Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Lemma 2. Let \( 1 \leq k \leq p - 1 \) . Then\n\n\[ \n{h}_{0}^{\left( k\right) } \sim \mathop{\prod }\limits_{{\theta \neq 1}}\frac{1}{k}{B}_{k,\theta {\omega }^{-k}} \n\] | Proof. The case when \( \theta = 1 \) combined with Lemma 1 shows that the factor \( p \) in the definition of \( {h}_{0}^{\left( k\right) } \) cancels the pole of order 1 at \( p \) of the single term with \( \theta = 1 \) in the product. What remains is the desired expression. | No |
Lemma 3. Let \( 1 \leq k \leq p - 1 \) . Then\n\n\[ \n{h}_{n}^{\left( k\right) } \sim {h}_{0}^{\left( k\right) }\mathop{\prod }\limits_{{\theta \neq 1}}\mathop{\prod }\limits_{{\psi \neq 1}}\frac{1}{k}{B}_{k,{\psi \theta }{\omega }^{-k}}\n\] | Proof. Write \( \chi = {\theta \psi } \) . Then\n\n\[ \n\left( {1 - {\chi }_{k}\left( p\right) {p}^{k - 1}}\right) \frac{1}{k}{B}_{k,\chi {\omega }^{-k}} = \frac{1}{1 - \chi \left( c\right) \langle c{\rangle }_{p}^{k}}{\int }_{{Z}^{ * }}\chi \left( a\right) \langle a{\rangle }_{p}^{k}{a}_{p}^{-1}d{E}_{1, c}\left( a\rig... | Yes |
Lemma 4. If \( \theta = 1 \) and \( \psi \neq 1 \) then\n\n\[ \frac{1}{k}{B}_{k,\psi {\omega }^{-k}} \sim \frac{1}{\zeta - 1} \]\n\nwhere \( \gamma = 1 + p \) and \( \zeta = \psi \left( \gamma \right) \) . Furthermore\n\n\[ {p}^{n}\mathop{\prod }\limits_{{\psi \neq 1}}\frac{1}{k}{B}_{k,\psi {\omega }^{-k}} \sim 1 \] | Proof. We also take \( c = 1 + p \) . Then\n\n\[ 1 - \psi \left( c\right) \langle c{\rangle }^{k} \sim 1 - \zeta \;\text{ and }\;\mathop{\prod }\limits_{\substack{{\zeta {p}^{n} = 1} \\ {\zeta \neq 1} }}\left( {1 - \zeta }\right) = {p}^{n}. \]\n\nWe note that \( \chi \left( c\right) = \psi \left( c\right) \), and we ob... | Yes |
Theorem 3.1. There is a constant \( {c}_{k} \) such that for all \( n \) sufficiently large, we have\n\n\[ \n{\operatorname{ord}}_{p}{h}_{n}^{\left( k\right) } = \lambda \left( k\right) n + {c}_{k} \n\] | This is merely a special case of Corollary 3 of Theorem 1.2, applied to the Bernoulli distributions, as discussed in \( §2 \) . | Yes |
Theorem 3.2. Let \( {h}_{n} \) be the class number of \( \mathbf{Q}\left( {\mathbf{\mu }}_{{p}^{n + 1}}\right) \) . Then there is a constant \( c \) such that for all \( n \) sufficiently large, we have\n\n\[{\operatorname{ord}}_{p}{h}_{n}^{ - } = \lambda \left( 1\right) n + c.\] | Proof. The classical class number formula asserts that\n\n\[{h}_{n}^{ - } = 2{p}^{n + 1}\mathop{\prod }\limits_{{\chi \text{ odd }}} - \frac{1}{2}{B}_{1,\chi }\]\n\nso that we can apply Theorem 3.1 with \( k = 1 \) to conclude the proof. | No |
