Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Theorem 1.1. Let \( {f}^{ - } \) be the function such that \( {f}^{ - }\left( x\right) = f\left( {-x}\right) \). Then \( {T}^{2}f = q{f}^{ - } \), that is\n\n\[ \n{T}^{2}f\left( z\right) = {qf}\left( {-z}\right) \n\] | Proof. We have\n\n\[ \n{T}^{2}f\left( z\right) = \mathop{\sum }\limits_{y}\mathop{\sum }\limits_{x}f\left( x\right) \lambda \left( {-{yx}}\right) \lambda \left( {-{zy}}\right) \n\]\n\n\[ \n= \mathop{\sum }\limits_{x}f\left( {x - z}\right) \mathop{\sum }\limits_{y}\lambda \left( {-{yx}}\right) \n\]\n\nIf \( x \neq 0 \) ... | Yes |
Theorem 1.2. For functions \( f, g \) on \( F \) we have\n\n\[ T\left( {f * g}\right) = \left( {Tf}\right) \left( {Tg}\right) \]\n\n\[ T\left( {fg}\right) = \frac{1}{q}{Tf} * {Tg} \] | Proof. For the first formula we have\n\n\[ T\left( {f * g}\right) \left( z\right) = \mathop{\sum }\limits_{y}\left( {f * g}\right) \left( y\right) \lambda \left( {-{zy}}\right) = \mathop{\sum }\limits_{y}\mathop{\sum }\limits_{x}f\left( x\right) g\left( {y - x}\right) \lambda \left( {-{zy}}\right) .\n\nWe change the or... | Yes |
Theorem 1.3. Assume that \( \chi \) has order \( m \) .\n\n(i) \( S{\left( \chi \right) }^{m} \) lies in \( \mathbf{Q}\left( {\mu }_{m}\right) \) .\n\n(ii) Let \( b \) be an integer prime to \( m \), and let \( {\sigma }_{b} = {\sigma }_{b,1} \) . Then \( S{\left( \chi \right) }^{b - {\sigma }_{b}} \) lies in \( \mathb... | Proof. In each case we operate on the given expression by an automorphism \( {\sigma }_{1, v} \) with an integer \( v \) prime to \( {pm} \) . Using GS 5, it is then obvious that the given expression is fixed under such an automorphism, and hence lies in \( \mathbf{Q}\left( {\mu }_{m}\right) \) . | No |
Theorem 2.2. We have the factorization\n\n\[ S\left( {\omega }^{-k}\right) \sim {\mathfrak{P}}^{\left( {p - 1}\right) \theta \left( {k,\mathfrak{p}}\right) } \sim {\mathfrak{p}}^{\theta \left( {k,\mathfrak{p}}\right) }.\] | Proof. We have\n\n\[ \operatorname{ord}{\sigma }_{c}^{-1}{PS}\left( {\omega }^{-k}\right) = {\operatorname{ord}}_{\mathfrak{P}}{\sigma }_{c}S\left( {\omega }^{-k}\right) \]\n\n\[ = {\operatorname{ord}}_{\mathfrak{P}}S\left( {\omega }^{-{kc}}\right) \]\n\n\[ = s\left( {kc}\right) \]\n\nby Theorem 2.1. On the other hand,... | No |
For any integer \( k \) we have\n\n\[ s\left( k\right) = \left( {p - 1}\right) \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\left\langle \frac{k{p}^{i}}{q - 1}\right\rangle . \] | Proof. We may assume that \( 1 \leq k < q - 1 \) since both sides are \( \left( {q - 1}\right) \) - periodic in \( k \), and the relation is obvious for \( k = 0 \) . Since \( {p}^{n} \equiv 1\left( {{\;\operatorname{mod}\;q} - 1}\right) \) we find:\n\n\[ k = {k}_{0} + {k}_{1}p + \cdots + {k}_{n - 1}{p}^{n - 1} \]\n\n\... | Yes |
Lemma 2. We have \( {I\theta } = {R\theta } \cap R \) . | Proof. Note that \( m \in I \) because\n\n\[ m = - \left( {{\sigma }_{1 + m} - \left( {1 + m}\right) }\right) . \]\n\nSuppose that an element of \( {R\theta } \) lies in \( R \), that is\n\n\[ \sum z\left( b\right) {\sigma }_{b}\theta \in R \]\n\nwith \( z\left( b\right) \in \mathbf{Z} \) . Then\n\n\[ \sum z\left( b\ri... | Yes |
Theorem 2.4. The Stickelberger ideal annihilates the ideal class group of \( \\mathbf{Q}\\left( {\\mu }_{m}\\right) \) . | Proof. Let\n\n\[ \n\\alpha = \\mathop{\\sum }\\limits_{r}z\\left( r\\right) {\\theta }_{r}\\left( m\\right) \\in R \n\] \n\nbe an element of the Stickelberger ideal, with \( z\\left( r\\right) \\in \\mathbf{Z} \), and the sum taken with only a finite number of coefficients \( \\neq 0 \) . Then\n\n\[ \n\\mathop{\\sum }\... | Yes |
Lemma 1. (i) If \( \chi = \omega \) is the Teichmuller character, then \( {I}_{\chi } = \left( p\right) \). (ii) If \( \chi \) is non-trivial and not equal to the Teichmuller character, then \( {I}_{\chi } = \left( 1\right) \). | Proof. For (i), we can take an integer \( b \) of the form \[ b = \zeta + {pu} \] where \( u \) is a \( p \) -adic unit, and \( \zeta = \omega \left( b\right) \) is a \( \left( {p - 1}\right) \) th root of unity. This makes (i) clear, and (ii) is obvious, from the definitions. | No |
Corollary 1. Assume that \( m = p \) is prime \( \geq 3 \) . If \( \chi \) is not equal to the Teichmuller character and is non-trivial, then\n\n\[ \text{ord}{B}_{1,\bar{\chi }}{I}_{\chi } = \text{ord}{B}_{1,\bar{\chi }}\text{.} \] | Proof. Immediate from the lemma and the theorem. | No |
Corollary 2. If \( \chi \) is equal to the Teichmuller character then \( {B}_{1,\bar{\chi }}{I}_{\chi } = \left( 1\right) \) , and \( {\mathcal{C}}^{\left( p\right) }\left( \chi \right) = 0 \) . | Proof. Mod \( {\mathbf{Z}}_{p} \), we have the congruence\n\n\[ \n{B}_{1,{\omega }^{-1}} = \frac{1}{p}\mathop{\sum }\limits_{{c = 1}}^{{p - 1}}{c\omega }{\left( c\right) }^{-1} \equiv \frac{1}{p}\mathop{\sum }\limits_{{c = 1}}^{{p - 1}}1 \equiv \frac{p - 1}{p}\;\left( {\;\operatorname{mod}\;{\mathbf{Z}}_{p}}\right) .\n... | No |
Corollary 3 (Herbrand’s theorem). Assume again that \( m = p \) . Let \( \chi = {\omega }^{1 - k} \) , with \( 2 \leq k \leq p - 2 \) . If \( {\mathcal{C}}^{\left( p\right) }\left( \chi \right) \neq 0 \), then \( p \mid {B}_{k} \), where \( {B}_{k} \) is the \( k \) th Bernoulli number. | Proof. In the next chapter Theorem 2.5, we shall prove the congruence\n\n\[ \frac{1}{n}{B}_{n,{\omega }^{k - n}} \equiv \frac{1}{k}{B}_{k}\left( {\;\operatorname{mod}\;p}\right) \]\n\nfor \( k \) in the given range, and any positive integer \( n \) . By Corollary 1, we know that \( {B}_{1,\bar{\chi }} \) annihilates \(... | No |
Theorem 4.1. The algebraic number \( w\left( {a,\alpha }\right) \) is a root of unity. | Proof. As (α) ranges over all principal fractional ideals, the numbers \( w\left( {a,\alpha }\right) \) form a group. It will therefore suffice to prove that these numbers have absolute value 1 , for then their conjugates also have absolute value 1 , and these numbers form a finite group. In case \( a \) is special the... | Yes |
Theorem 4.2. If \( \alpha \) is an algebraic integer in \( \mathbf{Q}\left( {\mu }_{m}\right) \), and \( \alpha \equiv 1\left( {\;\operatorname{mod}\;{m}^{2}}\right) \)\nthen for all a we have \( w\left( {a,\alpha }\right) = 1 \), that is,\n\n\[ J\left( {a,\left( \alpha \right) }\right) = {\alpha }^{\theta \left\lbrack... | Proof. We fix \( \alpha \) and view \( J, w \) as functions of \( a \), omitting \( \alpha \) from the notation. In the Fourier inversion relation, we know that the Fourier coefficients \( \widehat{J}\left( b\right) \) are integers. But \( \alpha \equiv 1\left( {\;\operatorname{mod}\;{m}^{2}}\right) \) implies that\n\n... | Yes |
