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Lemma 3. Let \( F = F\left( x\right) \) be a distribution function on the real line and let \( \varphi = \varphi \left( t\right) \) be its characteristic function. Then there is a constant \( K > 0 \) such that for every \( a > 0 \)\n\[{\int }_{\left| x\right| \geq 1/a}{dF}\left( x\right) \leq \frac{K}{a}{\int }_{0}^{a... | Proof. Since \( \operatorname{Re}\varphi \left( t\right) = {\int }_{-\infty }^{\infty }\cos {txdF}\left( x\right) \), we find by Fubini’s theorem that\n\n\[ \frac{1}{a}{\int }_{0}^{a}\left\lbrack {1 - \operatorname{Re}\varphi \left( t\right) }\right\rbrack {dt} = \frac{1}{a}{\int }_{0}^{a}\left\lbrack {{\int }_{-\infty... | Yes |
Theorem 2 (Khinchin’s Law of Large Numbers). Let \( {\xi }_{1},{\xi }_{2},\ldots \) be a sequence of independent identically distributed random variables with \( \mathrm{E}\left| {\xi }_{1}\right| < \infty ,{S}_{n} = {\xi }_{1} + \) \( \cdots + {\xi }_{n} \) and \( \mathrm{E}{\xi }_{1} = m \) . Then \( {S}_{n}/n\overse... | Proof. Let \( \varphi \left( t\right) = \mathrm{E}{e}^{{it}{\xi }_{1}} \) and \( {\varphi }_{{S}_{n}/n}\left( t\right) = \mathrm{E}{e}^{{it}{S}_{n}/n} \) . Since the random variables are independent, we have\n\n\[ {\varphi }_{{S}_{n}/n}\left( t\right) = {\left\lbrack \varphi \left( \frac{t}{n}\right) \right\rbrack }^{n... | Yes |
Theorem 3 (Central Limit Theorem for Independent Identically Distributed Random Variables). Let \( {\xi }_{1},{\xi }_{2},\ldots \) be a sequence of independent identically distributed (nondegenerate) random variables with \( \mathrm{E}{\xi }_{1}^{2} < \infty \) and \( {S}_{n} = {\xi }_{1} + \cdots + {\xi }_{n} \). Then... | Proof. Let \( \mathrm{E}{\xi }_{1} = m,\operatorname{Var}{\xi }_{1} = {\sigma }^{2} \) and\n\[ \varphi \left( t\right) = \mathrm{E}{e}^{{it}\left( {{\xi }_{1} - m}\right) }.\]\nThen if we put\n\[ {\varphi }_{n}\left( t\right) = \mathrm{E}\exp \left\{ {{it}\frac{{S}_{n} - \mathrm{E}{S}_{n}}{\sqrt{\operatorname{Var}{S}_{... | Yes |
Theorem 4 (Poisson’s Theorem). For each \( n \geq 1 \) let the independent random variables \( {\xi }_{n1},\ldots ,{\xi }_{nn} \) be such that\n\n\[ \mathrm{P}\left( {{\xi }_{nk} = 1}\right) = {p}_{nk},\;\mathrm{P}\left( {{\xi }_{nk} = 0}\right) = {q}_{nk} \]\n\nwith \( {p}_{nk} + {q}_{nk} = 1 \) . Suppose that\n\n\[ \... | Proof. Since\n\n\[ \mathrm{E}{e}^{{it}{\xi }_{nk}} = {p}_{nk}{e}^{it} + {q}_{nk} \]\n\nfor \( 1 \leq k \leq n \), we have\n\n\[ {\varphi }_{{S}_{n}}\left( t\right) = \mathrm{E}{e}^{{it}{S}_{n}} = \mathop{\prod }\limits_{{k = 1}}^{n}\left( {{p}_{nk}{e}^{it} + {q}_{nk}}\right) \]\n\n\[ = \mathop{\prod }\limits_{{k = 1}}^... | Yes |
Theorem 2. Let for each \( n \geq 1 \)\n\n\[ \n{\xi }_{n1},{\xi }_{n2},\ldots ,{\xi }_{nn} \]\n\nbe independent random variables such that \( \mathrm{E}{\xi }_{nk} = 0 \) and \( \operatorname{Var}{S}_{n} = 1 \), where \( {S}_{n} = \) \( {\xi }_{n1} + \cdots + {\xi }_{nn} \)\n\nThen Lindeberg’s condition (16) is suffici... | 4. Since\n\n\[ \n\mathop{\max }\limits_{{1 \leq k \leq n}}\mathrm{E}{\xi }_{nk}^{2} \leq {\varepsilon }^{2} + \mathop{\sum }\limits_{{k = 1}}^{n}\mathrm{E}\left\lbrack {{\xi }_{nk}^{2}I\left( {\left| {\xi }_{nk}\right| \geq \varepsilon }\right) }\right\rbrack \]\n\nit is clear that Lindeberg's condition (16) implies th... | No |
A random variable \( T \) can be a limit in distribution of sums \( {T}_{n} = \) \( \mathop{\sum }\limits_{{k = 1}}^{n}{\xi }_{n, k} \) if and only if \( T \) is infinitely divisible. | If \( T \) is infinitely divisible, for each \( n \geq 1 \) there are independent identically distributed random variables \( {\xi }_{n,1},\ldots ,{\xi }_{n, k} \) such that \( T \triangleq {\xi }_{n,1} + \cdots + {\xi }_{n, k} \), and this means that \( T\overset{d}{ = }{T}_{n}, n \geq 1 \) .\n\nConversely, let \( {T}... | Yes |
Theorem 4 (Lévy-Khinchin Representation). A random variable \( T \) is stable if and only if its characteristic function \( \varphi \left( t\right) \) has the form \( \varphi \left( t\right) = \exp \psi \left( t\right) \) | \[ \psi \left( t\right) = {it\beta } - d{\left| t\right| }^{\alpha }\left( {1 + {i\theta }\frac{t}{\left| t\right| }G\left( {t,\alpha }\right) }\right) ,\] where \( 0 < \alpha < 2,\beta \in R, d \geq 0,\left| \theta \right| \leq 1, t/\left| t\right| = 0 \) for \( t = 0 \), and \[ G\left( {t,\alpha }\right) = \left\{ \b... | Yes |
Theorem 1. The Lévy-Prokhorov metric \( L\left( {P,\widetilde{P}}\right) \) metrizes weak convergence:\n\n\[ L\left( {{P}_{n}, P}\right) \rightarrow 0 \Leftrightarrow {P}_{n}\overset{w}{ \rightarrow }P. \] | Proof. \( \left( \Rightarrow \right) \) Let \( L\left( {{P}_{n}, P}\right) \rightarrow 0, n \rightarrow \infty \) . Then for every specified closed set \( F \in \mathcal{E} \) and every \( \varepsilon > 0 \), we have, by (2) and equation (a) of Lemma 1,\n\n\[ \mathop{\limsup }\limits_{n}{P}_{n}\left( F\right) \leq P\le... | No |
Theorem 2. The metric \( \parallel P - \widetilde{P}{\parallel }_{BL}^{ * } \) metrizes weak convergence:\n\n\[ \n{\begin{Vmatrix}{P}_{n} - P\end{Vmatrix}}_{BL}^{ * } \rightarrow 0 \Leftrightarrow {P}_{n}\overset{w}{ \rightarrow }P. \n\] | Proof. The implication \( \left( \Leftarrow \right) \) follows directly from (13). To prove \( \left( \Rightarrow \right) \), it is enough to show that in the definition of weak convergence \( {P}_{n}\overset{w}{ \rightarrow }P \) as given by (12) for every continuous bounded function \( f = f\left( x\right) \), it is ... | Yes |
Lemma 2. Weak convergence \( {P}_{n}\overset{w}{ \rightarrow }P \) occurs if and only if property (12) is satisfied for every function \( f = f\left( x\right) \) of class \( {BL} \) . | Proof. The proof is obvious in one direction. Let us now consider the functions \( {f}_{A}^{\varepsilon } = {f}_{A}^{\varepsilon }\left( x\right) \) defined in (14). As was established above in the proof of Theorem 1, for each \( \varepsilon > 0 \) the class \( {\mathcal{G}}^{\varepsilon } = \left\{ {{f}_{A}^{\varepsil... | Yes |
Let \( P,{P}_{n}, n \geq 1 \), be probability measures on \( \left( {E,\mathcal{E},\rho }\right) \) such that \( {P}_{n}\overset{w}{ \rightarrow }P \) . Then there is a probability space \( \left( {{\Omega }^{ * },{\mathcal{F}}^{ * },{\mathbf{P}}^{ * }}\right) \) and random elements \( {X}^{ * },{X}_{n}^{ * }, n \geq \... | Proof of the THEOREM IN THE CASE \( E = R \) . Let \( F = F\left( x\right) \) and \( {F}_{n} = {F}_{n}\left( x\right) \) be the distribution functions corresponding to the measures \( P \) and \( {P}_{n} \) on \( \left( {R,\mathcal{B}\left( R\right) }\right) \) . We associate with a function \( F = F\left( x\right) \) ... | Yes |
Let \( \left( {E,\mathcal{E},\rho }\right) \) and \( \left( {{E}^{\prime },{\mathcal{E}}^{\prime },{\rho }^{\prime }}\right) \) be separable metric spaces, and let \( {X}_{n}\overset{\mathcal{D}}{ \rightarrow }X \) . Let the mapping \( h = h\left( x\right), x \in E \), have the property that\n\n\[ \mathrm{P}\left\{ {\o... | Proof. As in Theorem 1, it is enough to prove the validity of, for example, the first proposition.\n\nLet \( {X}^{ * } \) and \( {X}_{n}^{ * }, n \geq 1 \), be random elements constructed by the \ | No |
