\widehat{G}}_{h} \)... | No |
Proposition 5.10 (Hille and Tamarkin [34]). Let \( \\operatorname{Var}\\left( {\\widehat{G}}_{h}\\right) \) be a total variation of \( {\\widehat{G}}_{h} \). Then we have\n\n\[ \n\\mathop{\\sup }\\limits_{{0 \\leq h \\leq \\delta }}\\operatorname{Var}\\left( {\\widehat{G}}_{h}\\right) = 2 \n\]\n\nfor every \( \\delta >... | This holds because sets \( C \\cap \\left( {C \\pm {3}^{-n}}\\right) \) have finite numbers of elements for all positive integer \( n \) . | No |
Theorem 5.13. Let \( F : \mathbb{R} \rightarrow \mathbb{R} \) be a function of bounded variation with singular part \( \varphi \) . Then the limit relation\n\n\[ \n\mathop{\limsup }\limits_{{h \rightarrow 0}}\left( {\operatorname{Var}\left( {F}_{h}\right) }\right) = 2\operatorname{Var}\left( \varphi \right) \n\]\n\nhol... | The theorem is an immediate adaptation of the result that was proved by Wiener and Young [57]. | No |
Proposition 6.4. The following equality holds:\n\n\[ \n{m}_{n + 1} = \frac{\mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{matrix} n + 1 \\ k \end{matrix}\right) {2}^{n - k}{m}_{k}}{{3}^{n + 1} - 1} \]\n\n(6.5)\n\nfor every natural \( n \) and we have\n\n\[ \n2{m}_{n} = \mathop{\sum }\limits_{{k = 0}}^{{n - 1}}{\left(... | Proof. First we prove (6.5).\n\nThe Cantor set \( C \) can be defined as the intersection \( \mathop{\bigcap }\limits_{{k = 0}}^{\infty }{C}_{k} \), see (1.4). Each \( {C}_{k} \) consists of \( {2}^{k} \) disjoint closed intervals \( {C}_{k}^{i} = \left\lbrack {{a}_{k}^{i},{b}_{k}^{i}}\right\rbrack, i = 1,\ldots ,{2}^{... | Yes |
Proposition 6.15. The equality\n\n\[ \n{\int }_{0}^{1}{\mathrm{e}}^{ax}\mathrm{\;d}G\left( x\right) = \exp \left( \frac{a}{2}\right) \mathop{\prod }\limits_{{k = 1}}^{\infty }\cosh \left( \frac{a}{{3}^{k}}\right) \n\]\n\nholds for every \( a \in \mathbb{C} \) . | Proof. Write\n\n\[ \n\Phi \left( a\right) \mathrel{\text{:=}} {\int }_{0}^{1}{\mathrm{e}}^{ax}\mathrm{\;d}G\left( x\right) .\n\]\n\nIt follows from (4.9) to (4.11) that\n\n\[ \nG\left( \frac{x}{3}\right) = \frac{1}{2}G\left( x\right) ,\;G\left( {\frac{2}{3} + \frac{x}{3}}\right) = \frac{1}{2} + \frac{1}{2}G\left( x\rig... | Yes |
Proposition 6.17 (Hille and Tamarkin [34]). For all integers \( n \) , | \[ {\widehat{\mu }}_{n} = {\mathrm{e}}^{\pi \mathrm{i}n}\mathop{\prod }\limits_{{j = 1}}^{\infty }\cos \left( \frac{2\pi n}{{3}^{j}}\right) . \] | Yes |
Corollary 6.18 (Hille and Tamarkin [34]). Let \( k \) be a positive integer and let \( n = {3}^{k} \) . Then\n\n\[ \n{\widehat{\mu }}_{n} = - \mathop{\prod }\limits_{{v = 1}}^{\infty }\cos \left( \frac{2\pi }{{3}^{v}}\right) = \text{ const. } \neq 0.\n\] | Remark 6.19. The infinite product \( \mathop{\prod }\limits_{{v = 1}}^{\infty }\cos \left( {{2\pi }/{3}^{v}}\right) \) converges absolutely because\n\n\[ \n\left| {1 - \cos \frac{2\pi }{{3}^{v}}}\right| \leq 2{\sin }^{2}\left( \frac{\pi }{{3}^{v}}\right) < 2\frac{{\pi }^{2}}{{9}^{v}}\n\]\n\nand hence \( \mathop{\prod }... | Yes |
Proposition 6.20. The length of the arc of the curve \( y = G\left( x\right) \) between the points \( \left( {0,0}\right) \) and \( \left( {1,1}\right) \) is 2 . | Remark 6.21. The detailed proof of this proposition can be found in [13]. In fact, this follows rather easily from 2.1.4. Probably the length of the curve \( y = G\left( x\right) \) was first calculated in [34]. | No |