Theorem 3.3. Let \( K \) be a cyclotomic extension of the rationals (i.e. a subfield of a cyclotomic field). Let \( {K}_{\infty } \) be the cyclotomic \( {\mathbf{Z}}_{p} \) -extension of \( K \), and let \( {h}_{n} \) be the class number of \( {K}_{n} \) . Then there exists a constant \( {c}^{\prime } \) such that for... | Proof. It is an easy exercise from the class number formula of Chapter 3 to show that the minus part of the class number differs from the product giving \( {h}_{n}^{\left( 1\right) } \) only by a finite number of factors. Hence the same estimate holds as in Theorem 3.2. | No |
Theorem 4.2. Let \( K \) be a cyclotomic extension of \( \mathbf{Q} \). Let \( {K}_{\infty } \) be the cyclotomic \( {\mathbf{Z}}_{p} \)-extension of \( K \). Let \( l \) be a prime \( \neq p \). Then \[ {\operatorname{ord}}_{l}\left| {C\left( {K}_{n}\right) }\right| \] is bounded. | Proof. By lemma 2, \( §1 \) of Chapter 13 it suffices to prove the theorem when \( K = \mathbf{Q}\left( {\mathbf{\mu }}_{dq}\right) \) for some positive integer \( d \) not divisible by \( p \). Furthermore, we may also adjoin an \( l \)-th root of unity to the ground field, and thus assume without loss of generality t... | Yes |
Lemma 4.3. If \( \frac{1}{2}{B}_{1,{x\psi }} \equiv 0{\;\operatorname{mod}\;\mathfrak{L}} \) for infinitely many \( \psi \) (so of arbitrarily large conductor \( \left. {p}^{n + 1}\right) \), then there exist infinitely many \( n \) such that for such \( \psi \) and all \( \alpha \in {\mathbf{Z}}_{p}^{ * } \) we have\n... | Proof. Abbreviate \( T = {T}_{n, m} \) . From the irreducible equation of a \( p \) -power root of unity, we see at once that \( T\left( \varepsilon \right) = 0 \) for any \( p \) -power root of unity \( \varepsilon \) which does not lie in \( {F}_{m} \) . Thus if \( \beta \in {\mathbf{Z}}_{p}^{ * } \) and we write\n\n... | Yes |
Lemma 1.1. Let \( d \) , \( m \) be positive integers with \( d \) prime to \( p \) . For all \( n \) sufficiently large, there exists \( {\alpha }_{1},{\alpha }_{2} \in {\mathbf{Z}}_{p} \), with \( {\alpha }_{1},{\alpha }_{2} \equiv 1{\;\operatorname{mod}\;{p}^{m}} \), and an element \( {\eta }_{0} \in \mathcal{R} \) ... | The proof of this lemma will be given in the next sections. | No |
Lemma 2.1. Let \( \\left\\{ {{\\beta }_{1},\\ldots ,{\\beta }_{r}}\\right\\} \) be elements of \( {\\mathbf{Z}}_{p} \) which are linearly independent over the rationals. Then for almost all \( \\alpha \\in {\\dot{\\mathbf{Z}}}_{p} \) the family \( \\left\\{ {\\alpha {\\beta }_{1},\\ldots ,\\alpha {\\beta }_{r}}\\right\... | Proof. By Weyl’s criterion, we must show that for every \( r \) -tuple of integers \( \\left( {{a}_{1},\\ldots ,{a}_{r}}\\right) \) not all 0, and almost all \( \\alpha \\), we have\n\n\[ \n\\mathop{\\lim }\\limits_{{N \\rightarrow \\infty }}\\frac{1}{N}\\mathop{\\sum }\\limits_{{n = 1}}^{N}e\\left( {\\mathop{\\sum }\\... | Yes |