Theorem 3. Let \( P \) and \( {P}_{n}, n \geq 1 \), be probability measures on \( \left( {E,\mathcal{E},\rho }\right) \) for which \( {P}_{n}\overset{w}{ \rightarrow }P \) . Then\n\n\[ \mathop{\sup }\limits_{{g \in \mathcal{G}}}\left| {{\int }_{E}g\left( x\right) {P}_{n}\left( {dx}\right) - {\int }_{E}g\left( x\right) ... | Proof. Let (7) not occur. Then there are an \( a > 0 \) and functions \( {g}_{1},{g}_{2},\ldots \) from \( \mathcal{G} \)\n\nsuch that\n\n\[ \left| {{\int }_{E}{g}_{n}\left( x\right) {P}_{n}\left( {dx}\right) - {\int }_{E}{g}_{n}\left( x\right) P\left( {dx}\right) }\right| \geq a > 0 \]\n\nfor infinitely many values of... | No |
For each pair \( P,\widetilde{P} \) of measures we can find a probability space \( \left( {{\Omega }^{ * },{\mathcal{F}}^{ * },{\mathrm{P}}^{ * }}\right) \) and random elements \( X \) and \( \widetilde{X} \) on it with values in \( E \) such that their distributions coincide respectively with \( P \) and \( \widetilde... | Proof. By Theorem 1, we can find a probability space \( \left( {{\Omega }^{ * },{\mathcal{F}}^{ * },{\mathrm{P}}^{ * }}\right) \) and random elements \( X \) and \( \widetilde{X} \) such that \( {\mathrm{P}}^{ * }\left( {X \in A}\right) = P\left( A\right) \) and \( {\mathrm{P}}^{ * }\left( {\widetilde{X} \in A}\right) ... | Yes |
Lemma 1. The distance in variation is given by\n\n\\[ \n\\parallel P - \\widetilde{P}\\parallel = 2\\mathop{\\sup }\\limits_{{A \\in \\mathcal{F}}}\\left| {P\\left( A\\right) - \\widetilde{P}\\left( A\\right) }\\right| .\n\\]\n\n(2) | Proof. Since, for all \\( A \\in \\mathcal{F} \\) ,\n\n\\[ \nP\\left( A\\right) - \\widetilde{P}\\left( A\\right) = \\widetilde{P}\\left( \\bar{A}\\right) - P\\left( \\bar{A}\\right)\n\\]\n\nwe have\n\n\\[ \n2\\left| {P\\left( A\\right) - \\widetilde{P}\\left( A\\right) }\\right| = \\left| {P\\left( A\\right) - \\widet... | Yes |
Lemma 2. Let \( Q \) be a \( \sigma \) -finite measure such that \( P \ll Q,\widetilde{P} \ll Q \) and let \( z = {dP}/{dQ} \) , \( \widetilde{z} = d\widetilde{P}/{dQ} \) be the Radon-Nikodym derivatives of \( P \) and \( \widetilde{P} \) with respect to \( Q \) . Then\n\n\[ \n\parallel P - \widetilde{P}\parallel = {E}... | Proof. For all \( \mathcal{F} \) -measurable functions \( \psi = \psi \left( \omega \right) \) with \( \left| {\psi \left( \omega \right) }\right| \leq 1 \), we see from the definitions of \( z \) and \( \widetilde{z} \) that\n\n\[ \n\left| {{E\psi } - \widetilde{E}\psi }\right| = \left| {{E}_{Q}\psi \left( {z - \widet... | Yes |
The Hellinger integral of order \( \alpha \in \left( {0,1}\right) \) (and consequently also \( \rho \left( {P,\widetilde{P}}\right) ) \) is independent of the choice of the dominating measure \( Q \) . | 1. If the measure \( {Q}^{\prime } \) dominates \( P \) and \( \widetilde{P},{Q}^{\prime } \) also dominates \( Q = \left( {P + \widetilde{P}}\right) /2 \) . Hence, it is enough to show that if \( Q \ll {Q}^{\prime } \), we have\n\n\[ \n{E}_{Q}\left( {{z}^{\alpha }{\widetilde{z}}^{1 - \alpha }}\right) = {E}_{{Q}^{\prim... | Yes |
Theorem 1. We have the following inequalities:\n\n\[ \n2\left\lbrack {1 - H\left( {P,\widetilde{P}}\right) }\right\rbrack \leq \parallel P - \widetilde{P}\parallel \leq \sqrt{8\left\lbrack {1 - H\left( {P,\widetilde{P}}\right) }\right\rbrack } \n\] \n\n(21) \n\n\[ \n\parallel P - \widetilde{P}\parallel \leq 2\sqrt{1 - ... | Proof. Since \( H\left( {P,\widetilde{P}}\right) \leq 1 \) and \( 1 - {x}^{2} \leq 2\left( {1 - x}\right) \) for \( 0 \leq x \leq 1 \), the right-hand inequality in (21) follows from (22), the proof of which is provided by the following chain of inequalities (where \( Q = \left( {1/2}\right) \left( {P + \widetilde{P}}\... | Yes |
Corollary 4. Since by (5)\n\n\[ \mathcal{E}r\left( {P,\widetilde{P}}\right) = 1 - \frac{1}{2}\parallel P - \widetilde{P}\parallel \]\n\nwe have, by (21) and (22), | \[ \frac{1}{2}{H}^{2}\left( {P,\widetilde{P}}\right) \leq 1 - \sqrt{1 - {H}^{2}\left( {P,\widetilde{P}}\right) } \leq \mathcal{E}r\left( {P,\widetilde{P}}\right) \leq H\left( {P,\widetilde{P}}\right) . \] | Yes |
Theorem 3. The following conditions are equivalent:\n\n(a) \( \widetilde{P} \bot P \) ,\n\n(b) \( \widetilde{P}\left( {z > 0}\right) = 0 \) ,\n\n(c) \( H\left( {\alpha ;P,\widetilde{P}}\right) \rightarrow 0,\alpha \downarrow 0 \) ,\n\n(d) \( H\left( {\alpha ;P,\widetilde{P}}\right) = 0 \) for all \( \alpha \in \left( {... | Proof. The proofs of these theorems will be given simultaneously. By the definitions of \( z \) and \( \widetilde{z} \) ,\n\n\[ P\left( {z = 0}\right) = {E}_{Q}\left\lbrack {{zI}\left( {z = 0}\right) }\right\rbrack = 0, \]\n\n(27)\n\n\[ \widetilde{P}\left( {A\cap \{ z > 0\} }\right) = {E}_{Q}\left\lbrack {\widetilde{z}... | Yes |
Let \( P = {P}_{1} \times {P}_{2} \times \ldots ,\widetilde{P} = {\widetilde{P}}_{1} \times {\widetilde{P}}_{2}\ldots \), where \( {P}_{k} \) and \( {\widetilde{P}}_{k} \) are Gaussian measures on \( \left( {R,\mathcal{B}\left( R\right) }\right) \) with densities \[ {p}_{k}\left( x\right) = \frac{1}{\sqrt{2\pi }}{e}^{{... | Since \[ H\left( {\alpha ;P,\widetilde{P}}\right) = \mathop{\prod }\limits_{{k = 1}}^{\infty }H\left( {\alpha ;{P}_{k},{\widetilde{P}}_{k}}\right) \] where a simple calculation shows that \[ H\left( {\alpha ;{P}_{k},{\widetilde{P}}_{k}}\right) = {\int }_{-\infty }^{\infty }{p}_{k}^{\alpha }\left( x\right) {\widetilde{p... | Yes |
Example 2. Again let \( P = {P}_{1} \times {P}_{2} \times \ldots ,\widetilde{P} = {\widetilde{P}}_{1} \times {\widetilde{P}}_{2} \times \ldots \), where \( {P}_{k} \) and \( {\widetilde{P}}_{k} \) are Poisson distributions with respective parameters \( {\lambda }_{k} > 0 \) and \( {\widetilde{\lambda }}_{k} > 0 \) . Th... | \[ \widetilde{P} \ll P \Leftrightarrow P \ll \widetilde{P} \Leftrightarrow \widetilde{P} \sim P \Leftrightarrow \mathop{\sum }\limits_{{k = 1}}^{\infty }{\left( \sqrt{{\lambda }_{k}} - \sqrt{{\widetilde{\lambda }}_{k}}\right) }^{2} < \infty ,\] \[ \widetilde{P} \bot P \Leftrightarrow \mathop{\sum }\limits_{{k = 1}}^{\i... | Yes |
Under the above conditions the random variables\n\n\\[ \n{D}_{N}\left( \omega \right) = \mathop{\sup }\limits_{{x \in R}}\left| {{F}_{N}\left( {x;\omega }\right) - F\left( x\right) }\right| \n\\]\n\nconverge to zero with probability one:\n\n\\[ \n\mathrm{P}\left( {\mathop{\lim }\limits_{N}{D}_{N}\left( \omega \right) =... | Proof. Let \( Q \) be the set of rational numbers in \( R \) . Clearly,\n\n\\[ \n\mathop{\sup }\limits_{{r \in Q}}\left| {{F}_{N}\left( {r;\omega }\right) - F\left( r\right) }\right| \n\\]\n\nis a random variable. And since\n\n\\[ \n{D}_{N}\left( \omega \right) = \mathop{\sup }\limits_{{x \in R}}\left| {{F}_{N}\left( {... | Yes |
Theorem 2. Assume that \( F\left( x\right) \) is continuous. With the above notation, we have\n\n\[ \mathrm{P}\left( {\sqrt{N}{D}_{N}^{ + } \leq y}\right) \rightarrow 1 - {e}^{-2{y}^{2}},\;y \geq 0, \]\n\n(8)\n\n\[ \mathrm{P}\left( {\sqrt{N}{D}_{N} \leq y}\right) \rightarrow K\left( y\right) \]\n\n(9)\n\nwhere\n\n\[ K\... | The rigorous proof of this theorem goes beyond the scope of this book. It can be found in Billingsley [9], Sect. 13. Here we give an outline of the proof and a heuristic derivation of the formulas in (8) and (10). | No |