Theorem 6.1. Let \( N \) be the number of points of \( V\left( d\right) \) (in affine space) in the field \( F \) . Then\n\n\[ N = {q}^{2} - \left( {q - 1}\right) \sum {\chi }^{a + b}\left( {-1}\right) J\left( {{\chi }^{a},{\chi }^{b}}\right) .\n\]\nThe sum is taken over integers \( a, b \) satisfying \( 0 < a < d \) a... | Proof. We have\n\n\[ N = \mathop{\sum }\limits_{{a, b, c}}\mathop{\sum }\limits_{{L\left( {u, v, w}\right) = 0}}{\chi }^{a}\left( u\right) {\chi }^{b}\left( v\right) {\chi }^{c}\left( w\right) \]\n\nwhere the sum over \( u, v, w \) is taken over triples of elements of \( F \) lying on the line\n\n\[ u + v + w = 0. \]\n... | Yes |
Lemma 1. We have \( {R}^{ - } = 2{\varepsilon }^{ - }R = \left( {1 - {\sigma }_{-1}}\right) R \) and\n\n\[ \left( {{\varepsilon }^{ - }R : {R}^{ - }}\right) = {2}^{M}\text{.} \] | Proof. The inclusion \( \left( {1 - {\sigma }_{-1}}\right) R \subset {R}^{ - } \) is clear. Conversely, let \( P \) be a set of representatives in \( \mathbf{Z}{\left( m\right) }^{ * } \) for \( \mathbf{Z}{\left( m\right) }^{ * }/ \pm 1 \) . Let\n\n\[ \alpha = \sum z\left( c\right) {\sigma }_{c}^{-1} \in {R}^{ - } \]\n... | Yes |
Lemma 2. \[ {R\theta } \cap R = {\left( R{\theta }^{\prime } \cap R\right) }^{ - }. \] | Proof. Let \( T = R{\theta }^{\prime } \cap R \) . Clearly \[ {T}^{ - } \subset {\varepsilon }^{ - }{R\theta } = {R\theta }\text{ and }{T}^{ - } \subset R, \] so the inclusion \( \supset \) is obvious. Conversely, let \( \alpha \in {R\theta } \cap R \) . It will suffice to prove that \( \alpha \in R{\theta }^{\prime } ... | Yes |
Lemma 3. \[ \left( {{R\theta } : {R\theta } \cap R}\right) = w. \] | Proof. We define a homomorphism \[ T : {R\theta } \rightarrow \frac{1}{w}\mathbb{Z}/\mathbb{Z} \] by mapping an element of the group algebra on its first coefficient \( {\;\operatorname{mod}\;\mathbf{Z}} \) . In other words, if \[ \alpha = \sum a\left( c\right) {\sigma }_{c} \] we let \( {T\alpha } = a\left( 1\right) \... | Yes |
Lemma 4. \[ \left( {{R\theta } : {Rm\theta }}\right) = {m}^{M}. \] | Proof. This is obvious if one can show that \( {R\theta } \) is a free abelian group of rank \( M \) . When \( m \) is a prime power, this results from the fact that for odd \( \chi \) we have \[ \chi \left( \theta \right) = {B}_{1,\chi } \neq 0. \] | No |
Lemma 5.\n\[ \n\\left( {{\\varepsilon }^{ - }R : {\\varepsilon }^{ - }{Rm\\theta }}\\right) = \\pm {m}^{M}\\mathop{\\prod }\\limits_{{\\chi \\text{ odd }}}{B}_{1,\\chi }.\n\] | Proof. First observe that the sign is whatever is needed to make the righthand side positive. Multiplication by \( {\\varepsilon }^{ - }{m\\theta } \) is an endomorphism of \( \\mathbf{Q}{R}^{ - } \) , which is a semisimple algebra, decomposing into a product of 1-dimensional algebras corresponding to the odd character... | Yes |
Theorem 2.1. (i) The values of \( {E}_{k, c}^{\left( N\right) } \) are \( N \) -integral.\n\n(ii) We have the congruence for every prime \( p \) dividing \( N \) :\n\n\[ {E}_{k, c}^{\left( N\right) }\left( x\right) \equiv {x}^{k - 1}{E}_{1, c}^{\left( N\right) }\left( x\right) {\;\operatorname{mod}\;N}{\mathbf{Z}}_{p}.... | Proof. For large integer \( v \) the values \( {N}^{v}/{kD}\left( k\right) \) are \( N \) -integral. Let \( M = {N}^{v} \) . The distribution relation yields\n\n\[ {E}_{k, c}^{\left( N\right) }\left( x\right) = \mathop{\sum }\limits_{y}{E}_{k, c}^{\left( M\right) }\left( y\right) \]\n\nwhere the sum is taken over those... | Yes |
Theorem 2.3. Let \( c \in {\mathbf{Z}}_{p}^{ * } \) and let \( k \) be an integer \( \geq 1 \) such that \( {c}^{k} \neq 1 \) . Then\n\n\[ \frac{1}{k}{B}_{k} = \frac{1}{1 - {c}^{k}}{\int }_{{\mathbf{z}}_{p}}{x}^{k - 1}d{E}_{1, c}\left( x\right) \] | Proof. By definition,\n\n\[ \frac{1}{k}{B}_{k} = {\int }_{{\mathbf{Z}}_{p}}d{E}_{k} = {\int }_{{\mathbf{Z}}_{p}}d{E}_{k, c} + {\int }_{{\mathbf{Z}}_{p}}{c}^{k}d{E}_{k}\left( {{c}^{-1}x}\right) . \]\n\nOn the last integral to the right, we make the change of variable\n\n\[ x \mapsto {cx} \]\nwhich gives\n\n\[ {\int }_{{... | No |
Corollary 1 (Kummer Congruence). Let \( \alpha \) be a residue class \( {\;\operatorname{mod}\;p} - 1 \) and \( \alpha \neq 0 \) . Then for even positive integers \( k \equiv \alpha {\;\operatorname{mod}\;p} - 1 \), the values \( \left( {1/k}\right) {B}_{k} \) are all congruent \( {\;\operatorname{mod}\;p} \), and are ... | Proof. Select \( c \) to be a primitive root \( {\;\operatorname{mod}\;p} \) so that\n\n\[
{c}^{k} ≢ 1{\;\operatorname{mod}\;p}\text{.}
\]\n\nThen \( 1 - {c}^{k} \) is a unit at \( p \) . The values \( 1 - {c}^{k} \) and \( {x}^{k - 1}{\;\operatorname{mod}\;p} \) are independent of the choice of \( k \) in the residue ... | No |
Corollary 2 (Von Staudt Congruence). Let \( k \equiv 0{\;\operatorname{mod}\;p} - 1 \), and \( k \) even.\n\nThen\n\n\[ \n{B}_{k} \equiv - \frac{1}{p}{\;\operatorname{mod}\;{\mathbf{Z}}_{p}}.\n\] | Proof. Suppose \( p \) odd for simplicity. Let \( c = 1 + p \) . An easy induction shows that\n\n\[ \n{c}^{k} \equiv 1 + {pk}{\;\operatorname{mod}\;{p}^{2}}k{\mathbf{Z}}_{p}\n\]\n\nHence\n\n\[ \n\frac{1}{1 - {c}^{k}} = - \frac{1}{pk}\left( {1 + O\left( p\right) }\right)\n\]\n\nand so\n\n\[ \n{B}_{k} \equiv - \frac{1}{p... | No |
Theorem 2.4. Let \( \psi \) be a character of finite order on \( {\mathbf{Z}}_{p}^{ * } \) . Then \[ \frac{1}{n}{B}_{n,\psi } = \frac{1}{1 - \psi \left( c\right) {c}^{n}}{\int }_{{\mathbf{Z}}_{p}^{ * }}\psi \left( a\right) {a}^{n - 1}d{E}_{1, c}\left( a\right) . \] | Proof. We write \( d{E}_{n} = d{E}_{n, c} + {c}^{n}d{E}_{n} \circ {c}^{-1} \), or in other words \[ \frac{1}{n}{B}_{n,\psi } = \int {\psi d}{E}_{n, c} + \int \psi \left( x\right) {c}^{n}d{E}_{n}\left( {{c}^{-1}x}\right) . \] Integrals are taken over \( {\mathbf{Z}}_{p}^{ * } \) . We let \( x \mapsto {cx} \) in the seco... | Yes |
Theorem 2.5. Let \( 2 \leq k \leq p - 2 \) . Let \( \omega : \mathbf{Z}{\left( p\right) }^{ * } \rightarrow {\mathbf{Z}}_{p}^{ * } \) be the Teichmuller character such that\n\n\[ \omega \left( a\right) \equiv a\left( {\;\operatorname{mod}\;p}\right) \]\n\nFor any integer \( n \geq 1 \) we have\n\n\[ \frac{1}{n}{B}_{n,{... | Proof. Let \( \psi = {\omega }^{k - n} \) . Choose \( c \) to be a primitive root \( {\;\operatorname{mod}\;p} \), so that \( {c}^{k} ≢ 1{\;\operatorname{mod}\;p} \) . By Theorem 2.3 we get\n\n\[ \frac{1}{k}{B}_{k} \equiv \frac{1}{1 - {c}^{k}}{\int }_{{\mathbf{Z}}_{p}^{ * }}{x}^{k - 1}d{E}_{1, c}\left( x\right) \left( ... | Yes |