Lemma 3.1. Let \( \\left\\{ {{\\beta }_{1},\\ldots ,{\\beta }_{r}}\\right\\} \) be p-adic integers, linearly independent over the rationals. Suppose we are given \( \\varepsilon > 0 \) ; an integer \( m > 0 \) ; an integer \( d \) with \( \\left( {p, d}\\right) = 1 \) ; real numbers \( {x}_{1},\\ldots ,{x}_{r} \\in \\l... | Proof. We use vector notation and put \( x = \\left( {{x}_{1},\\ldots ,{x}_{r}}\\right) ,\\beta = \\left( {{\\beta }_{1},\\ldots ,{\\beta }_{r}}\\right) \) . We let \( \\parallel \\parallel \) be the sup norm on the torus \( {\\mathbf{R}}^{r}/{\\mathbf{Z}}^{r} \) . For each \( n \) we define the residue\n\n\[ \n{\\oper... | Yes |
Lemma 4.1. Let \( A = \left( {a}_{ji}\right) \) be a real matrix such that no row equals + or - another, and in every row there are at least two non-zero elements. After an admissible change of sign, we can find a vector \( \left( {{x}_{1},\ldots ,{x}_{r}}\right) = {}^{t}{X}^{\left( r\right) } \in {\mathbf{R}}^{r} \) s... | Proof. In \( R \) -space, we consider the conditions:\n\n\[ \n{x}_{j} = 0\text{for some}j = 1,\ldots, R\text{;}\n\]\n\n\[ \n{x}_{j} - {x}_{{j}^{\prime }} = 0\text{or}{x}_{j} + {x}_{{j}^{\prime }} = 0\text{for some pair}\left( {j,{j}^{\prime }}\right) \text{with}j \neq {j}^{\prime }\text{.\n\]\n\nEach such condition def... | Yes |
Lemma 2.1. Assume that \( \mu \) is a measure on \( Z \) such that each \( {\mu }_{{\zeta }_{0}} \) is rational, for every \( d \) -th root of unity \( {\zeta }_{0} \) . Assume also that the functions\n\n\[ \n{R}_{{\zeta }_{0}}\left( {{\zeta }_{0}^{-1}T}\right) = R\left( T\right) \n\]\n\nare independent of \( {\zeta }_... | Proof. This is again immediate from the integral formula (12) of \( §1 \) . | No |
Let \( c \) be a positive integer prime to \( {dp} \) . Then \( {E}_{1, c} \) on \( \mathbf{Z}\left( d\right) \times {\mathbf{Z}}_{p} \) has an associated rational function, equal to \( {R}_{1, c}\left( T\right) \) above. | To extend the above result from \( {\mathbf{Z}}_{p} \) to \( \mathbf{Z}\left( d\right) \times {\mathbf{Z}}_{p} \), it suffices to prove that for every root of unity \( \zeta \in {\mathbf{\mu }}_{N} \), and \( \dot{N} = d{p}^{n} \), we have\n\n\[{\int }_{Z}{\zeta }^{x}d{E}_{1, c}\left( x\right) = {R}_{1, c}\left( \zeta ... | Yes |
Proposition 3.2. Let \( \chi \) have conductor \( N = d{p}^{n} \) with \( n \geq 0 \) . Then \( \chi {E}_{1, c} \) has an associated rational function \( {R}_{\chi, c} \) given by the formula\n\n\[ \n{R}_{\chi, c}\left( T\right) = {G}_{\chi }\left( T\right) - {c\chi }\left( c\right) {G}_{\chi }\left( {T}^{c}\right) , \... | Proof. Special case of \( \mathbf{R}\mathbf{6} \) . | Yes |
Theorem 3.3. Let \( \chi \) have conductor \( N = d{p}^{n}, n \geq 0 \) . Let \( \zeta \) be a primitive \( N \) -th root of unity. Then\n\n\[ \n{L}_{p}\left( {1,\chi }\right) = - \left( {1 - \frac{\chi \left( p\right) }{p}}\right) \frac{S\left( {\chi ,\zeta }\right) }{N}\mathop{\sum }\limits_{{a \in \mathbf{Z}{\left( ... | Proof. Let \( \xi \) range over the \( p \) -th roots of unity. By R 6,\n\n\[ \n- \left( {1 - \chi \left( c\right) }\right) {L}_{p}\left( {1,\chi }\right) = \mathbf{U}{H}_{\chi, c}\left( 1\right) .\n\]\n\nHence, following exactly the proof of Chapter 4, Theorem 3.6:\n\n\[ \n- \left( {1 - \chi \left( c\right) }\right) {... | Yes |