Proposition 1. Let \( n \) be a fixed squarefree positive integer. Let \( X, Y, Z, x \) always denote rational numbers, with \( X < Y < Z \) . There is a one-to-one correspondence between right triangles with legs \( X \) and \( Y \), hypotenuse \( Z \), and area \( n \) ; and numbers \( x \) for which \( x, x + n \), ... | Proof. First suppose that \( X, Y, Z \) is a triple with the desired properties: \( {X}^{2} + {Y}^{2} = {Z}^{2},\frac{1}{2}{XY} = n \) . If we add or subtract four times the second equation from the first, we obtain: \( {\left( X \pm Y\right) }^{2} = {Z}^{2} \pm {4n} \) . If we then divide both sides by four, we see th... | No |
Proposition 2. Let \( \left( {x, y}\right) \) be a point with rational coordinates on the curve \( {y}^{2} = {x}^{3} - {n}^{2}x \) . Suppose that \( x \) satisfies the two conditions: (i) it is the square of a rational number and (ii) its denominator is even. Then there exists a right triangle with rational sides and a... | Proof. Let \( u = \sqrt{x} \in {\mathbb{Q}}^{ + } \) . We work backwards through the sequence of steps at the beginning of this section. That is, set \( v = y/u \), so that \( {v}^{2} = {y}^{2}/x = \) \( {x}^{2} - {n}^{2} \), i.e., \( {v}^{2} + {n}^{2} = {x}^{2} \) . Now let \( t \) be the denominator of \( u \), i.e.,... | Yes |
Proposition 3. A function \( f\left( z\right) \in {\mathcal{E}}_{L}, L = \left\{ {m{\omega }_{1} + n{\omega }_{2}}\right\} \), which has no pole in the fundamental parallelogram \( \Pi \) must be a constant. | Proof. Since \( \Pi \) is compact, any such function must be bounded on \( \Pi \), say by a constant \( M \) . But then \( \left| {f\left( z\right) }\right| < M \) for all \( z \), since the values of \( f\left( z\right) \) are determined by the values on \( \Pi \) . By Liouville’s theorem, a meromorphic function which... | Yes |
Proposition 4. With the same notation as above, let \( \alpha + \Pi \) denote the translate of \( \Pi \) by the complex number \( \alpha \), i.e., \( \{ \alpha + z \mid z \in \Pi \} \) . Suppose that \( f\left( z\right) \in {\mathcal{E}}_{L} \) has no poles on the boundary \( C \) of \( \alpha + \Pi \) . Then the sum o... | Proof. By the residue theorem, this sum is equal to\n\n\[ \frac{1}{2\pi i}{\int }_{c}f\left( z\right) {dz} \]\n\nBut the integral over opposite sides cancel, since the values of \( f\left( z\right) \) at corresponding points are the same, while \( {dz} \) has opposite signs, because the path of integration is in opposi... | Yes |
Proposition 5. Under the conditions of Proposition 4, suppose that \( f\left( z\right) \) has no zeros or poles on the boundary of \( \alpha + \Pi \) . Let \( \left\{ {m}_{i}\right\} \) be the orders of the various zeros in \( \alpha + \Pi \), and let \( \left\{ {n}_{j}\right\} \) be the orders of the various poles. Th... | Proof. Apply Proposition 4 to the elliptic function \( {f}^{\prime }\left( z\right) /f\left( z\right) \) . Recall that the logarithmic derivative \( {f}^{\prime }\left( z\right) /f\left( z\right) \) has a pole precisely where \( f\left( z\right) \) has a zero or pole, such a pole is simple, and the residue there is equ... | Yes |
Proposition 6. The sum in (4.1) converges absolutely and uniformly for \( z \) in any compact subset of \( \mathbb{C} - L \) . | Proof. The sum in question is taken over a two-dimensional lattice. The proof of convergence will be rather routine if we keep in mind a one-dimensional analog. If instead of \( L \) we take the integers \( \mathbb{Z} \), and instead of reciprocal squares we take reciprocals, we obtain a real function \( f\left( x\righ... | Yes |
Lemma 2. \( \sum {\left| l\right| }^{-s} \) converges if \( s > 2 \) . | The proof of Lemma 1 is routine, and will be omitted. We give a sketch of the proof of Lemma 2. We split the sum into sums over \( l \) satisfying \( n - 1 < \left| l\right| \leq n \), as \( n = 1,2,\ldots \) It is not hard to show that the number of \( l \) in that annulus has order of magnitude \( n \) . Thus, the su... | No |
Proposition 7. \( \wp \left( z\right) \in {\mathcal{E}}_{L} \), and its only pole is a double pole at each lattice point. | Proof. The same argument as in the proof of Proposition 6 shows that for any fixed \( l \in L \), the function \( \wp \left( z\right) - {\left( z - l\right) }^{-2} \) is continuous at \( z = l \) . Thus, \( \wp \left( z\right) \) is a meromorphic function with a double pole at all lattice points and no other poles. Nex... | Yes |
Proposition 8. \( {\mathcal{E}}_{L} = \mathbb{C}\left( {\wp ,{\wp }^{\prime }}\right) \), i.e., any elliptic function for \( L \) is a rational expression in \( \wp \left( {z;L}\right) \) and \( {\wp }^{\prime }\left( {z;L}\right) \) . More precisely, given \( f\left( z\right) \in {\mathcal{E}}_{L} \), there exist two ... | Proof. If \( f\left( z\right) \) is an elliptic function for \( L \), then so are the two even functions\n\n\[ \frac{f\left( z\right) + f\left( {-z}\right) }{2}\text{ and }\frac{f\left( z\right) - f\left( {-z}\right) }{2{\wp }^{\prime }\left( z\right) }.\]\n\nSince \( f\left( z\right) \) is equal to the first of these ... | No |
Proposition 9. The subfield \( {\mathcal{E}}_{L}^{ + } \subset {\mathcal{E}}_{L} \) of even elliptic functions for \( L \) is generated by \( \wp \left( z\right) \), i.e., \( {\mathcal{E}}_{L}^{ + } = \mathbb{C}\left( \wp \right) \) . | Proof. The idea of the proof is to cook up a function which has the same zeros and poles as \( f\left( z\right) \) using only functions of the form \( \wp \left( z\right) - u \) with \( u \) a constant.\n\n\n\nFigure I... | Yes |
Proposition 10. The map (6.10) is an analytic one-to-one correspondence between \( \mathbb{C}/L \) and the elliptic curve \( {y}^{2} = 4{x}^{3} - {g}_{2}\left( L\right) x - {g}_{3}\left( L\right) \) in \( {\mathbb{P}}_{\mathbb{C}}^{2} \). | One might be interested in how the inverse map from the elliptic curve to \( \mathbb{C}/L \) can be constructed. This can be done by taking path integrals of \( {dx}/y = {\left( 4{x}^{3} - {g}_{2}x - {g}_{3}\right) }^{-1/2}{dx} \) from a fixed starting point to a variable endpoint. The resulting integral depends on the... | No |
Proposition 12. If \( {P}_{1} + {P}_{2} = {P}_{3} \), then \( - {P}_{3} \) is the third point of intersection of \( l = \overline{{P}_{1}{P}_{2}} \) with the elliptic curve. If \( {P}_{1} = {P}_{2} \), then by \( \overline{{P}_{1}{P}_{2}} \) we mean the tangent line at \( {P}_{1} \) . | Proof. We have already treated the case when \( {P}_{1} \) or \( {P}_{2} \) is the point at infinity 0, and when \( {P}_{2} = - {P}_{1} \) . So suppose that \( l = \overline{{P}_{1}{P}_{2}} \) has the form \( y = {mx} + b \) . Let \( {P}_{1} = {P}_{{z}_{1}},{P}_{2} = {P}_{{z}_{2}} \) . To say that a point \( {P}_{z} = ... | Yes |