Theorem 3.1. (i) We have\n\n\[ R{\theta }_{k}^{\prime } \cap R = {I}^{\left( k\right) }{\theta }_{k}^{\prime } \]\n\nIn fact, if an element \( \xi \in R \) is such that \( \xi {\theta }^{\prime } \in R \), then \( \xi \in {I}^{\left( k\right) } \). | Proof. First we prove that for any prime \( \geq 2 \), we have\n\n\[ I{\theta }^{\prime } \subset R,\text{ and }{I}_{p}\theta \subset {R}_{p}. \]\n\nA similar property is due to Mazur and Coates-Sinnott, as mentioned before. Indeed, we have\n\n\[ {\sigma }_{c}^{-1}\left( {{\sigma }_{c} - {c}^{k}}\right) {\theta }_{k} =... | No |
Lemma 1. The polynomial \( \left( {1/k}\right) \left( {{\mathbf{B}}_{k}\left( X\right) - {\mathbf{B}}_{k}\left( 0\right) }\right) \) maps \( \mathbf{Z} \) into \( \mathbf{Z} \) and maps \( {\mathbf{Z}}_{l} \) into \( {\mathbf{Z}}_{l} \) for every prime \( l \) . | Proof. A standard property of Bernoulli polynomials states that\n\n\[ \n\frac{1}{k}\left( {{\mathbf{B}}_{k}\left( {X + 1}\right) - {\mathbf{B}}_{k}\left( X\right) }\right) = {X}^{k - 1}.\n\]\n\nHence for any integer \( m \) we see recursively that the first assertion of the lemma is true. The second, concerning \( l \)... | No |
Lemma 2. (i) Let \( \xi \in R \) and suppose that \( \xi {\theta }^{\prime } \in {\mathbf{Z}}_{p}\left\lbrack G\right\rbrack = {R}_{p} \) . Then \( \xi \in J \) . | Proof. Write \( \xi = \sum z\left( b\right) {\sigma }_{b} \) with integral coefficients \( z\left( b\right) \) . Then\n\n\[ \xi {\theta }^{\prime } = {N}^{k - 1}\mathop{\sum }\limits_{c}\mathop{\sum }\limits_{b}z\left( b\right) \frac{1}{k}{\mathbf{B}}_{k}^{\prime }\left( \left\langle \frac{bc}{N}\right\rangle \right) {... | Yes |
Lemma 3. Let \( {p}^{s} \) be the smallest power of \( p \) such that \( {p}^{s}{\theta }_{k}^{\prime } \) is p-integral. Then\n\n\[ s = n + {\operatorname{ord}}_{p}k. \]\n\nWe have \( {I}^{\left( k\right) } \cap \mathbb{Z} = \left( {p}^{s}\right) \). | Proof. The argument uses the same expression for the Bernoulli polynomial as in the previous lemma. We see that\n\n\[ {p}^{s}\sum \frac{{N}^{k - 1}}{k}\left( \begin{array}{l} k \\ i \end{array}\right) {B}_{i}{\left( \frac{1}{N}\right) }^{k - i}\text{is}p\text{-integral.} \]\n\nThe leading term is \( {p}^{s}/{kN} \) . T... | Yes |
Lemma 4. We have \( J = I + \mathbf{Z}N \), and \( \left( {J : I}\right) = {p}^{s - n} = {p}^{\text{ord }k} \) . | Proof. It is clear that \( N \in J \) . Conversely, write an element of \( J \) in the form\n\n\[ \sum m\left( c\right) \left( {{\sigma }_{c} - {c}^{k}}\right) + \sum m\left( c\right) {c}^{k} \]\n\nThe first term is in \( I \), and the second term is an integral multiple of \( N \) . This proves the lemma. | No |
Theorem 5.1. \[ \left( {{R}_{0} : {R}_{0} \cap {R\theta }}\right) = N{p}^{\operatorname{ord}k - t}\mathop{\prod }\limits_{{\chi \neq 1}} \pm \frac{1}{k}{B}_{k,\chi }. \] | First observe that since \( \deg \theta \) and \( \deg {\theta }^{\prime } \neq 0 \) we have \[ {R}_{0} \cap {R\theta } = {R}_{0} \cap R{\theta }^{\prime }. \] By Theorem 2.1, we conclude that \[ R{\theta }^{\prime } \cap R = I{\theta }^{\prime },\text{ and hence }R{\theta }^{\prime } \cap {R}_{0} = {I}_{0}{\theta }^{\... | Yes |
Lemma 1. We have \( {R}^{ - } = 2{\varepsilon }^{ - }R \) and \( \left( {{\varepsilon }^{ - }R : {R}^{ - }}\right) = {2}^{\phi \left( N\right) /2} \) . | Proof. This is the same as Lemma 1 of \( §1 \) . | No |
Lemma 2. \( \left( {{R}^{ - } : 2{\varepsilon }^{ - }I}\right) = {p}^{s} \) where \( s = n + {\operatorname{ord}}_{p}k \) . | Proof. The group \( 2{\varepsilon }^{ - }I \) is generated by elements of the form\n\n\[ \left( {{\sigma }_{c} - {\sigma }_{-c}}\right) - {c}^{k}\left( {{\sigma }_{1} - {\sigma }_{-1}}\right) \]\n\nAn element \( \xi \in {R}^{ - } \) lies in \( \mathbf{Z}\left( {{\sigma }_{1} - {\sigma }_{-1}}\right) {\;\operatorname{mo... | No |
Theorem 8.1. The function \( g : \mathbf{Q}/\mathbf{Z} \rightarrow \lim F\left\lbrack {G\left( N\right) }\right\rbrack \) is an ordinary distribution. | Proof. Immediate from the definitions. | No |
Theorem 8.2. The dimension of \( {A}_{N} \) is equal to the cardinality of \( {\widehat{G}}_{h}\left( N\right) \) . | Proof. The space generated by the elements \( {g}_{N}\left( r\right) \) with \( r \in {Z}_{N} \) is clearly a \( G\left( N\right) \) -module since\n\n\[ \n{\sigma }_{b}{g}_{N}\left( r\right) = {g}_{N}\left( {rb}\right) ,\;\text{ for }b \in G\left( N\right) .\n\] \n\nWe let the idempotent associated with \( \chi \) be t... | Yes |
Theorem 9.1. (i) The elements \( g\left( {T}_{N}\right) \) generate the abelian group generated by \( g\left( {Z}_{N}\right) \) . | Proof. The first statement is obvious from the preceding remarks. | No |
Theorem 10.1. (Davenport-Hasse) We have\n\n\[ \mathop{\prod }\limits_{{{\chi }^{m} = 1}}\tau \left( {\chi \psi }\right) = \tau \left( {\psi }^{m}\right) C\left( {\psi, m}\right) \]\n\nwhere \( C\left( {\psi, m}\right) = \psi \left( {m}^{-m}\right) \mathop{\prod }\limits_{{{\chi }^{m} = 1}}\tau \left( \chi \right) \) . | Proof. Let \( {u}_{m}\left( \psi \right) \) be the quotient of the left-hand side by the right-hand side, that is\n\n\[ {u}_{m}\left( \psi \right) = \frac{\prod \tau \left( {\chi \psi }\right) }{\tau \left( {\psi }^{m}\right) C\left( {\psi, m}\right) }.\]\n\nWe have to show \( {u}_{m}\left( \psi \right) = 1 \) . First ... | Yes |
Lemma 1. Let \( 0 \leq k < q - 1 \) . Then\n\n\[ k! \equiv {\left( -p\right) }^{\frac{k - s\left( k\right) }{p - 1}}\gamma \left( k\right) {\;\operatorname{mod}\; * }p. \] | Proof. By induction. Suppose first that \( p \nmid k \) . Then \( {k}_{0} \geq 1 \), and\n\n\[ s\left( k\right) = s\left( {k - 1}\right) + 1,\;\gamma \left( k\right) = \gamma \left( {k - 1}\right) {k}_{0}. \]\n\nThe assertion is then obvious from the inductive step for \( k - 1 \) . Next suppose \( p \mid k \), so \( k... | Yes |
Theorem 1.1. If \( \chi \) is primitive and \( d \) is not prime to \( m \), then\n\n\[ S\left( {\chi ,\lambda \circ d}\right) = 0. \] | Proof. Using the prime power decomposition, we may assume without loss of generality that \( m = {p}^{n} \) is a prime power. Abbreviate\n\n\[ A = \mathbb{Z}\left( {p}^{n}\right) \]\n\nAlso without loss of generality, we may assume \( d = {p}^{r} \) for some integer \( r \geq 1 \), and \( r < n \) . Form a coset decomp... | Yes |
Theorem 1.2. (i) We have \( {T}^{2}f = m{f}^{ - } \) .\n\n(ii) If \( \chi \) is primitive, then\n\n\[ \n{T\chi } = \chi \left( {-1}\right) S\left( \chi \right) {\chi }^{-1}.\n\]\n\n(iii) Again if \( \chi \) is primitive, then\n\n\[ \nS\left( \chi \right) \overline{S\left( \chi \right) } = m \n\] | Proof. Part (i) is proved as for the finite field case. For (ii), if \( y \) is not prime to \( m \), then \( {T\chi }\left( y\right) = 0 \) by Theorem 1.1. If \( y \) is prime to \( m \) then we can make the usual change of variables to get the right answer. Part (iii) is then proved as in the finite field case. | No |