Lemma 1. Let \( F \subset K \) be number fields. Let \( {F}_{\infty } \) be a \( {\mathbf{Z}}_{p} \) -extension of \( F \). Then \[ m\left( {{F}_{\infty }/F}\right) \leq m\left( {K{F}_{\infty }/K}\right) \;\text{ and }\;\lambda \left( {{F}_{\infty }/F}\right) \leq \lambda \left( {K{F}_{\infty }/K}\right) . \] | Proof. The degree \( \left\lbrack {{K}_{n} : K{F}_{n}}\right\rbrack \) is bounded by a fixed power \( {p}^{r} \). By class field theory, we have \[ \left( {C\left( {F}_{n}\right) : {N}_{{K}_{n}/{F}_{n}}C\left( {K}_{n}\right) }\right) \leq \left\lbrack {K : F}\right\rbrack {p}^{r}. \] Hence \[ \left| {C\left( {K}_{n}\ri... | Yes |
Lemma 2. Let \( {F}_{\infty } \) be a \( {\mathbf{Z}}_{p} \) -extension of a number field \( F \) . Let \( K \) be a finite extension of \( F \) and let \( {K}_{\infty } = K{F}_{\infty } \) . Let \( l \) be any prime number. If \( {\operatorname{ord}}_{l}\left| {C\left( {K}_{n}\right) }\right| \) is bounded, then \( {\... | Proof. The argument using the norm index in the first part of the proof of Lemma 1.1 applies equally well to prove the result stated in Lemma 1.2. | No |
Lemma 1. \( {Q}_{K} = 1 \) or 2 . | The proof given for Theorem 4.1 of Chapter 3 applies here. In fact one verifies immediately that the map \( u \mapsto \bar{u}/u \) gives an injection\n\n\[ \n{E}_{K}/{W}_{K}{E}_{K}^{ + } \rightarrow {W}_{K}/{W}_{K}^{2} \n\] | No |
Theorem 2.2. Let \( K = \mathbf{Q}\left( {\mathbf{\mu }}_{p}\right) \), and let \( {h}_{p} \) be the class number of \( K \) . If \( {h}_{p}^{ - } \) is prime to \( p \), then \( {h}_{p}^{ + } \) is prime to \( p \), and so \( {h}_{p} \) is prime to \( p \) . | Proof. Obvious, since the p-rank is 0 , and the stronger of the two inequalities applies. | No |
Theorem 2.3. Let \( {K}_{\infty } \) be a \( {\mathbf{Z}}_{p} \) -extension of \( {K}_{0}\left( {p\text{odd}}\right) \), such that each \( {K}_{n} \) is a CM field. Then\n\n\[ \n{r}_{1}^{ + } \leq {r}_{1}^{ - } \n\]\n\nIn particular, if \( {m}^{ - } = 0 \) then \( {m}^{ + } = 0 \) . | Proof. Immediate from Theorem 2.1 and Lemma 3 of \( §1 \) . | No |
Proposition 2.4. Let \( K \) be a CM field, and \( C = C\left( K\right) \) . Then\n\n\[ \n{\operatorname{rank}}_{2}{C}_{{K}^{ + }} \leq {\operatorname{ord}}_{2}\left| {C}_{K}^{ - }\right| + 1 \n\] | Proof. Let \( r = {\operatorname{rank}}_{2}{C}_{{K}^{ + }} \) . Then by Lemma 2,\n\n\[ \n{2}^{r} = \left( {{C}_{{K}^{ + }} : {C}_{{K}^{ + }}^{2}}\right) = \left( {N{C}_{K} : N{C}^{ + }}\right) , \n\]\n\nwhere \( {C}^{ + } \) denotes the image of \( C\left( {K}^{ + }\right) \) in \( C\left( K\right) \) . This last index... | Yes |