Proposition 13. Let \( {K}^{\prime } \) be any field extension of \( K \) (not necessarily algebraic), and let \( \sigma : {K}^{\prime } \rightarrow \sigma {K}^{\prime } \) be any field isomorphism which leaves fixed all elements of \( K \) . Let \( P \in {\mathbb{P}}_{{K}^{\prime }}^{2} \) be a point of exact order \(... | Proof. It follows from the addition formulas that \( \sigma {P}_{1} + \sigma {P}_{2} = \sigma \left( {{P}_{1} + {P}_{2}}\right) \) , and hence \( N\left( {\sigma P}\right) = \sigma \left( {NP}\right) = ▞ = 0 \) (since \( \sigma \left( {0,1,0}\right) = \left( {0,1,0}\right) \) ). Hence \( {\sigma P} \) has order \( N \)... | No |
Proposition 14. In the situation of Proposition 13, with \( K \) a subfield of \( \mathbb{C} \), let \( {K}_{N} \subset \mathbb{C} \) denote the field obtained by adjoining to \( K \) the \( x \) - and \( y \) -coordinates of all points of order \( N \) . Let \( {K}_{N}^{ + } \) denote the field obtained by adjoining j... | Proof. In each case \( {K}_{N} \) and \( {K}_{N}^{ + } \), we are adjoining a finite set of complex numbers which are permuted by any automorphism of \( \mathbb{C} \) which fixes \( K \) . This immediately implies the proposition. | No |
Proposition 16. Let \( q = {p}^{f}, p/{2n} \). Suppose that \( q \equiv 3\left( {\;\operatorname{mod}\;4}\right) \). Then there are \( q + 1{\mathbb{F}}_{q} \) -points on the elliptic curve \( {y}^{2} = {x}^{3} - {n}^{2}x \). | Proof. First, there are four points of order 2: the point at infinity, \( \left( {0,0}\right) \), and \( \left( {\pm n,0}\right) \). We now count all pairs \( \left( {x, y}\right) \) where \( x \neq 0, n, - n \). We arrange these \( q - {3x} \) ’s in pairs \( \{ x, - x\} \). Since \( f\left( x\right) = {x}^{3} - {n}^{2... | Yes |
Proposition 18. \( n \) is a congruent number if and only if \( {E}_{n}\left( \mathbb{Q}\right) \) has nonzero rank \( r \) . | Proof. First suppose that \( n \) is a congruent number. At the beginning of \( §2 \) , we saw that the existence of a right triangle with rational sides and area \( n \) leads to a rational point on \( {E}_{n} \) whose \( x \) -coordinate lies in \( {\left( {\mathbb{Q}}^{ + }\right) }^{2} \) . Since the \( x \) -coord... | Yes |
Proposition 19. There is a one-to-one correspondence between right triangles with rational sides \( X < Y < Z \) and area \( n \), and pairs of points \( \left( {x, \pm y}\right) \in \) \( 2{E}_{n}\left( \mathbb{Q}\right) - 0 \) . The correspondence is: | \[ \left( {x, \pm y}\right) \mapsto \sqrt{x + n} - \sqrt{x - n},\sqrt{x + n} + \sqrt{x - n},2\sqrt{x}; \] \[ X, Y, Z \mapsto \left( {{Z}^{2}/4, \pm \left( {{Y}^{2} - {X}^{2}}\right) Z/8}\right) . \] | Yes |
Proposition 20. Let \( E \) be the elliptic curve \( {y}^{2} = \left( {x - {e}_{1}}\right) \left( {x - {e}_{2}}\right) \left( {x - {e}_{3}}\right) \) with \( {e}_{1},{e}_{2},{e}_{3} \in \mathbb{Q} \) . Let \( P = \left( {{x}_{0},{y}_{0}}\right) \in E\left( \mathbb{Q}\right) - 0 \) . Then \( P \in {2E}\left( \mathbb{Q}\... | Proof. We first note that, without loss of generality, we may assume that \( {x}_{0} = 0 \) . To see this, make the change of variables \( {x}^{\prime } = x - {x}_{0} \) . By simply translating the geometrical picture for adding points, we see that the point \( {P}^{\prime } = \left( {0,{y}_{0}}\right) \) on the curve ... | Yes |
Lemma 1. Let \( q \equiv 1\\left( {\\;\\operatorname{mod}\\,4}\\right) \), and let \( {\\chi }_{2} \) and \( {\\chi }_{4} \) be characters of \( {\\mathbb{F}}_{q}^{ * } \) of exact order 2 and 4, respectively. Then \( 1 + J\\left( {{\\chi }_{2},{\\chi }_{4}}\\right) \) is divisible by \( 2 + {2i} \) in the ring \( \\ma... | Proof. We first relate \( J\\left( {{\\chi }_{2},{\\chi }_{4}}\\right) \) to \( J\\left( {{\\chi }_{4},{\\chi }_{4}}\\right) \) by expressing both in terms of Gauss sums. By property (3), we have: \( J\\left( {{\\chi }_{2},{\\chi }_{4}}\\right) = J\\left( {{\\chi }_{4},{\\chi }_{4}}\\right) g{\\left( {\\chi }_{2}\\righ... | Yes |
Lemma 1. | \[ g\left( {\chi }_{n}^{\prime }\right) = \left\{ \begin{array}{ll} \left( \frac{-2}{n}\right) g\left( {\chi }_{1}^{\prime }\right) g\left( \left( \frac{}{n}\right) \right) , & n\text{ odd }; \\ \left( \frac{-1}{{n}_{0}}\right) g\left( {\chi }_{2}^{\prime }\right) g\left( \left( \frac{}{{n}_{0}}\right) \right) , & n = ... | No |
Proposition 5. If \( f\left( x\right) = {e}^{-\pi {x}^{2}} \), then \( \widehat{f} = f \) . | Proof. Differentiating under the integral sign, we have\n\n\[ \n{\widehat{f}}^{\prime }\left( y\right) = \frac{d}{dy}{\int }_{-\infty }^{\infty }{e}^{-{2\pi ixy}}f\left( x\right) {dx} = - {2\pi i}{\int }_{-\infty }^{\infty }{e}^{-{2\pi ixy}}x{e}^{-\pi {x}^{2}}{dx}.\n\]\n\nIntegrating by parts gives\n\n\[ \n{\widehat{f}... | Yes |
Proposition 6 (Poisson Summation Formula). If \( g \in \mathcal{S} \), then\n\n\[ \mathop{\sum }\limits_{{m = - \infty }}^{\infty }g\left( m\right) = \mathop{\sum }\limits_{{m = - \infty }}^{\infty }\widehat{g}\left( m\right) \] | Proof. Define \( h\left( x\right) = \mathop{\sum }\limits_{{k = - \infty }}^{\infty }g\left( {x + k}\right) \) . The function \( h\left( x\right) \) is periodic with period 1, and has Fourier series \( h\left( x\right) = \mathop{\sum }\limits_{{m = - \infty }}^{\infty }{c}_{m}{e}^{2\pi imx} \), where\n\n\[ {c}_{m} = {\... | Yes |
Proposition 7. The theta-function satisfies the functional equation\n\n\\[ \n\\theta \\left( t\\right) = \\frac{1}{\\sqrt{t}}\\theta \\left( {1/t}\\right) \n\\]\n\n(4.10) | Proof. We apply Poisson summation to \\( g\\left( x\\right) = {e}^{-{\\pi t}{x}^{2}} \\) for fixed \\( t > 0 \\) . We write \\( g\\left( x\\right) = f\\left( {\\sqrt{t}x}\\right) \\) with \\( f\\left( x\\right) = {e}^{-\\pi {x}^{2}} \\) . By Proposition 5 and property (3) of the Fourier transform (with \\( b = \\sqrt{t... | Yes |
Proposition 8. As \( t \) approaches zero from above, we have\n\n\[ \left| {\theta \left( t\right) - {t}^{-1/2}}\right| < {e}^{-C/t} \]\n\nfor some positive constant \( C \) . | Proof. By (4.10) and (4.9), the left side is equal to \( 2{t}^{-1/2}\mathop{\sum }\limits_{{n = 1}}^{\infty }{e}^{-\pi {n}^{2}/t} \) . Suppose \( t \) is small enough so that \( \sqrt{t} > 4{e}^{-1/t} \) and also \( {e}^{-{3\pi }/t} < \frac{1}{2} \) . Then\n\n\[ \left| {\theta \left( t\right) - {t}^{-1/2}}\right| < \fr... | Yes |
Proposition 10 (Poisson Summation Formula). If \( g \in \mathcal{S} \), then\n\n\[ \mathop{\sum }\limits_{{m \in {\mathbb{Z}}^{n}}}g\left( m\right) = \mathop{\sum }\limits_{{m \in {\mathbb{Z}}^{n}}}\widehat{g}\left( m\right) \] | The proofs of Propositions 9 and 10 are completely similar to those of properties (1)-(3) of the Fourier transform in one variable and Propositions 5 and 6 of the last section. One simply has to proceed one variable at a time. | No |
Proposition 11. If \( f \in \mathcal{S} \) and \( g = w \cdot \frac{\partial }{\partial x}f \), then \( \widehat{g}\left( y\right) = {2\pi iw} \cdot {yf}\left( y\right) \) . | Proof. Since both sides of the equality are linear in \( w \), it suffices to prove the proposition when \( w \) is the \( j \) -th standard basis vector, i.e., to prove that the Fourier transform of \( \frac{\partial }{\partial {x}_{j}}f\left( x\right) \) is \( {2\pi i}{y}_{j}f\left( y\right) \) . This is easily done ... | No |