Theorem 2.1. Assume that \( \chi \) is a primitive character \( {\;\operatorname{mod}\;m} \) . Then\n\n\[ L\left( {s,\chi }\right) = \frac{1}{m}S\left( \chi \right) \mathop{\sum }\limits_{{b \in \mathbf{Z}{\left( m\right) }^{ * }}}\bar{\chi }\left( b\right) \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{\zeta }^{-{nb}... | Proof. If \( x \) is not prime to \( m \) then the Gauss sum is 0 by Theorem 1.1. If \( b \) is prime to \( m \), we can make the change of variables which yields the desired expression. | No |
Theorem 2.2. If \( \chi \) is a primitive character, then\n\n\[ L\left( {1,\chi }\right) = - \frac{S\left( \chi \right) }{m}\mathop{\sum }\limits_{{b \in \mathbf{Z}{\left( m\right) }^{ * }}}\bar{\chi }\left( b\right) \log \left( {1 - {\zeta }^{-b}}\right) . \] | Case 1. \( \chi \) is even.\n\nIn this case, adding the sum with \( b \) and \( - b \) yields\n\n\[ 2\sum \bar{\chi }\left( b\right) \log \left( {1 - {\zeta }^{-b}}\right) = \sum \bar{\chi }\left( b\right) \left\lbrack {\log \left( {1 - {\zeta }^{b}}\right) + \log \left( {1 - {\zeta }^{-b}}\right) }\right\rbrack . \]\n... | Yes |
Theorem 3.1. We have product expressions:\n\n(i)\n\n\\[ \n\\mathop{\\prod }\\limits_{{\\chi \\neq 1}}m\\left( \\chi \\right) = d \n\\]\n\n(ii)\n\n\\[ \n\\mathop{\\prod }\\limits_{{\\chi \\neq 1}}S\\left( \\chi \\right) = \\left\\{ \\begin{array}{ll} {d}^{1/2} & \\text{ if }K\\text{ is real } \\\\ {i}^{{r}_{2}}{d}^{1/2}... | Proof. It is possible to give essentially algebraic proofs for these facts (although the sign of the Gauss sums is always a little delicate, involving something about the complex numbers). The best way to see the theorem, however, is probably as in Hasse [Ha 1], using the functional equations of the zeta function and \... | Yes |
Theorem 3.2. For imaginary \( K \) , \[ h = {h}^{ + }{Qw}{2}^{-N/2}\mathop{\prod }\limits_{{\chi \text{ odd }}} - {B}_{1,\chi } \] | In the next section, we shall analyze more closely the decomposition \[ h = {h}^{ + }{h}^{ - } \] where \( {h}^{ - } \) is defined as \( h/{h}^{ + } \), and we shall see that \( {h}^{ - } \) is an integer. In any case, we have the class number formula: \( {\mathrm{{CNF}}}^{ - } \). \[ {h}^{ - } = {Qw}\mathop{\prod }\li... | No |
Theorem 3.3. If \( m \) is a prime power, \( K = \mathbf{Q}\left( {\mu }_{m}\right), G = \operatorname{Gal}\left( {K/\mathbf{Q}}\right) \), and \( \mathcal{S} \) is the Stickelberger ideal, then\n\n\[ \n{h}^{ - } = \left( {\mathbf{Z}{\left\lbrack G\right\rbrack }^{ - } : \mathcal{S}}\right) \n\] | Let \( p \) be a prime number. If \( A \) is an abelian group, we denote by \( {A}^{\left( p\right) } \) its \( p \) -primary part. As Iwasawa observed [Iw 7], knowing the index immediately shows that:\n\nThe group \( {C}_{K}^{-\left( p\right) } \) is generated by one element over \( \mathbf{Z}\left\lbrack G\right\rbra... | No |
Theorem 4.1. Let \( K = \mathbf{Q}\left( {\mu }_{m}\right) \) . Then \( {Q}_{K} = 1 \) if \( m \) is a prime power, and 2 if \( m \) is not a prime power. | Proof. Let \( E = {E}_{K} \) be the unit group in \( K \) . For each unit \( u \) in \( E \), the quotient \( \bar{u}/u \) is a unit, of absolute value 1, and for any automorphism \( \sigma \) of \( K \) over \( \mathbf{Q} \), we have\n\n\[ \sigma \left( {\bar{u}/u}\right) = \overline{\sigma u}/{\sigma u} \]\n\nbecause... | Yes |
Theorem 4.2. Let \( K = \mathbf{Q}\left( {\mu }_{m}\right) \) . The natural map\n\n\[ \n{C}_{K} + \rightarrow {C}_{K} \n\]\n\nof ideal classes in \( {K}^{ + } \) into the ideal class group of \( K \) is injective. | Proof. Let \( \mathfrak{a} \) be an ideal of \( {K}^{ + } \) and suppose \( \mathfrak{a} = \left( \alpha \right) \) with \( \alpha \) in \( K \) . Then \( \bar{\alpha }/\alpha \) is a unit, and in fact a root of unity as one sees by an argument similar to that in Theorem 4.1. Suppose that \( m \) is composite. By Theor... | Yes |
Theorem 4.3. Let \( K \) be an imaginary abelian extension of \( \mathbf{Q} \). Then the norm map\n\n\[ \n{N}_{K/{K}^{ + }} : {C}_{K} \rightarrow {C}_{{K}^{ + }} \n\]\n\non the ideal class group is surjective. | Proof. We have to use class field theory, which gives the more general statement:\n\nLemma. Let \( K \) be an abelian extension of a number field \( F \). Let \( H \) be the\n\nHilbert class field of \( F \) (maximal abelian unramified extension of \( F \)). If\n\n\( K \cap H = F \) then the norm map \( {N}_{K/F} : {C}... | Yes |
Theorem 4.4. Let \( K = \mathbf{Q}\left( {\mu }_{m}\right) \) . Then the sequence\n\n\[ 1 \rightarrow {C}_{K}^{ - } \rightarrow {C}_{K}\xrightarrow[]{\text{ norm }}{C}_{{K}^{ + }} \rightarrow 1 \]\n\nis exact. | Proof. We consider the norm map followed by the injection,\n\n\[ {C}_{K}\xrightarrow[]{\text{ norm }}{C}_{{K}^{ + }}\xrightarrow[]{\text{ inj }}{C}_{K \]\n\nThe kernel of this composite map is \( {C}_{K}^{ - } \) by definition, so the theorem is obvious by what had already been proved. | No |
Theorem 5.1. Let \( K = \mathbf{Q}\left( {\mu }_{m}\right) \) and \( h = {h}_{K} \) . Assume \( m = {p}^{n} \) is a prime power.\n\nThen\n\n\[ \n{h}^{ + } = \left( {{E}^{ + } : {\mathcal{E}}^{ + }}\right) = \left( {E : \mathcal{E}}\right) .\n\] | Proof. Let \( G \) be any finite abelian group. Then we have the Frobenius determinant formula for any function \( f \) on \( G \) :\n\n\[ \n\mathop{\prod }\limits_{{\chi \neq 1}}\mathop{\sum }\limits_{{a \in G}}\chi \left( a\right) f\left( {a}^{-1}\right) = {\det }_{a, b \neq 1}\left\lbrack {f\left( {a{b}^{-1}}\right)... | No |
Lemma 1. We have for \( G = \mathbf{Z}{\left( m\right) }^{ * }/ \pm 1 \) :\n\n\[ \pm {\det }_{a, b \neq 1}\log \left| {{\sigma }_{a}{g}_{b}}\right| = \mathop{\prod }\limits_{{\chi \neq 1}}\mathop{\sum }\limits_{{b \in G}}\chi \left( b\right) \log \left| {1 - {\zeta }^{b}}\right| \]\n\n\[ = \mathop{\prod }\limits_{{\chi... | Proof. The first expression comes from the Frobenius determinant formula (Theorem 6.2), and the second comes from the fact that for non-trivial \( \chi \) ,\n\n\[ \sum \chi \left( b\right) \log \left| {1 - \zeta }\right| = 0 \] | Yes |