Lemma 1. Let \( F \) be a number field and \( K \) a Galois extension of degree \( d \) . Let \( l \) be a prime number not dividing \( d \) . Let \( {C}_{F} \) denote the \( l \) -primary part of the ideal class group. Then the natural homomorphism \( {C}_{F} \rightarrow {C}_{K} \) is injective, the norm \( {N}_{K/F} ... | Proof. For the first assertion, suppose an ideal \( \mathfrak{a} \) of \( F \) becomes principal in \( K \), say \( \mathfrak{a} = \left( \alpha \right) \) . Taking the norm yields\n\n\[{\mathfrak{a}}^{d} = \left( {{N}_{K/F}\alpha }\right)\]\n\nand since \( d \) is prime to \( l \), it follows that \( \mathfrak{a} \) i... | Yes |
Lemma 2. Let \( p \) be a prime number \( \neq l \) . Let \( {K}_{n} \) be a cyclic extension of a number field \( {K}_{0} \), of degree \( {p}^{n} \) . Let \( f \) be the order of \( l{\;\operatorname{mod}\;{p}^{n}} \) . Let \( {C}_{v} \) be the l-primary part of the ideal class group in the subfield of degree \( {p}^... | Proof. Let \( G = \operatorname{Gal}\left( {{K}_{n}/{K}_{0}}\right) \) . We have a representation of \( G \) on the \( \mathbf{Z}\left( l\right) \) -vector space \( D = {D}_{n} \), and we first show that if \( D \neq 0 \), then the representation is faithful. If not, it is not injective on the unique cyclic subgroup of... | Yes |
Theorem 3.1. Let \( {K}_{\infty } \) be a \( {\mathbf{Z}}_{p} \) -extension of \( {K}_{0} \), and let \( {C}_{n} \) be the l-primary part of the ideal class group in \( {K}_{n} \), where \( l \) is a prime \( \neq p \) . If the l-ranks of \( {C}_{n} \) are bounded, then the orders of \( {C}_{n} \) are also bounded for ... | Proof. Otherwise, we must have \( {D}_{n} \neq 0 \) for arbitrarily large \( n \), and the order \( {f}_{n} \) of \( l{\;\operatorname{mod}\;{p}^{n}} \) satisfies\n\n\[ \n{f}_{n} \gg {p}^{n} \n\]\n\nThe preceding lemma would then imply that the ranks tend to infinity, a contradiction, which proves the theorem. | No |
Theorem 3.2. Let \( {K}_{\infty } \) be a \( {\mathbf{Z}}_{p} \) -extension of \( {K}_{0} \) . Assume that each \( {K}_{n} \) is a CM field. Let \( l \) be a prime number \( \neq p \), and assume that the \( l \) -th roots of unity are in \( {K}_{0} \) . If \( {\operatorname{ord}}_{l}\left| {C}_{n}^{ - }\right| \) is b... | Proof. By Theorem 2.1 the \( l \) -rank of \( {C}_{n}^{ + } \) is bounded, so the \( l \) -rank of \( {C}_{n} \) is bounded. Then Theorem 3.1 concludes the proof. | No |
Lemma 4.2. Let \( l \) be an integer \( \geq 2 \) . Let \( {K}_{d} \) be an extension of a number field \( K \) of degree d. Let \( {\mathfrak{q}}_{1},\ldots ,{\mathfrak{q}}_{t} \) be prime ideals of \( K \) which split completely in \( {K}_{d} \) . Let \( {K}^{\prime } \) be a cyclic extension of \( K \), of degree \(... | Proof. We have the diagram\n\n\n\nThe extensions \( {K}_{d} \) and \( {K}^{\prime } \) are linearly disjoint because of the way any one of the primes \( {\mathrm{q}}_{i} \) splits in them. Thus \( {K}_{d}^{\prime } \) ... | Yes |