Proposition 12. If \( n \equiv 5,6 \) or \( 7\left( {\;\operatorname{mod}\;8}\right) \), and if the weak Birch-Swinnerton-Dyer conjecture holds for \( {E}_{n} \), then \( n \) is a congruent number. | Proof. According to the theorem in \( §5 \), if \( n \equiv 5,6,7\left( {\;\operatorname{mod}\;8}\right) \), then \( \Lambda \left( s\right) = \) \( - \Lambda \left( {2 - s}\right) \), where \( \Lambda \left( s\right) \) is given by (5.11). Substituting \( s = 1 \), we conclude that \( \Lambda \left( 1\right) = - \Lamb... | Yes |
The critical value of the Hasse-Weil L-function of the elliptic curve \( {E}_{n} : {y}^{2} = {x}^{3} - {n}^{2}x \) for squarefree \( n \equiv 1,2,3\left( {\;\operatorname{mod}\;8}\right) \) is given by:\n\n\[ L\left( {{E}_{n},1}\right) = 2\mathop{\sum }\limits_{{m = 1}}^{\infty }\frac{{b}_{m, n}}{m}{e}^{-{\pi m}/\sqrt{... | Proof. We have already proved all except for the bound on \( {b}_{m, n} \) . If we write the Euler factor in the form \( {\left( 1 - {\alpha }_{p}{p}^{-s}\right) }^{-1}{\left( 1 - {\bar{\alpha }}_{p}{p}^{-s}\right) }^{-1} \), expand each factor in a geometric series, and collect coefficients of \( {p}^{-{es}} \) for ea... | Yes |
Proposition 3. If \( z \in F \), then \( {\Gamma }_{z} = \pm I \) except in the following three cases:\n\n(i) \( \Gamma = \pm \{ I, S\} \) if \( z = i \) ;\n\n(ii) \( \Gamma = \pm \left\{ {I,{ST},{\left( ST\right) }^{2}}\right\} \) if \( z = \omega = - \frac{1}{2} + \frac{\sqrt{-3}}{2} \) ;\n\n(iii) \( \Gamma = \pm \le... | Both Propositions 2 and 3 follow from the second part of the proof of Proposition 1. | No |
Proposition 5.\n\n\[ \n{G}_{k} \in {M}_{k}\left( \Gamma \right) \n\] | We now compute the \( q \) -expansion coefficients for \( {G}_{k} \) . We shall find that these coefficients are essentially the arithmetic functions of \( n \)\n\n\[ \n{\sigma }_{k - 1}\left( n\right) \underset{\text{ def }}{ = }\mathop{\sum }\limits_{{d \mid n}}{d}^{k - 1} \n\]\n\n(2.6) | No |
\[ {z}^{-2}{E}_{2}\left( {-1/z}\right) = {E}_{2}\left( z\right) + \frac{12}{2\pi iz} \] | Proof. The proposition says that \( {12}/{2\pi iz} \) is the difference between the double sum (2.18) and the double sum (2.16). Suppose we introduce a \ | No |
Proposition 9. Let \( k \) be an even integer, \( \Gamma = S{L}_{2}\left( \mathbb{Z}\right) \). (a) The only modular forms of weight 0 for \( \Gamma \) are constants, i.e., \( {M}_{0}\left( \Gamma \right) = \mathbb{C} \) . | Proof. Note that for a modular form all terms on the left in (2.21) are nonnegative. (a) Let \( f \in {M}_{0}\left( \Gamma \right) \), and let \( c \) be any value taken by \( f\left( z\right) \). Then \( f\left( z\right) - c \in {M}_{0}\left( \Gamma \right) \) has a zero, i.e., one of the terms on the left in (2.21) i... | Yes |
Proposition 10. Any \( f \in {M}_{k}\left( \Gamma \right) \) can be written in the form\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{{4i} + {6j} = k}}{c}_{i, j}{E}_{4}{\left( z\right) }^{i}{E}_{6}{\left( z\right) }^{j}. \]\n\n(2.27) | Proof. We use induction on \( k \) . For \( k = 4,6,8,{10},{14} \) we note that \( {E}_{4},{E}_{6} \) , \( {E}_{4}^{2},{E}_{4}{E}_{6},{E}_{4}^{2}{E}_{6} \), respectively, is an element of \( {M}_{k}\left( \Gamma \right) \), and so, by Proposition 9(c), must span \( {M}_{k}\left( \Gamma \right) \) . Now suppose that \( ... | Yes |
Proposition 11. The function \( j \) gives a bijection from \( \Gamma \smallsetminus \bar{H} \) (the fundamental domain with \( \Gamma \) -equivalent sides identified and the point at infinity included) and the Riemann sphere \( {\mathbb{P}}_{\mathbb{C}}^{1} = \mathbb{C} \cup \{ \infty \} \) . | Proof. In the proof of Proposition 9(d) we saw that \( \Delta \left( z\right) \) has a simple zero at infinity and no other zero. Since \( {g}_{2} \) does not vanish at infinity, this means that \( j\left( z\right) \) has a simple pole at infinity and is holomorphic on \( H \) . For any \( c \in \mathbb{C} \) the modul... | Yes |
Proposition 12. The modular functions of weight zero for \( \Gamma \) are precisely the rational functions of \( j \) . | Proof. A rational function of \( j\left( z\right) \) is a modular function of weight zero (see Remark 3 at the beginning of this section). Conversely, suppose that \( f\left( z\right) \) is a modular function of weight zero for \( \Gamma \) . If \( {z}_{j} \) are the poles of \( f\left( z\right) \) in \( \Gamma \smalls... | Yes |
Proposition 13. For any \( A, B \in \mathbb{C} \) such that \( {A}^{3} \neq {27}{B}^{2} \) there exists \( L = \lambda {L}_{z} \) such that\n\n\[ \n{g}_{2}\left( L\right) = A \n\]\n\n(2.31)\n\n\[ \n{g}_{3}\left( L\right) = B \n\]\n\n(2.32) | Proof. It follows immediately from the definition of \( {g}_{2} \) that \( {g}_{2}\left( {\lambda {L}_{z}}\right) = \) \( {\lambda }^{-4}{g}_{2}\left( {L}_{z}\right) \), and similarly \( {g}_{3}\left( {\lambda {L}_{z}}\right) = {\lambda }^{-6}{g}_{3}\left( {L}_{z}\right) \) .\n\nBy (2.14), we can restate the propositio... | Yes |
Proposition 14. Let \( \sqrt{} \) denote the branch of the square root having nonnegative real part. Then\n\n\[ \eta \left( {-1/z}\right) = \sqrt{z/i}\eta \left( z\right) \] | Proof. The product (2.33) clearly converges to a nonzero value for any \( z \in H \), and defines a holomorphic function on \( H \) . Suppose we show that the logarithmic derivatives of the left and right sides of (2.34) are equal. Then (2.34) must hold up to a multiplicative constant; but substituting \( z = i \) show... | Yes |
\[ {\left( 2\pi \right) }^{-{12}}\Delta \left( z\right) = q\mathop{\prod }\limits_{{n = 1}}^{\infty }{\left( 1 - {q}^{n}\right) }^{24},\;q = {e}^{2\pi iz}. \] | Proof. Let \( f\left( z\right) \) be the product on the right of (2.37). The function \( f \) is holomorphic on \( H \), periodic of period 1, and vanishes at infinity. Moreover, \( f\left( z\right) \) is the 24 -th power of \( \eta \left( z\right) \), by definition; and, raising both sides of (2.34) to the 24-th power... | Yes |
Proposition 16. The condition (3.8) depends only on the \( {\Gamma }^{\prime } \) -equivalence class of \( s = {\gamma }_{0}\infty \) . More precisely, if \( {\gamma }_{1}\infty = {\gamma }^{\prime }{\gamma }_{2}\infty \) for some \( {\gamma }^{\prime } \in {\Gamma }^{\prime } \), then the smallest power of \( {q}_{N} ... | Proof. If \( {\gamma }_{1}\infty = {\gamma }^{\prime }{\gamma }_{2}\infty \), then the element \( {\gamma }_{1}^{-1}{\gamma }^{\prime }{\gamma }_{2} \in \Gamma \) keeps \( \infty \) fixed, in which case it must be of the form \( \pm {T}^{j} \) . (Note that \( \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \... | Yes |
Proposition 17. (a) Let \( {\Gamma }^{\prime } \) be a congruence subgroup of \( \Gamma \), let \( \alpha \in G{L}_{2}^{ + }\left( \mathbb{Q}\right) \) , and set \( {\Gamma }^{\prime \prime } = {\alpha }^{-1}{\Gamma }^{\prime }\alpha \cap \Gamma \) . Then \( {\Gamma }^{\prime \prime } \) is a congruence subgroup of \( ... | Proof. (a) We need two lemmas. | No |