Lemma 2. Let \( {G}_{\chi } = \mathbf{Z}{\left( m\left( \chi \right) \right) }^{ * }/ \pm 1 \) . For prime power \( m = {p}^{n} \), we have\n\n\[ \mathop{\sum }\limits_{{b \in {G}_{\chi }}}\chi \left( b\right) \log \left| {1 - {\zeta }_{m\left( \chi \right) }^{b}}\right| = \mathop{\sum }\limits_{{b \in G}}\chi \left( b... | Proof. Let \( m\left( \chi \right) = {p}^{s} \) . We write residue classes in \( \mathbf{Z}{\left( {p}^{n}\right) }^{ * } \) in the form\n\n\[ y = b + {p}^{s}c,\;\text{ with }0 \leq c < {p}^{n - s}, \]\n\nand \( b \) ranges over a fixed set of representatives for residue classes of \( \mathbf{Z}{\left( {p}^{s}\right) }... | Yes |
Theorem 5.3.\n\[ \n{h}_{F} = \left( {{E}_{F} : {\mathcal{E}}_{F}}\right) \n\] | Proof. Let\n\n\[ \n\alpha = \mathop{\prod }\limits_{{\chi \left( a\right) = 1}}\left( {1 - {\zeta }^{a}}\right) \;\text{ and }\;{\alpha }^{\prime } = \mathop{\prod }\limits_{{\chi \left( a\right) = - 1}}\left( {1 - {\zeta }^{a}}\right) \n\]\n\nwhere \( \zeta \) is a fixed primitive \( D \) th root of unity. Note that t... | Yes |
Theorem 6.1. Let \( f \) be any (complex valued) function on \( G \) . Then\n\n\[ \mathop{\prod }\limits_{{\chi \in G}}\mathop{\sum }\limits_{{a \in G}}\chi \left( a\right) f\left( {a}^{-1}\right) = \mathop{\det }\limits_{{a, b}}f\left( {{a}^{-1}b}\right) \] | Proof. Let \( F \) be the space of functions on \( G \) . It is a finite dimensional vector space whose dimension is the order of \( G \) . It has two natural bases. First, the characters \( \{ \chi \} \), and second the functions \( \left\{ {\delta }_{b}\right\}, b \in G \), where\n\n\[ {\delta }_{b}\left( x\right) = ... | Yes |
Theorem 6.2. The determinant in Theorem 4.1 splits into\n\n\\[ \n\\mathop{\\det }\\limits_{{a, b}}f\\left( {a{b}^{-1}}\\right) = \\left\\lbrack {\\mathop{\\sum }\\limits_{{a \\in G}}f\\left( a\\right) }\\right\\rbrack \\mathop{\\det }\\limits_{{a, b \\neq 1}}\\left\\lbrack {f\\left( {a{b}^{-1}}\\right) - f\\left( a\\ri... | Proof. Let \\( {a}_{1} = 1,\\ldots ,{a}_{n} \\) be the elements of \\( G \\) . In the determinant\n\n\\[ \n\\det f\\left( {{a}_{i}{a}_{j}^{-1}}\\right) = \\left| \\begin{matrix} f\\left( {{a}_{1}{a}_{1}^{-1}}\\right) & f\\left( {{a}_{1}{a}_{2}^{-1}}\\right) \\cdots f\\left( {{a}_{1}{a}_{n}^{-1}}\\right) \\\\ \\vdots & ... | Yes |
Theorem 7.1.\n\[ \pm {D}_{p} = {\left( 2p\right) }^{\left( {p - 3}\right) /2}{h}_{p}^{ - } \] | where\n\[ {D}_{p} = \det \left\lbrack {R\left( {\zeta }^{i + j}\right) - R\left( {-{\zeta }^{i + j}}\right) }\right\rbrack .\n\]\nObserve that each entry in the determinant \( {D}_{p} \) is an integer of absolute value \( \leq p - 1 \) .\n\nThe absolute value of the determinant is the volume of the fundamental domain o... | Yes |
Theorem 1.1. Let \( f\left( X\right) = \sum {c}_{k}{X}^{k} \in \mathfrak{o}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) . Then | \[ {c}_{k} = {\int }_{{\mathbb{Z}}_{p}}\left( \begin{array}{l} x \\ k \end{array}\right) d{\mu }_{f}\left( x\right) \] | No |
Theorem 1.2. The power series \( {P\mu } \) is the unique power series \( f \) such that for \( z \) in the maximal ideal of \( \mathfrak{o} \), we have\n\n\[ \n{\int }_{{\mathbf{Z}}_{p}}{\left( 1 + z\right) }^{x}{d\mu }\left( x\right) = f\left( z\right) \n\] | Proof. We have\n\n\[ \n{\int }_{{\mathbf{Z}}_{p}}{\left( 1 + z\right) }^{x}d{\mu }_{f}\left( x\right) = {\int }_{{\mathbf{Z}}_{p}}\mathop{\sum }\limits_{{k = 0}}^{\infty }\left( \begin{array}{l} x \\ k \end{array}\right) {z}^{k}d{\mu }_{f}\left( x\right) . \n\]\n\nWe can interchange the sum and integral, apply Theorem ... | No |
Let \( v \) be a measure on \( {\mathbf{Z}}_{p} \) whose support lies in the open closed subset \( 1 + p{\mathbf{Z}}_{p} \). Let \( \gamma \) be a topological generator of \( 1 + p{\mathbf{Z}}_{p} \), for instance \( \gamma = 1 + p \). There is an isomorphism\n\n\[ \n{\mathbf{Z}}_{p} \rightarrow 1 + p{\mathbf{Z}}_{p} \... | By Theorem 1.2, writing \( {\gamma }^{s} = 1 + z \), we get\n\n\[ \n{\int }_{1 + p{\mathbf{Z}}_{p}}{u}^{s}{dv}\left( u\right) = f\left( {{\gamma }^{s} - 1}\right) \n\] | Yes |
Theorem 1.4. We have \( \parallel f\parallel = \begin{Vmatrix}{\mu }_{f}\end{Vmatrix} \) . | Proof. Since\n\n\[ \n{c}_{n} = \int \left( \begin{array}{l} x \\ n \end{array}\right) d{\mu }_{f}\left( x\right) \n\] \n\nwe get trivially \( \parallel f\parallel \leq \begin{Vmatrix}{\mu }_{f}\end{Vmatrix} \) . Conversely, given a level \( {p}^{n} \), let \( {x}_{0} \in \mathbf{Z}\left( {p}^{n}\right) \) and let \( \v... | Yes |
Theorem 3.1. Let \( g \in \mathfrak{o}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) be such that \( \mathbf{U}g = g \), and let \( h \) be a power series such that \( {Dh} = g \) . Then \( \mathbf{U}h \in \mathfrak{o}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) and\n\n\[{\Gamma }_{p}\mathbf{U}h... | Proof. This is an immediate application of Meas 7, after integrating the function \( \langle a{\rangle }^{s} \) . | No |
Theorem 3.2. The value of \( {L}_{p}\left( {1 - s,\chi }\right) \) is independent of the choice of \( c \) , and for any positive integer \( k \) ,\n\n\[ \n{L}_{p}\left( {1 - k,\chi }\right) = - \frac{1}{k}{B}_{k,\chi {\omega }^{-k}} \n\] \n\nIn particular, if \( k \equiv 0{\;\operatorname{mod}\;p} - 1 \), and \( p \) ... | Proof. Since the set of sufficiently large integers \( k \equiv 0{\;\operatorname{mod}\;p} - 1 \) is dense \nin \( {\mathbf{Z}}_{p} \), we see that the first assertion follows from the explicit values given at integers of the form \( 1 - k \) as described. For these, we have: \n\n\[ \n{\mathbf{M}}_{p}\left( {\chi {E}_{... | Yes |
Theorem 3.3. Let \( g = \\mathbf{U}g \) and let \( h \) be the power series such that\n\n\[ \n{Dh} = g\\text{ and }h\\left( 0\\right) = 0.\n\]\n\nThen\n\n\[ \n{\\mathbf{M}}_{p}{\\mu }_{g}\\left( 0\\right) = - \\frac{1}{p}\\mathop{\\sum }\\limits_{{{\\xi }^{p} = 1}}h\\left( {\\xi - 1}\\right)\n\] | Proof. By Meas 7 we have\n\n\[ \n{\\mathbf{M}}_{p}{\\mu }_{g}\\left( 0\\right) = \\int {a}^{-1}d{\\mu }_{g}\\left( a\\right) = \\int d{\\mu }_{\\mathbf{U}h}\\left( a\\right) = \\mathbf{U}h\\left( 0\\right) .\n\]\n\nThe formula is then clear from the definition of \( \\mathbf{U} \) . | No |
Proposition 3.4. Let \( c \in {\mathbf{Z}}_{p}^{ * } \) . The power series associated with the measure \( {E}_{1, c} \) is \[ {f}_{1, c} = \frac{1}{T - 1} - \frac{c}{{T}^{c} - 1},\;\text{ with }T = 1 + X. \] | Proof. It is immediate to verify that as power series in \( X \) the expression on the right-hand side is holomorphic at \( X = 0 \), and that its coefficients are \( p \) -integral because \( c \) is a \( p \) -unit. Let \[ f\left( T\right) = \frac{\log T}{T - 1} - \frac{c\log T}{{T}^{c} - 1} \] Putting \( T = {e}^{Z}... | Yes |