Theorem 6.22. Let \( F : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R} \) be a continuous, increasing function for which \( F\left( 0\right) = 0 \) and \( F\left( 1\right) = 1 \) . Then the following two statements are equivalent.\n\n6.22.1. The length of the arc \( y = F\left( x\right) ,0 \leq x \leq 1 \), is... | This theorem follows from the results of Pelling [46]. See also [19]. | No |
Proposition 7.1. The Cantor function \( G \) is isomorphic (as a map from \( \left\lbrack {0,1}\right\rbrack \) into \( \left\lbrack {0,1}\right\rbrack \) ) to a continuous monotone function \( q : \left\lbrack {0,1}\right\rbrack \rightarrow \left\lbrack {0,1}\right\rbrack \) if and only if the set of constancy \( {L}_... | \[ g\left( {\{ 0,1\} }\right) = \{ 0,1\} \subseteq \left\lbrack {0,1}\right\rbrack \smallsetminus {L}_{g}. \] | No |
Lemma 7.3. Let \( A, B \) be everywhere dense subsets of \( \left\lbrack {0,1}\right\rbrack \) and let \( f : A \rightarrow B \) be an increasing bijective map. Then \( f \) is a homeomorphism and, moreover, \( f \) can be extended to a self-homeomorphism of the closed interval \( \left\lbrack {0,1}\right\rbrack \) . | Proof. It is easy to see that\n\n\[ 0 = \mathop{\lim }\limits_{\substack{{x \rightarrow 0} \\ {x \in A} }}f\left( x\right) = 1 - \mathop{\lim }\limits_{\substack{{x \rightarrow 1} \\ {x \in A} }}f\left( x\right) \]\n\nand\n\n\[ \mathop{\lim }\limits_{\substack{{x \rightarrow t} \\ {x \in A \cap \left\lbrack {0, t}\righ... | Yes |
Proposition 7.6. If \( F \) is topologically isomorphic to the Cantor function \( G \), then \( F \) is not continuously differentiable. | Let \( {G}_{C} \) be the restriction of \( G \) to the Cantor set \( C \) . As it is easy to see, \( {G}_{C} \) is continuous, closed but not open. For example we have\n\n\[ \n{G}_{C}\left( {\left\lbrack {0,\frac{1}{2}}\right) \cap C}\right) = G\left( \left\lbrack {0,\frac{1}{2}}\right) \right) = \left\lbrack {0,\frac{... | No |
Proposition 7.7. Let \( A \) be a nonempty subset of \( C \) . If the interior of \( A \) is nonempty in \( C \) , then the interior of \( {G}_{C}\left( A\right) \) is nonempty in \( \left\lbrack {0,1}\right\rbrack \), or in other words\n\n\[ \left( {{\operatorname{Int}}_{C}A \neq \varnothing }\right) \Rightarrow \left... | Proof. Suppose that \( A \subseteq C \) and \( {\operatorname{Int}}_{C}A \neq \varnothing \) . Since \( {C}^{ \circ } \) is a dense subset of \( C \), there is an interval \( \left( {a, b}\right) \) such that both \( a \) and \( b \) are in \( {C}^{ \circ } \) and\n\n\[ A \supseteq \left( {a, b}\right) \cap C\text{.} \... | Yes |
Proposition 7.11. If \( B \) is a residual (first category) subset of \( \left\lbrack {0,1}\right\rbrack \), then \( {G}_{C}^{-1}\left( B\right) \) is residual (first category) in \( C \) . | Proof. Let \( B \) be a residual subset of \( \left\lbrack {0,1}\right\rbrack \) . Write\n\n\[ \n{K}_{1} \mathrel{\text{:=}} \left\lbrack {0,1}\right\rbrack \smallsetminus {C}^{ \circ },\;W \mathrel{\text{:=}} {G}^{-1}\left( B\right) .\n\]\n\nIt is sufficient to show that \( W \cap {C}^{ \circ } \) is residual in \( C ... | Yes |
Proposition 7.17. Let \( B \subseteq C \) . Then \( B \) is an everywhere dense subset of \( C \) if and only if \( {G}_{C}\left( B\right) \) is an everywhere dense subset of \( \left\lbrack {0,1}\right\rbrack \) . | Proof. Since \( {G}_{C} \) is continuous, the image \( {G}_{C}\left( B\right) \) is a dense subset of \( \left\lbrack {0,1}\right\rbrack \) for every dense B. Suppose that\n\n\[{\operatorname{Clo}}_{\left\lbrack 0,1\right\rbrack }\left( {{G}_{C}\left( B\right) }\right) = \left\lbrack {0,1}\right\rbrack \]\n\nbut\n\n\[C... | Yes |