Lemma 1. Let \( \alpha \in G{L}_{2}^{ + }\left( \mathbb{Q}\right) \) have integer entries, and let \( D = \det \alpha \) . If \( {\Gamma }^{\prime } \supset \) \( \Gamma \left( N\right) \), then \( {\alpha }^{-1}{\Gamma }^{\prime }\alpha \supset \Gamma \left( {ND}\right) \) . | Proof of Lemma 1. Suppose \( \gamma \in \Gamma \left( {ND}\right) \), i.e., \( \gamma = 1 + {ND\beta } \) for some \( 2 \times 2 \) -matrix \( \beta \) with integer entries and det \( \gamma = 1 \) . We must show that \( \gamma \in {\alpha }^{-1}{\Gamma }^{\prime }\alpha \), i.e., that \( {\Gamma }^{\prime } \ni {\alph... | Yes |
Lemma 2. Suppose that \( f\left( z\right) \) has the property (3.8) for all \( {\gamma }_{0} \in \Gamma \), i.e., \( f\left( z\right) \mid {\left\lbrack {\gamma }_{0}\right\rbrack }_{k} \) \( = \mathop{\sum }\limits_{{n = {n}_{0}}}^{\infty }{a}_{n}{q}_{N}^{n} \) (where \( {n}_{0} = 0 \) if \( f\left( z\right) \) is hol... | Proof of Lemma 2. Since \( \alpha \) can be multiplied by a positive scalar without affecting \( {\left\lbrack \alpha \right\rbrack }_{k} \), without loss of generality we may suppose that \( \alpha \) has integer entries. It is an easy exercise in linear algebra to show that there exists \( {\gamma }_{0} \in \Gamma = ... | No |
Proposition 18. \( {M}_{0}\left( {\Gamma }^{\prime }\right) = \mathbb{C} \) for any congruence subgroup \( {\Gamma }^{\prime } \subset \Gamma \) . That is, there are no non-constant modular forms of weight zero. | Proof. Let \( f \in {M}_{0}\left( {\Gamma }^{\prime }\right) \), and let \( a = f\left( {z}_{0}\right) \) for some fixed \( {z}_{0} \in H \) . Let \( \Gamma = \bigcup {\alpha }_{j}{\Gamma }^{\prime } \) be a disjoint union of cosets, and consider \( {g}_{\overline{\operatorname{def}}}\Pi \left( {f \mid {\left\lbrack {\... | Yes |
Proposition 19. Let \( f\left( z\right) \) be a nonzero element of \( {S}_{k}\left( {{\Gamma }_{0}\left( N\right) }\right) \), where \( N = 2 \) , 3,5, or 11 and \( k = 8,6,4 \), or 2, respectively, so that \( k\left( {N + 1}\right) = {24} \) . Then \( f\left( z\right) \) is a constant multiple of \( g\left( z\right) \... | Proof. By Proposition 17(a), \( g{\left( z\right) }^{N + 1} = \Delta \left( z\right) \Delta \left( {Nz}\right) \) is an element of \( {S}_{24}\left( {{\Gamma }_{0}\left( N\right) }\right) \) . In addition, \( g{\left( z\right) }^{N + 1} \) is nonzero on \( H \), since \( \Delta \left( z\right) \neq 0 \) on \( H \) . At... | Yes |
Proposition 20. \( {\left( \eta \left( z\right) \eta \left( 2z\right) \right) }^{8} \in {S}_{8}\left( {{\Gamma }_{0}\left( 2\right) }\right) \) . | Proof. Clearly, \( g\left( z\right) = {\left( \eta \left( z\right) \eta \left( 2z\right) \right) }^{8} \) is holomorphic on \( H \) . Its \( q \) -expansion at \( \infty \) is: \( {\left( {e}^{{2\pi iz}/{24} + {2\pi i2z}/{24}}\right) }^{8}\Pi {\left( 1 - {q}^{n}\right) }^{8}{\left( 1 - {q}^{2n}\right) }^{8} = {q\Pi }{\... | Yes |
Proposition 23. If \( 2\underline{a} \equiv \left( {0,0}\right) {\;\operatorname{mod}\;N} \) and \( k \) is odd, then \( {G}_{k}^{a{\;\operatorname{mod}\;N}} = 0 \) . Otherwise, \( {G}_{k}^{\left( 0,{a}_{2}\right) } \) is nonzero at \( \infty \), and \( {G}_{k}^{\left( {a}_{1},0\right) } \) has a zero of order \( \min ... | Proof. The first assertion we already saw as a result of (3.15). We now check that (3.14) is nonzero (unless \( N \mid 2{a}_{2} \) and \( k \) is odd). If \( k \) is even, then we have a sum of positive terms. If \( k \) is odd and we take \( 0 < {a}_{2} < N \), then the sum in (3.14) is equal to\n\n\[ \mathop{\sum }\l... | Yes |
Proposition 26. The function \( f\left( z\right) \) defined in (3.27) is a constant multiple of \( {\left( {\eta }^{p}\left( z\right) /\eta \left( pz\right) \right) }^{3} \) . | Because each of the \( {G}_{3}^{\left( 0,{a}_{2}\right) } \) in (3.27) is in \( {M}_{3}\left( {{\Gamma }_{1}\left( p\right) }\right) \) by Proposition 21, it follows that \( f \in {M}_{3\left( {p - 1}\right) /2}\left( {{\Gamma }_{1}\left( p\right) }\right) \) . However, unlike \( h\left( z\right), f\left( z\right) \) i... | No |
Proposition 27. Let \( f\left( z\right) \) be defined by (3.27), and let \( \gamma = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \in {\Gamma }_{0}\left( p\right) \) . Then \( f \mid {\left\lbrack \gamma \right\rbrack }_{3\left( {p - 1}\right) /2} = \left( \frac{d}{p}\right) f \), where \( \left( \frac{d}... | Proof. Since \( \left( {0,{a}_{2}}\right) \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \equiv \left( {0, d{a}_{2}}\right) {\;\operatorname{mod}\;p} \), it follows by (3.13) that\n\n\[ f \mid {\left\lbrack \gamma \right\rbrack }_{3\left( {p - 1}\right) /2} = \mathop{\prod }\limits_{{{a}_{2} = 1}}^{{\left( ... | Yes |
Proposition 28. \( {M}_{k}\left( {{\Gamma }_{1}\left( N\right) }\right) = \oplus {M}_{k}\left( {N,\chi }\right) \), where the sum is over all Dirichlet characters modulo \( N \) . | Proof. As mentioned before, this proposition is actually a special case of the basic fact from representation theory that any representation of a finite abelian group decomposes into a direct sum of characters. However, we shall give an explicit proof anyway.\n\nFirst, any function \( f \) that satisfies \( f \mid {\le... | Yes |
Proposition 30. \( {\Theta }^{2} \in {M}_{1}\left( {{\Gamma }_{1}\left( 4\right) }\right) = {M}_{1}\left( {4,\chi }\right) \), where \( \chi \left( d\right) = {\left( -1\right) }^{\left( {d - 1}\right) /2} \). | Proof. It suffices to verify the transformation rule for \( - I, T \), and \( S{T}^{4}S = \) \( \left( \begin{array}{rr} - 1 & 0 \\ 4 & - 1 \end{array}\right) \), which generate \( {\Gamma }_{0}\left( 4\right) \) (see Problem 13 of §III.1). This is immediate for \( T \), since \( {\Theta }^{2} \) has period 1 . Next, t... | Yes |
The theorem follows if we prove the following transformation formula for \( \phi \left( z\right) \) :\n\n\[ \phi \left( {\gamma z}\right) = {i}^{\left( {1 - c}\right) /2}\left( \frac{d}{c}\right) \sqrt{-i\left( {{cz} + d}\right) }\phi \left( z\right) \]\n\nfor \( \gamma \in \Gamma \) such that \( \gamma = \left( \begin... | Proof. Suppose that we have the transformation rule for \( \phi \left( z\right) \) . We must show that \( \Theta \) satisfies (4.3) for all \( \gamma = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \in {\Gamma }_{0}\left( 4\right) \) . We first note that if \( c = 0 \), then (4.3) holds trivially, since in... | Yes |
Lemma 2. \[ \phi \left( {pz}\right) /{\phi }^{p}\left( z\right) = \psi \left( z\right) /{\psi }^{2}\left( \frac{z + 1}{2}\right) . \] | Proof. By Problem 11(e) in the preceding section, with \( z \) replaced by \( \frac{z}{2} \) and by \( \frac{pz}{2} \), we find that the left side of (4.6) is equal to \[ \frac{{e}^{-{2\pi i}/{24}}{\eta }^{2}\left( \frac{{pz} + 1}{2}\right) /\eta \left( {pz}\right) }{{\left( {e}^{-{2\pi i}/{24}}{\eta }^{2}\left( \frac{... | Yes |