Proposition 3.5. Let \( \\chi \) be a non-trivial character on \( {\\mathbf{Z}}_{p}^{ * } \) with conductor \( N \). The power series associated with \( \\chi {E}_{1, c} \) is \[ {g}_{\\chi, c} = {G}_{\\chi }\\left( T\\right) - {c\\chi }\\left( c\\right) {G}_{\\chi }\\left( {T}^{c}\\right) \] where \[ {G}_{\\chi }\\lef... | Proof. Immediate from Meas 5. | No |
Theorem 3.6. Let \( \chi \) be a primitive Dirichlet character with conductor \( N \) equal to a power of p. Then\n\n\[ \n{L}_{p}\left( {1,\chi }\right) = - \frac{S\left( {\chi ,\zeta }\right) }{N}\mathop{\sum }\limits_{{a \in \mathbf{Z}{\left( N\right) }^{ * }}}\bar{\chi }\left( a\right) \log \left( {1 - {\zeta }^{a}}... | Proof. By Theorem 3.3, Proposition 3.5, and the definition of \( {L}_{p}\left( {1,\chi }\right) \), we find:\n\n\[ \n\left( {1 - \chi \left( c\right) }\right) {L}_{p}\left( {1,\chi }\right) = \frac{1}{p}\frac{S\left( {\chi ,\zeta }\right) }{N}\mathop{\sum }\limits_{\xi }\mathop{\sum }\limits_{a}\mathop{\sum }\limits_{{... | No |
Let \( K \) be the real subfield of \( \mathbf{Q}\left( {\mu }_{m}\right) \) . Then\n\n\[ \n{R}_{p}\left( \mathcal{E}\right) = \left( {E : \mathcal{E}}\right) {R}_{p}\left( E\right) = \left( {E : \mathcal{E}}\right) {R}_{p}.\n\] | We know from Theorem 5.1 of the preceding chapter that\n\n\[ \n{h}^{ + } = \left( {E : \mathcal{E}}\right) \n\]\n\nLet \( {g}_{a} \) ( \( a \) prime to \( m \) ) be the cyclotomic units, and \( {g}_{a}^{ + } \) the corresponding real cyclotomic units. From our definition of the \( p \) -adic log, we know that for any e... | Yes |
Theorem 4.2 (Brumer). We have \( {R}_{p} \neq 0 \) for the real cyclotomic field \( \mathbf{Q}{\left( {\mu }_{m}\right) }^{ + } \) . | Proof. The cyclotomic units are algebraic, and it is a known theorem from the theory of transcendental numbers that the logs ( \( p \) -adic or otherwise) of multiplicatively independent algebraic numbers are linearly independent over the algebraic numbers. The proof is the \( p \) -adic analogue of Baker’s proof for t... | Yes |
Theorem 4.3 (Leopoldt p-adic Class Number-regulator Formula). Let \( m = {p}^{n} \) be a prime power, and \( {K}^{ + } = \mathbf{Q}{\left( {\mu }_{m}\right) }^{ + } \) . Then\n\n\[ \mathop{\prod }\limits_{\substack{{\chi \neq 1} \\ {\chi \text{ even }} }}\frac{1}{2}{L}_{p}\left( {1,\chi }\right) = \frac{{h}^{ + }}{\sqr... | Proof. From Theorem 4.1 and the complexly derived index\n\n\[ {h}^{ + } = \left( {E : \mathcal{E}}\right) \]\n\nof Theorem 5.1 in the preceding chapter, we find:\n\n\[ \pm {h}^{ + }{R}_{p}\left( E\right) = \pm {R}_{p}\left( \mathcal{E}\right) = \mathop{\prod }\limits_{\substack{{\chi \neq 1} \\ {\chi \text{ even }} }}\... | Yes |
Theorem 6.1. If \( f \in \mathcal{L} \) then \( f \) converges on the disc of elements\n\n\[ x \in {\mathbb{C}}_{p}\;\text{ and }\;\left| x\right| \leq {\left| p\right| }^{1/\left( {p - 1}\right) }.\]\n\nFor such \( x \) we have\n\n\[ \left| {f\left( x\right) }\right| \leq \parallel f{\parallel }_{\mathcal{L}} \] | Proof. Obvious, because\n\n\[ {\left| p\right| }^{n/\left( {p - 1}\right) } \leq \left| {n!}\right| \text{ and so }\left| {{a}_{n}\frac{{x}^{n}}{n!}}\right| \leq \left| {a}_{n}\right| . \] | No |
Theorem 6.2. Let \( \alpha \) be a residue class \( {\;\operatorname{mod}\;p} - 1 \) . There exists a unique continuous linear map\n\n\[ \n{\Gamma }_{\alpha } : {\mathcal{L}}_{K} \rightarrow C\left( {{\mathbb{Z}}_{p}, K}\right) \n\] \n\nsatisfying any one of the following three equivalent conditions:\n\n\( {\Gamma }_{\... | Proof. Any continuous linear map on the space of polynomials (with Leopoldt norm) extends uniquely by continuity to the Leopoldt Banach algebra. We shall prove that the linear map\n\n\[ \n{\Gamma }_{\alpha } : K\left\lbrack X\right\rbrack \rightarrow C\left( {{\mathbf{Z}}_{p}, K}\right) \n\] \n\nwith values\n\n\[ \n{\G... | Yes |
Theorem 6.3. Let \( m \) be an integer \( \geq 0 \), and \( m \equiv \alpha {\;\operatorname{mod}\;p} - 1 \) . Then \[ {\Gamma }_{\alpha }f\left( m\right) = \Gamma \mathrm{U}f\left( m\right) \] | Proof. The two maps \[ f \mapsto {\Gamma }_{\alpha }f\text{ and }f \mapsto \Gamma \mathbb{U}f \] of \( K\left\lbrack X\right\rbrack \rightarrow C\left( {{\mathbf{Z}}_{p}, K}\right) \) are equal on the polynomials \( {\left( 1 + X\right) }^{v} \) . For a fixed \( m \) the maps \[ f \mapsto {\Gamma }_{\alpha }f\left( m\r... | No |
Theorem 6.4. For \( f \in {\mathcal{L}}_{K} \) we have\n\n\[{\Gamma }_{0}f\left( 0\right) = \mathbf{U}f\left( 0\right) = f\left( 0\right) - \frac{1}{p}\mathop{\sum }\limits_{{{\zeta p} = 1}}f\left( {\zeta - 1}\right) .\] | Proof. The power series for \( \mathbf{U}f \) in terms of \( X \) or \( Z \) have the same constant term. Hence\n\n\[ \Gamma \mathrm{U}f\left( 0\right) = \mathrm{U}f\left( 0\right) \]\n\nTaking \( \alpha = 0 \), the theorem is obvious from Theorem 6.3, and the fact that \( \zeta - 1 \) lies in the domain of convergence... | No |
Theorem 6.5. For \( s \in {\mathbf{Z}}_{p} \) and \( f \in \mathfrak{o}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) we have\n\n\[{\int }_{{\mathbf{Z}}_{p}^{ * }}\langle a{\rangle }^{s}d{\mu }_{f}\left( a\right) = {\Gamma }_{0}f\left( s\right)\] | Proof. By continuity in \( s \), it suffices to prove the theorem when \( s = k \) is an integer \( \geq 1 \) and \( k \equiv 0{\;\operatorname{mod}\;p} - 1 \) . Let \( \varphi \) be the characteristic function of \( {\mathbf{Z}}_{p}^{ * } \) . Then\n\n\[{\int }_{{\mathbf{Z}}_{p}^{ * }}\langle a{\rangle }^{k}d{\mu }_{f... | Yes |
Theorem 1.1. The homomorphism \( \varepsilon \) is an isomorphism. | Proof. A trivial induction shows that\n\n\[ \n{h}_{n} = {\left( 1 + X\right) }^{{p}^{n}} - 1 \in {\left( p, X\right) }^{n + 1} \n\] \n\nwhere \( \left( {p, X}\right) \) denotes the maximal ideal of \( {\mathbf{Z}}_{p}\left\lbrack X\right\rbrack \), generated by \( p \) and \( X \) . It follows that the intersection of ... | Yes |
Theorem 1.2. (i) If \( V = \Lambda /{p}^{m} \) then \( {e}_{n} = m{p}^{n} \) . | Proof. In case (i) we have\n\n\[ \mathbf{Z}\left( {p}^{m}\right) \left\lbrack \left\lbrack X\right\rbrack \right\rbrack /\left( {{\left( X + 1\right) }^{{p}^{n}} - 1}\right) \approx \mathbf{Z}\left( {p}^{m}\right) \left\lbrack X\right\rbrack /\left( {{\left( X + 1\right) }^{{p}^{n}} - 1}\right) ,\]\n\nand this is just ... | Yes |
Theorem 1.3. Assume that \( V \) is of Iwasawa type. Then the conclusions of Theorem 1.2(i), (ii), (iii) remain valid, except that in 1.2(ii) we have to write the exponent\n\n\[ \n{e}_{n} = {dn} + {c}_{0} \n\]\n\nwith some constant \( {c}_{0} \) . | Proof. Note that Case (i) is unchanged, only Case (ii) is now slightly different, but the proof runs along entirely similar lines as follows. In this case, \( V \) is \( {\mathbf{Z}}_{p} \) -free of rank \( d \) . An argument similar to that of Theorem 1.2(ii) shows that\n\n\[ \n{g}_{n}V = {p}^{n - {n}_{0}}{g}_{{n}_{0}... | Yes |