Theorem 7.20. The Cantor function \( G \) is isomorphic (as a map from \( \left\lbrack {0,1}\right\rbrack \) into \( \mathbb{R} \) ) to a continuous monotone function \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R} \) with \( \{ 0,1\} \subseteq \left( {\left\lbrack {0,1}\right\rbrack \smallsetminus {L}_{f... | Proof. Let \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R} \) be a function which is topologically isomorphic to \( G \) and let \( A \subseteq \mathbb{R} \) . Then it follows from Proposition 7.1 that the set of constancy \( {L}_{f} \) is everywhere dense in \( \left\lbrack {0,1}\right\rbrack \) . Reason... | Yes |
Proposition 8.1. The set \( G\left( M\right) \) has a (linear Lebesgue) measure zero. | Proof. See Theorem 7.2 in Chapter IX of Saks [48]. | No |
Corollary 8.2. Let \( {\mathcal{H}}^{{s}_{c}} \) be a Hausdorff measure with \( {s}_{c} = \lg 2/\lg 3 \) . Then we have\n\n\[ \n{\mathcal{H}}^{{s}_{c}}\left( M\right) = 0.\n\] | Proof. This follows from Propositions 8.1 and 5.5. | No |
Corollary 8.5. The set \( W \cap C \) is residual in \( C \) . | Proof. The proof follows from 8.3.1 and Proposition 7.11. | No |
Corollary 8.14. If \( x \in {C}^{ \circ } \), then \( {D}_{ - }G\left( x\right) = {D}_{ + }G\left( x\right) = 0 \) if and only if | \[ {\lambda }_{x} = {\mu }_{x} = 0 \] | No |
Corollary 8.16. For each \( \mu \in \left\lbrack {0,\infty }\right\rbrack \) and \( \lambda \in \left\lbrack {0,\infty }\right\rbrack \) there exists a nonempty, compact perfect subset in \( \left\{ {z \in C : {D}_{ - }G\left( z\right) = \lambda }\right. \) and \( \left. {{D}_{ + }G\left( z\right) = \mu }\right\} \) . | Proof. Since \( G \) is continuous on \( \left\lbrack {0,1}\right\rbrack \), each of the Dini derivatives of \( G \) is in a Baire class two [6, Chapter IV, Theorem 2.2]. Hence, the set \( \left\{ {z \in C : {D}_{ - }G\left( z\right) = \lambda }\right. \) and \( \left. {{D}_{ + }G\left( z\right) = \mu }\right\} \) is a... | Yes |
Theorem 9.3. For all sufficiently large \( b \) the inequality\n\n\[ \mathop{\limsup }\limits_{{{\Delta x} \rightarrow 0 + }}\frac{G\left( {x + {b\Delta x}}\right) - G\left( {x - {b\Delta x}}\right) }{G\left( {x + {\Delta x}}\right) - G\left( {x - {\Delta x}}\right) } < b \]\n\n(9.4)\n\nholds for all \( x \in C \smalls... | Remark 9.5. It was shown in [9] that (9.4) is true for all \( b \geq {81} \) and false if \( b = 4 \) . Moreover, as it follows from [9] (see, also [53, Theorem 7.32]) that Theorem 9.3 implies Proposition 9.1. | No |
Proposition 10.1. The function \( G \) satisfies a Hölder condition of order \( {s}_{c} = \lg 2/\lg 3 \), the Hölder coefficient being not greater than 1. In other words\n\n\[ \left| {G\left( x\right) - G\left( y\right) }\right| \leq {\left| x - y\right| }^{{s}_{c}} \]\n\nfor all \( x, y \in \left\lbrack {0,1}\right\rb... | Proof. Using lemma from [32] we obtain the inequality\n\n\[ {\left( \frac{1}{2}t\right) }^{{s}_{c}} \leq G\left( t\right) \leq {t}^{{s}_{c}} \]\n\nfor all \( t \in \left\lbrack {0,1}\right\rbrack \) . Since \( G \) is its first modulus of continuity (see Proposition 3.2), the last inequality implies (10.2). Moreover, i... | Yes |
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