Lemma 3. Let \( p \) be an odd prime, let \( \gamma = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \in \mathfrak{G}\left( 2\right) \cap {\Gamma }_{0}\left( p\right) \), and let \( f\left( z\right) = \) \( {\psi }^{3}\left( \frac{z + 1}{2}\right) \) . Then \( f \mid {\left\lbrack \gamma \right\rbrack }_{3\... | Proof. Let \( \alpha = \left( \begin{array}{ll} 1 & 1 \\ 0 & 2 \end{array}\right) \), so that \( \psi \left( {\left( {z + 1}\right) /2}\right) = \psi \left( {\alpha z}\right) \) . Then \( \psi \left( {\left( {{\gamma z} + 1}\right) /2}\right) = \) \( \psi \left( {\alpha \gamma z}\right) = \psi \left( {\left( {{\alpha \... | Yes |
Lemma 4. Let \( p \) be an odd prime, let \( \gamma = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \in \mathfrak{G}\left( 2\right) \cap {\Gamma }_{0}\left( p\right) \), and let \( g\left( z\right) = \) \( \phi \left( {pz}\right) /{\phi }^{p}\left( z\right) \) . Then \( g \mid {\left\lbrack \gamma \right\r... | Proof. We first claim that \( {g}^{8} \) transforms trivially under \( \gamma \) . Let \( \alpha = \left( \begin{array}{ll} p & 0 \\ 0 & 1 \end{array}\right) \) , and let \( {\gamma }^{\prime } = {\alpha \gamma }{\alpha }^{-1} \) . Then both \( \gamma \) and \( {\gamma }^{\prime } \) are in \( \mathfrak{G}\left( 2\righ... | Yes |
Lemma 5. Let \( n \) be a positive odd integer, let \( \gamma = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \in \mathfrak{G}\left( 2\right) \cap {\Gamma }_{0}\left( n\right) \), and let \( g\left( z\right) = \phi \left( {nz}\right) /{\phi }^{n}\left( z\right) \) . Then \( g \mid {\left\lbrack \gamma \rig... | Proof. We write \( n = {p}_{1}\cdots {p}_{r} \) as a product of primes (not necessarily distinct), and we use induction on the number \( r \) of prime factors. Lemma 4 is the case \( r = 1 \) . Now suppose we know Lemma 5 for \( n \) ; we shall prove the corresponding equality for a product \( {n}^{\prime } = {np} \) o... | Yes |
Proposition 32. (a) If g.c.d. \( \left( {m, n}\right) = 1 \), then \( {T}_{mn} = {T}_{m}{T}_{n} \) ; in particular, \( {T}_{m} \) and \( {T}_{n} \) commute. | Proof. (a) In the sum (5.2) for \( {T}_{mn} \), the \( {L}^{\prime } \) correspond to certain subgroups \( {S}^{\prime } \) of order \( {mn} \) in \( \frac{1}{mn}L/L \), namely, those which have trivial intersection with the subgroup \( \mathbb{Z}t \subset \mathbb{C}/L \) . Since g.c.d. \( \left( {m, n}\right) = 1 \), ... | Yes |
Proposition 33. Suppose that \( F\left( {L, t}\right) \) corresponds to a function \( f\left( z\right) \) on \( H \) which is in \( {M}_{k}\left( {{\Gamma }_{1}\left( N\right) }\right) \) . Then \( \left\lbrack d\right\rbrack F,{T}_{n, n}F \), and \( {T}_{n}F \) also correspond to functions (denoted \( \left\lbrack d\r... | Proof. To show that \( \left\lbrack d\right\rbrack f,{T}_{n, n}f \) and \( {T}_{n}f \) are invariant under \( {\left\lbrack \gamma \right\rbrack }_{k} \) for \( \gamma \in {\Gamma }_{1}\left( N\right) \) , by Proposition 31 it suffices to show that \( \left\lbrack d\right\rbrack F\left( {L, t}\right) ,{T}_{n, n}F\left(... | Yes |
Proposition 34. The operators \( {T}_{n} \) and \( {T}_{n, n} \) commute with \( \left\lbrack d\right\rbrack \), and preserve the space of \( F\left( {L, t}\right) \) of weight \( k \) which satisfy (5.10). If \( F\left( {L, t}\right) \) has weight \( k \) and satisfies (5.10), then \( {T}_{n, n}F = {n}^{k - 2}\chi \le... | Proof. That the operators commute follows directly from the definitions. Next, if \( \left\lbrack d\right\rbrack F = \chi \left( d\right) F \), it follows that \( \left\lbrack d\right\rbrack {T}_{n}F = {T}_{n}\left\lbrack d\right\rbrack F = \chi \left( d\right) {T}_{n}F \) and \( \left\lbrack d\right\rbrack {T}_{n, n}F... | Yes |
Proposition 36. The operators \( {T}_{n} \) on \( {M}_{k}\left( {N,\chi }\right) \) satisfy the formal power series identity\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{T}_{n}{n}^{-s} = \mathop{\prod }\limits_{{\text{all }p}}{\left( 1 - {T}_{p}{p}^{-s} + \chi \left( p\right) {p}^{k - 1 - {2s}}\right) }^{-1}. \] | Proof. We simply use (5.8) and observe that for \( p/N \) we have \( {T}_{p, p}f = \) \( {p}^{k - 2}\chi \left( p\right) f \), while for \( p \mid N \) the term on the right in (5.11) becomes (1- \( {\left. {T}_{p}{p}^{-s}\right) }^{-1} \) because \( \chi \left( p\right) = 0 \) . | No |
Proposition 37. Let \( f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{q}^{n}, q = {e}^{2\pi iz}, f \in {M}_{k}\left( {N,\chi }\right) \), and let \( {T}_{p}f\left( z\right) = \) \( {\sum }_{n = 0}^{\infty }{b}_{n}{q}^{n} \) . Then\n\n\[ \n{b}_{n} = {a}_{pn} + \chi \left( p\right) {p}^{k - 1}{a}_{n/... | Proof. We have \( {T}_{p}f\left( z\right) = \frac{1}{p}{\sum }_{{L}^{\prime }}F\left( {{L}^{\prime },\frac{1}{N}}\right) \), where \( F \) is the function on modular points which corresponds to \( f \) and the sum is over all lattices \( {L}^{\prime } \) containing \( {L}_{z} \) with index \( p \) such that \( \frac{1}... | Yes |
Proposition 38. We have the following formal identity:\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{T}_{n}{n}^{-s} = \left( {\mathop{\sum }\limits_{{n = 1}}^{\infty }\chi \left( n\right) {n}^{k - 1}{V}_{n}{n}^{-s}}\right) \left( {\mathop{\sum }\limits_{{n = 1}}^{\infty }{U}_{n}{n}^{-s}}\right) ,\ ]\n\nor, equivalent... | Proof. By (5.11) and (5.16) we find that the left side of (5.17) is equal to\n\n\[ \mathop{\prod }\limits_{p}{\left( \left( 1 - {U}_{p}{p}^{-s}\right) \left( 1 - \chi \left( p\right) {p}^{k - 1}{V}_{p}{p}^{-s}\right) \right) }^{-1}. \]\n\nSince the \( {U}_{p} \) and \( {V}_{p} \) do not commute, we must be careful abou... | Yes |
Proposition 39. Under the conditions of Proposition 37, if \( {T}_{m}f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n}{q}^{n} \) , then\n\n\[ {b}_{n} = \mathop{\sum }\limits_{{d \mid \text{ g. c. d. }\left( {m, n}\right) }}\chi \left( d\right) {d}^{k - 1}{a}_{{mn}/{d}^{2}}. \] | Proof. According to (5.18), we have\n\n\[ {T}_{m}\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{q}^{n} = \mathop{\sum }\limits_{{d \mid m}}\chi \left( d\right) {d}^{k - 1}{V}_{d} \circ {U}_{m/d}\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{q}^{n} \]\n\n\[ = \mathop{\sum }\limits_{{d|m}}\chi \left( d\right) {d}^{k ... | Yes |
Proposition 40. Suppose that \( f\left( z\right) \in {M}_{k}\left( {N,\chi }\right) \) is an eigenform for all of the operators \( {T}_{m} \) with eigenvalues \( {\lambda }_{m}, m = 1,2,\ldots : {T}_{m}f = {\lambda }_{m}f \) . Let \( {a}_{m} \) be the \( q \) -expansion coefficients: \( f\left( z\right) = \mathop{\sum ... | Proof. Using (5.19) with \( n = 1 \), we find that the coefficient of the first power of \( q \) in \( {T}_{m}f \) is \( {a}_{m} \) . If \( {T}_{m}f = {\lambda }_{m}f \), then this coefficient is also equal to \( {\lambda }_{m}{a}_{1} \) . This proves the first assertion. If we had \( {a}_{1} = 0 \), then it would foll... | Yes |
Proposition 41. Let \( {\Gamma }^{\prime } \subset G \) be any subgroup of a group, and let \( \alpha \in G \) be any element such that \( {\Gamma }^{\prime } \) and \( {\alpha }^{-1}{\Gamma }^{\prime }\alpha \) are commensurable. Let \( {\Gamma }^{\prime \prime } = {\Gamma }^{\prime } \cap {\alpha }^{-1}{\Gamma }^{\pr... | Proof. Given an element \( {\gamma }_{1}\alpha {\gamma }_{2} \) with \( {\gamma }_{1},{\gamma }_{2} \in {\Gamma }^{\prime } \), we can write \( {\gamma }_{2} = {\gamma }^{\prime \prime }{\gamma }_{j}^{\prime } \) with \( {\gamma }^{\prime \prime } \in {\Gamma }^{\prime \prime } \) for some \( j \) . Since \( {\gamma }^... | No |