Theorem 2.1. Let \( \mathfrak{o} \) be a complete local ring with maximal ideal \( \mathfrak{m} \) . Let\n\n\[ f\left( X\right) = \mathop{\sum }\limits_{{i = 0}}^{\infty }{a}_{i}{X}^{i} \]\n\nbe a power series in \( \mathfrak{o}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \), such that not all \( {a}_{i} \) l... | Proof. Let \( \alpha \) and \( \tau \) be the projections on the beginning and tail end of the series, given by\n\n\[ \alpha : \sum {b}_{i}{X}^{i} \mapsto \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{b}_{i}{X}^{i} = {b}_{0} + {b}_{1} + \cdots + {b}_{n - 1}{X}^{n - 1} \]\n\n\[ \tau : \sum {b}_{i}{X}^{i} \mapsto \mathop{\su... | Yes |
Theorem 2.2 (Weierstrass Preparation). The power series \( f \) in the previous theorem can be written in the form\n\n\[ f\left( X\right) = \left( {{X}^{n} + {b}_{n - 1}{X}^{n - 1} + \cdots + {b}_{0}}\right) u, \]\n\nwhere \( {b}_{i} \in \mathfrak{m} \), and \( u \) is a unit in \( \mathfrak{o}\left\lbrack \left\lbrack... | Proof. Write\n\n\[ {X}^{n} = {qf} + r \]\n\nby the Euclidean algorithm. Then \( q \) is invertible because\n\n\[ q = {c}_{0} + {c}_{1}X + \cdots \]\n\n\[ f = \cdots + {a}_{n}{X}^{n} + \cdots \]\n\nso that\n\n\[ 1 \equiv {c}_{0}{a}_{n}\left( {\;\operatorname{mod}\;\mathfrak{m}}\right) \]\n\nand \( {c}_{0} \) is a unit i... | Yes |
Theorem 3.2. If \( R \) is a matrix of relations, we can transform \( R \) with a finite number of admissible operations into a matrix \( {R}^{\prime } \) of the form\n\n\[ \left( \begin{matrix} {\lambda }_{11} & 0 & \cdots & 0 & 0 & \cdots & 0 \\ \vdots & \vdots & & \vdots & \vdots & & \vdots \\ 0 & 0 & \cdots & {\lam... | Proof. By the lemma, we can replace \( R \) by a matrix \( {R}^{\prime } \) of the form\n\n\[ \left( \begin{matrix} {\lambda }_{11} & 0 & \cdots & 0 & 0 & \cdots & 0 \\ \vdots & \vdots & & \vdots & \vdots & & \vdots \\ 0 & 0 & \cdots & {\lambda }_{rr} & 0 & \cdots & 0 \\ * & * & \cdots & * & 0 & \cdots & 0 \end{matrix}... | Yes |
Theorem 4.1. Assume first that IW is satisfied with \( s = 1 \) . Let \( I \) be the inertia group of any prime above \( \mathfrak{p} \) in \( G \) . Then:\n\n(i) \( G = I{G}_{C} \) is a semidirect product, and the restriction of \( I \) to \( {K}_{\infty } \) gives an isomorphism of \( I \) and \( \Gamma \) .\n\n(ii) ... | Proof. We have an exact sequence\n\n\[ 1 \rightarrow {G}_{c} \rightarrow G \rightarrow \Gamma \rightarrow 1 \]\n\nThe image of \( I \) in \( \Gamma \) by restriction to \( {K}_{\infty } \) is surjective because \( {K}_{\infty } \) is totally ramified over \( {K}_{0} \) . It is injective because \( {M}_{\infty } \) is u... | Yes |
Theorem 4.2. Assume that IW is satisfied, with primes \( {\mathfrak{p}}_{1},\ldots ,{\mathfrak{p}}_{s} \) . Let \( {I}_{j} \) be the inertia group of \( {\mathfrak{p}}_{j} \) in \( G \) . Then:\n\n(i) There is a semidirect product decomposition\n\n\[ G = {I}_{1}{G}_{C} \]\n\nand \( {G}^{\prime } = {G}_{C}^{\gamma - 1} ... | Proof. Identical with that of Theorem 4.1, except that in the present more general situation, we have to look at the smallest subgroup of \( G \) containing the commutator group \( {G}^{\prime } \) and all the inertia groups \( {I}_{j} \) instead of a single inertia group \( I \) . | No |
Theorem 4.3. Assume that IW is satisfied with one prime. If \( {C}_{0} = \{ 1\} \), then \( {C}_{n} = \{ 1\} \) for all \( n \) . | Proof. If \( {C}_{0} = 1 \), then Theorem 4.1 shows that \( C = {C}^{\gamma - 1} \) . Viewing \( C \) as module over \( {\mathbf{Z}}_{p}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \), this means that \( C = {XC} \) . But \( X \) is contained in the maximal ideal of \( {\mathbf{Z}}_{p}\left\lbrack \left\lbrac... | Yes |
Theorem 4.4. For any \( {\mathbf{Z}}_{p} \) -extension the module \( C \) over \( {\mathbf{Z}}_{p}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) is a finitely generated torsion module. | Proof. Suppose first for simplicity that condition IW is satisfied with only one prime. Then \( C/\mathfrak{m}C \) is a factor group of \( C/{C}^{\gamma - 1} \), which is none other than \( {C}_{0} \) by Theorem 4.1, and is therefore finite. That \( C \) is finitely generated is a special case of Nakayama's lemma.\n\nI... | Yes |
Theorem 5.1. Let \( H \) be the p-Hilbert class field of \( K \) . Then we have an isomorphism\n\n\[ \text{Gal}\left( {{M}_{p}\left( K\right) /H}\right) \approx p\text{-part of}{U}_{p}/\overline{{\sigma }_{p}E} \]\n\n\[ = {U}_{p}^{\left( 1\right) }/\left( {{U}_{p}^{\left( 1\right) } \cap \overline{{\sigma }_{p}E}}\righ... | Again, as \( p \) is fixed, we write simply \( {U}_{p}/\bar{E} \) . By a quasi-isomorphism, we shall mean a homomorphism with finite kernel and cokernel. We denote a quasi-isomorphism by a single \( \sim \) . The theorem yields a quasi-isomorphism\n\n\[ {G}_{p}^{\mathrm{{ab}}}\left( K\right) \sim {U}_{p}/\bar{E} \]\n\n... | No |
Theorem 5.2. Assume the Leopoldt conjecture for \( K \) . Then we have a quasi-isomorphism\n\n\[ \n{G}_{p}^{\mathrm{{ab}}}\left( K\right) \sim {\mathbb{Z}}_{p}^{{r}_{2} + 1} \approx \operatorname{Gal}\left( {{Z}_{p}\left( K\right) /K}\right) \n\] | Proof. The first statement comes from the definitions and\n\n\[ \n\left\lbrack {K : \mathbf{Q}}\right\rbrack = {r}_{1} + 2{r}_{2} \n\]\n\nFor the second statement, we note that the composite of all \( {\mathbf{Z}}_{p} \) -extensions of\n\n\( K \) has a Galois group embedded in the product of \( {\mathbf{Z}}_{p} \) with... | Yes |
Theorem 6.1. Let \( \Omega \) be the maximal p-abelian p-ramified extension of \( {K}_{\infty } \) . Then:\n\n(i) \( G = \operatorname{Gal}\left( {\Omega /{K}_{\infty }}\right) \) is finitely generated over the Iwasawa algebra, and in fact\n\n\[ G/{G}^{\gamma - 1} \sim {\mathbb{Z}}_{p}^{\rho }\;\text{ where }\rho = \le... | Proof. By definition,\n\n\[ {\Omega }_{0} = {M}_{p}\left( {K}_{0}\right) \]\n\nand the rank over \( {\mathbf{Z}}_{p} \) of a subgroup of finite index in its Galois group was determined to be \( \left\lbrack {{K}_{0} : \mathbf{Q}}\right\rbrack - {r}_{p} \) in Theorem 5.2. Taking into account \( \Gamma \) itself shows th... | Yes |