Proposition 42. \( f\left( z\right) \mid {\left\lbrack {\Gamma }^{\prime }\alpha {\Gamma }^{\prime }\right\rbrack }_{k} \) does not change if \( \alpha \) is replaced by any other representative \( {\alpha }^{\prime } \) of the same double coset: \( {\Gamma }^{\prime }{\alpha }^{\prime }{\Gamma }^{\prime } = {\Gamma }^... | Proof. We first prove the second assertion, that is, that (5.25) is unchanged if \( {\gamma }_{i}^{\prime } \) is replaced by \( {\gamma }_{i}^{\prime \prime }{\gamma }_{i}^{\prime } \), where \( {\gamma }_{j}^{\prime \prime } \in {\Gamma }^{\prime \prime } \) . Since \( {\Gamma }^{\prime \prime } \subset {\alpha }^{-1... | Yes |
Proposition 43. In the case \( {\Gamma }^{\prime } = {\Gamma }_{1}\left( N\right) \), the definition (5.26) agrees with our earlier definition of the Hecke operators \( {T}_{n} \) . | Proof. Let \( {\Delta }^{n} = {\Delta }^{n}\left( {N,\{ 1\} ,\mathbb{Z}}\right) \) . For each \( a \in {\left( \mathbb{Z}/N\mathbb{Z}\right) }^{ * } \) we fix \( {\sigma }_{a} \in \Gamma \) such that \( {\sigma }_{a} \equiv \left( \begin{matrix} 1/a & 0 \\ 0 & a \end{matrix}\right) {\;\operatorname{mod}\;N}. \) | No |
Proposition 44. Let \( {\Gamma }^{\prime } \subset \Gamma \) be a congruence subgroup, let \( {F}^{\prime } \subset H \) be any fundamental domain for \( {\Gamma }^{\prime } \), and define\n\n\n\nFigure III. 5\n\n\[ ... | Proof. First let \( \left\lbrack {\bar{\Gamma } : {\bar{\Gamma }}^{\prime }}\right\rbrack = d \), let \( \bar{\Gamma } = \mathop{\bigcup }\limits_{{j = 1}}^{d}{\alpha }_{j}{\bar{\Gamma }}^{\prime } \), and take \( {F}^{\prime } = \bigcup {\alpha }_{j}^{-1}F \) . Notice that the integral for \( \mu \left( \Gamma \right)... | Yes |
Proposition 46. Let \( f, g \in {M}_{k}\left( {\Gamma }^{\prime }\right) \) with for \( g \) a cusp form. Let \( \alpha \in G{L}_{2}^{ + }\left( \mathbb{Q}\right) \) . Then\n\n\[ \langle f, g\rangle = \left\langle {f\left| {{\left\lbrack \alpha \right\rbrack }_{k}, g}\right| {\left\lbrack \alpha \right\rbrack }_{k}}\ri... | Proof. Let \( {\Gamma }^{\prime \prime } = {\Gamma }^{\prime } \cap \alpha {\Gamma }^{\prime }{\alpha }^{-1} \) . Then \( f, g \in {M}_{k}\left( {\Gamma }^{\prime \prime }\right) \), and \( f\left| {{\left\lbrack \alpha \right\rbrack }_{k}, g}\right| {\left\lbrack \alpha \right\rbrack }_{k} \in \) \( {M}_{k}\left( {{\a... | Yes |
Proposition 47. With \( f, g,\alpha \) as in Proposition 46,\n\n\[ \left\langle {f \mid {\left\lbrack \alpha \right\rbrack }_{k}, g}\right\rangle = \left\langle {f, g \mid {\left\lbrack {\alpha }^{\prime }\right\rbrack }_{k}}\right\rangle . \]\n\n(5.33)\n\nIn addition, \( \left\langle {f \mid {\left\lbrack \alpha \righ... | Proof. If in (5.32) we replace \( g \) by \( g \mid {\left\lbrack {\alpha }^{-1}\right\rbrack }_{k} \) and replace \( {\Gamma }^{\prime } \) by \( {\Gamma }^{\prime } \cap \alpha {\Gamma }^{\prime }{\alpha }^{-1} \) , then Proposition 46 gives (5.33). Now suppose we replace \( \alpha \) by \( {\gamma }_{1}\alpha {\gamm... | Yes |
Proposition 50. Let \( n \) be a positive integer prime to \( N \), and let \( \chi \) be a Dirichlet character modulo \( N \) . Let \( {c}_{n} \) be either square root of \( \bar{\chi }\left( n\right) \) . Then the operator \( {c}_{n}{T}_{n} \) on \( {S}_{k}\left( {N,\chi }\right) \) is hermitian, i.e., \( \left\langl... | Proof. \( \left\langle {{c}_{n}{T}_{n}f, g}\right\rangle = {c}_{n}\left\langle {{T}_{n}f, g}\right\rangle = {c}_{n}\chi \left( n\right) \langle f,{T}_{n}g\rangle = {c}_{n}{\bar{c}}_{n}^{2}\langle f,{T}_{n}g\rangle = \n\n\( {\bar{c}}_{n}\left\langle {f,{T}_{n}g}\right\rangle = \left\langle {f,{c}_{n}{T}_{n}g}\right\rang... | Yes |
Proposition 51. There exists a basis of the \( \mathbb{C} \) -vector space \( {S}_{k}\left( {N,\chi }\right) \) whose elements are eigenforms for all of the \( {T}_{n} \) for which g.c.d. \( \left( {n, N}\right) = 1 \) . | Proof. For any fixed \( {T}_{n} \) with g.c.d. \( \left( {n, N}\right) = 1 \) and any subspace \( S \subset {S}_{k}\left( {N,\chi }\right) \) which is preserved by \( {T}_{n} \), there exists a basis of \( S \) consisting of eigenforms of \( {T}_{n} \) . To see this, we apply the following basic fact from linear algebr... | Yes |
Proposition 1. \( G \) is a group under the operation (1.4). | Proof. We first check closure, i.e., that the right side of (1.4) belongs to \( G \) . If \( \alpha = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) ,\beta = \left( \begin{array}{ll} e & f \\ g & h \end{array}\right) \), we have\n\n\[ \n{\left( \phi \left( \beta z\right) \psi \left( z\right) \right) }^{2} =... | Yes |
Proposition 2. The element \( \left( {\left( \begin{array}{ll} 1 & h \\ 0 & 1 \end{array}\right), t}\right) \in {G}_{\infty }^{1} \) depends only on the \( {\Gamma }^{\prime } \) -equivalence class of the cusp s. | Proof. First suppose that \( \xi \) is replaced by another element \( {\xi }_{1} = \left( {{\alpha }_{1},{\phi }_{1}\left( z\right) }\right) \in \) \( {G}^{1} \) such that \( s = {\alpha }_{1}\infty \) . Then \( {\xi }^{-1}{\xi }_{1} \in {G}^{1} \) fixes \( \infty \), and so it is of the form \( \left( {\pm \left( \beg... | Yes |
Proposition 4. Let \( \Theta \left( z\right) = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{q}^{{n}^{2}}, F\left( z\right) = \mathop{\sum }\limits_{{n > 0\text{ odd }}}{\sigma }_{1}\left( n\right) {q}^{n}, q = {e}^{2\pi iz} \) . Assign weight \( 1/2 \) to \( \Theta \) and weight 2 to \( F \) . Then \( {M}_{k/2}\le... | Proof. In Chapter III we found that for \( k/2 \in \mathbb{Z} \), the space \( {M}_{k/2}\left( {N,{\chi }_{-1}^{k/2}}\right) \) consists of polynomials in \( \Theta \) and \( F \) having pure weight \( k/2 \) (see Problems 17–18 of §III.3). This gives Proposition 4 when \( k/2 \in \mathbb{Z} \) . Next, by the definitio... | Yes |
Proposition 6. Let \( \lambda = \left( {k - 1}\right) /2 \geq 2 \), and let \( {E}_{k/2},{F}_{k/2} \in {M}_{k/2}\left( {{\widetilde{\Gamma }}_{0}\left( 4\right) }\right) \) be defined by (2.4)–(2.5). Then \[ {H}_{k/2}\underset{\text{ def }}{ = }\zeta \left( {1 - {2\lambda }}\right) \left( {{E}_{k/2} + \left( {1 + {i}^{... | This proposition is due to H. Cohen [1975]. (His notation is slightly different from ours.) It can be viewed as a prototype for the theorem of Waldspurger-Tunnell which we shall discuss later. | No |
Let \( {s}_{r}\left( n\right) \) denote the number of ways \( n \) can be written as a sum of \( r \) squares; thus \( {\Theta }^{5} = \sum {s}_{5}\left( n\right) {q}^{n} \) . Let \( D \) be the discriminant of a real quadratic field, and let \( {\chi }_{D} \) be the corresponding character. Then | \[ L\left( {{\chi }_{D}, - 1}\right) = \frac{1}{120}{s}_{5}\left( D\right) - \frac{1}{6}\mathop{\sum }\limits_{\substack{{\left| j\right| < \sqrt{D}} \\ {D - {j}^{2}\text{ odd }} }}{\sigma }_{1}\left( {D - {j}^{2}}\right) . \] | Yes |
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