Theorem 6.2. Assume that \( {K}_{0} \) is totally imaginary, and that each \( {K}_{n} \) satisfies the Leopoldt conjecture (namely\n\n\[ \n\\left. {{r}_{p}\left( {E}_{n}\right) = {r}_{2}\left( {K}_{n}\right) }\\right) \n\]\n\nThen there is a quasi-isomorphism\n\n\[ \nG \\sim {\\Lambda }^{{r}_{2}} \\times {G}_{\\mathrm{... | Proof. From the structure theorem, we know that\n\n\[ \nG \\sim {\\Lambda }^{t} \\times {G}_{\\text{tor }} \n\]\n\nOn the other hand,\n\n\[ \n{r}_{2}\left( {K}_{n}\right) = {r}_{2}{p}^{n} \n\]\n\nBy Theorem 5.2 we know that\n\n\[ \n\\operatorname{Gal}\left( {{\\Omega }_{n}/{K}_{n}}\\right) \\sim {\\mathbb{Z}}_{p}^{{r}_... | Yes |
Lemma 1. \( \Omega = {\Omega }_{A}{\Omega }_{B} \) . | Proof. By Kummer theory, \( \Omega \) is a composite of cyclic extensions. Let \( {K}_{\infty }\left( {\alpha }^{1/{p}^{m}}\right) \subset \Omega \) for some \( \alpha \in {K}_{\infty } \) . Then \( \alpha \in {K}_{n} \) for some \( n \) . We take \( n \geq m \) and also such that\n\n\[ \n{K}_{n}\left( {\alpha }^{1/{p}... | Yes |
Theorem 2.1. The Galois groups \( \operatorname{Gal}\left( {{\Omega }_{A}/{\Omega }_{E}}\right) \) and \( \operatorname{Gal}\left( {{\Omega }_{B}/{\Omega }_{E}}\right) \) are \( \Lambda \) -torsion modules. So \( \operatorname{Gal}\left( {\Omega /{\Omega }_{E}}\right) \) is a \( \Lambda \) -torsion module. | Proof. We shall analyze each Galois group separately, and get a closer view of its structure.\n\nThe extension \( {\Omega }_{A}/{\Omega }_{E} \).\n\nLet \( {G}_{A/E} = \operatorname{Gal}\left( {{\Omega }_{A}/{\Omega }_{E}}\right) \). For now abbreviate \( {G}_{A/E} = G \), and let\n\n\[ {G}_{n} = \operatorname{Gal}\lef... | Yes |
Theorem 2.3. Assume that there is only one prime in \( {K}_{\infty } \) lying above \( p \) . Then\n\n\[ \n{\Omega }_{E} = {\Omega }_{{E}_{p}} = {\Omega }_{B} \n\] \n\nwhere \( {E}_{p} \) is the group of p-units in \( {K}_{\infty } \), and \n\n\[ \n{\Omega }_{{E}_{p}} = \Omega \left( {E}_{p}^{1/{p}^{\infty }}\right) \n... | Proof. We consider the diagram of fields:\n\n\n\nThe ideal above \( p \) in \( {\mathbf{Q}}_{n} \) is principal, say generated by the element \( {\lambda }_{n} = \) \( 1 - {\zeta }_{n} \) . The degree \( \left\lbrack... | Yes |
Theorem 3.1. The association of \( {\left( \mathbf{Q}/\mathbf{Z}\right) }^{\left( p\right) } \rightarrow V \) given by\n\n\[ a \mapsto {g}_{a}\text{ (mod roots of unity) } \]\n\nsatisfies the distribution relations except at 0 . | The theorem means that for \( a \neq 0 \) we have\n\n\[ \mathop{\prod }\limits_{{{pb} = a}}{g}_{b} = {g}_{a} \]\n\nand is obvious in the light of \( \mathbf{{CU}}2 \) . | No |
Theorem 3.2. The group \( {\mathcal{G}}_{n}^{ + } \) operates simply transitively on the primitive elements of \( {V}_{n} \), and the induced homomorphism \[ \mathbb{Z}\left\lbrack {\mathcal{G}}_{n}^{ + }\right\rbrack \rightarrow {V}_{n}\text{such that}{\sigma }_{c} \mapsto {g}_{c/{p}^{n}} \] is an isomorphism. | Proof. The homomorphism is obviously surjective. It is injective because \( \mathbf{Z}\left\lbrack {\mathcal{G}}_{n}^{ + }\right\rbrack \) is torsion free, and the ranks of the two groups are equal. This proves the theorem. | Yes |
Theorem 3.3. The factor group \( {V}_{m}/{V}_{n} \) for \( m \geq n \) has no torsion. | Proof. The embedding of \( {V}_{n} \) into \( {V}_{m} \) corresponds to the embedding of group rings\n\n\[ \mathbf{Z}\left\lbrack {\mathcal{G}}_{n}^{ + }\right\rbrack \rightarrow \mathbf{Z}\left\lbrack {\mathcal{G}}_{m}^{ + }\right\rbrack \]\n\nwhich sends an element \( {\sigma }_{c} \) on the element \( \sum {\sigma }... | Yes |
Theorem 3.4. Let \( p \) be odd, and let \( c \) be a primitive root \( {\;\operatorname{mod}\;{p}^{2}} \). Then \( {E}_{n} \) is generated over \( \mathbf{Z}{\left\lbrack {\mathcal{G}}_{n}\right\rbrack }_{0} \) by the element \[ v = {\sigma }_{c}\pi /\pi = \frac{{\zeta }^{c} - 1}{\zeta - 1} \] | Proof. We write an element \( \alpha \) of degree 0 in the form \[ \alpha = \sum k\left( b\right) \left( {{\sigma }_{b} - 1}\right) \] and observe that \( {\sigma }_{b} - 1 \) is divisible in the integral group ring by \( {\sigma }_{c} - 1 \) because \( {\sigma }_{c} \) is a generator of the cyclic group \( {\mathcal{G... | No |
Theorem 3.5. Let \( c \) be a generator of \( 1 + 4{\mathbf{Z}}_{2} \) if \( p = 2 \), and a primitive root \( {\;\operatorname{mod}\;{p}^{2}} \) if \( p > 2 \) . Let\n\n\[ \n{v}_{n} = \text{ image of }\frac{{\zeta }^{c} - 1}{\zeta - 1}\text{ in }{V}_{n} \]\n\nand let \( {V}_{n}^{0} \) be the subgroup of \( {V}_{n} \) ... | Proof. Clear. | No |
Theorem 3.6. This distribution is the universal even ordinary distribution with value 0 at 0, and values into abelian groups on which multiplication by 2 is invertible. | Proof. On \( \left( {1/N}\right) \mathbf{Z}/\mathbf{Z} \) the group generated by the image of \( h \) has rank\n\n\[ \frac{1}{2}\left| {\mathbf{Z}{\left( N\right) }^{ * }}\right| - 1 \]\n\nwhich according to Kubert's Theorem 9.1(iii) of Chapter 2 is the maximal possible rank (the value 0 at 0 gives rise to the -1 ). Th... | No |
Theorem 4.1. Assuming the Vandiver conjecture, we have \( G = {G}^{ - } \), and \( G \) is a 1-dimensional free module over \( {R}^{ - } \) . | Proof. By the Vandiver conjecture and Theorem 3.3 we have\n\n\[ \n{E}_{p, n} \cap {K}_{\infty }^{*{p}^{n}} = {E}_{p, n}^{{p}^{n}}{\mu }^{\left( p\right) }.\n\] \n\nLet us abbreviate for simplicity\n\n\[ \n{V}_{n}^{1/{p}^{n}} = {E}_{p, n}^{1/{p}^{n}}/{E}_{p, n} \cap {K}_{\infty }^{ * }.\n\] \n\nThen the Kummer theory pa... | Yes |
Theorem 4.2. Let \( {C}_{n} = C{l}^{\left( p\right) }\left( {K}_{n}\right) \) be the p-primary part of the ideal class group of \( {K}_{n} \) and let \( C = \) projective limit of the \( {C}_{n} \) under the norm map. Under the Vandiver conjecture, we have \( C = {C}^{ - } \), and \( {C}^{ - } \) is cyclic as a \( \Lam... | Proof. The field diagram (once the theorem is proved) is as follows. \[ G\left\{ \begin{array}{l} \Omega \\ \mid \\ {\Omega }^{\mathrm{{nr}}} \\ \mid \\ {K}_{\infty } \end{array}\right\} {G}_{C} \] What we have to do is to show that the maximal \( p \) -abelian unramified extension of \( {K}_{\infty } \) is in fact con... | Yes |
Theorem 4.3. (i) For \( m \geq n \) we have an injection\n\n\[ \operatorname{Ker}\left( {{C}_{n} \rightarrow {C}_{m}}\right) \rightarrow {H}^{1}\left( {{\Gamma }_{m, n},{E}_{m}}\right) \]\n\n(ii) Under the Vandiver conjecture we have \( {H}^{1}\left( {{\Gamma }_{m, n},{E}_{m}}\right) = 0 \), so\n\n\[ {C}_{n} \rightarro... | Proof. Let \( \mathfrak{a} \) be an ideal representing an element of \( {C}_{n} \), becoming principal in \( {K}_{m} \), say \( \mathfrak{a} = \left( \alpha \right) \) with \( \alpha \in {K}_{m} \) . For any element \( \sigma \in {\Gamma }_{m, n} \) we have \( \sigma \mathfrak{a} = \mathfrak{a} \) . Hence \( {\sigma \a... | Yes |
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