Theorem 5.1. Let \( {K}_{\infty }/K \) be a \( {\mathbf{Z}}_{p} \) -extension. Let \( {\mathfrak{q}}_{1},\ldots ,{\mathfrak{q}}_{t} \) be prime ideals of \( K \) which split completely in \( {K}_{\infty } \) . Let \( {K}^{\prime } \) be a cyclic extension of \( K \) of degree \( l \), in which \( {\mathfrak{q}}_{1},\ld... | Proof. This is merely a special case of Lemma 4.2. 13. Divisibility of Ideal Class Numbers | No |
Theorem 5.2. Let \( K \) be a CM field. Then:\n\n(i) There exists a \( {\mathbf{Z}}_{p} \)-extension \( {K}_{\infty } \) of \( K \), Galois over \( {K}^{ + } \), such that if \( \Gamma = \operatorname{Gal}\left( {{K}_{\infty }/K}\right) \), then\n\n\[ \Gamma = {\Gamma }^{ - }\text{.}\]\n\n(ii) For any such extension, l... | Proof. Let \( {M}_{p}\left( K\right) \) be the maximal \( p \)-abelian \( p \)-ramified extension of \( K \). By class field theory, e.g. Chapter 5, §5, there is a quasi-isomorphism\n\n\[ \operatorname{Gal}\left( {{M}_{p}\left( K\right) /K}\right) \sim {U}_{p}/\bar{E} \]\n\nwhere \( {U}_{p} \) is the product of the loc... | Yes |
Theorem 6.1. Let \( p \) be an odd prime. Let \( u \) be a unit in \( \mathbf{Q}\left( {\mathbf{\mu }}_{p}\right) \) . Suppose there exists an integer \( a \in \mathbf{Z} \) such that \( u \equiv a\left( {\;\operatorname{mod}\;p}\right) \) . If \( p \) does not divide the class number \( {h}_{p} \), then \( u \) is a \... | Proof. Let \( K = \mathbf{Q}\left( {\mathbf{\mu }}_{p}\right) \) . By class field theory, it suffices to show that the extension \( K\left( {u}^{1/p}\right) \) is unramified, because the hypothesis then implies that \( K\left( {u}^{1/p}\right) = K \), so \( u \) is a \( p \) -th power in \( K \) . Raising \( u \) to th... | Yes |
Lemma 1.1. For any positive integers \( N, n, k \) we have\n\n\[ f\left( {n + {p}^{N}k}\right) \equiv f\left( n\right) \;\left( {\;\operatorname{mod}\;{p}^{N}}\right) . \] | Proof. Let \( G = \mathbf{Z}{\left( {p}^{N}\right) }^{ * } \) . Pairing an element and its inverse in \( G \) we find:\n\n\[ \mathop{\prod }\limits_{{j \in G}}j \equiv \left\{ \begin{array}{ll} - 1 & {\;\operatorname{mod}\;{p}^{N}}\text{ if }p\text{ is odd } \\ 1 & {\;\operatorname{mod}\;{p}^{N}}\text{ if }p = 2. \end{... | No |
Theorem 1.2. For any integer \( n \geq 1 \), we have\n\n\[ \Gamma \left( n\right) \Gamma \left( {1 - n}\right) = {\left( -1\right) }^{n + \left\lbrack {\left( {n - 1}\right) /p}\right\rbrack } \]\n\nwhere the bracket, as usual, is the greatest integer function. | Proof. The theorem is true for \( n = 1 \) by the above. We can then proceed inductively, using \( \Gamma \left( {1 - n}\right) = \left\{ \begin{array}{ll} n & \Gamma \left( {-n}\right) ,\text{ to get : } \end{array}\right. \)\n\n\[ \Gamma \left( 0\right) = \left( {\mathop{\prod }\limits_{\substack{{j = 1} \\ {\left( {... | Yes |
Theorem 1.3. If \( p \neq 2 \) then\n\n\[ \Gamma \left( x\right) \Gamma \left( {1 - x}\right) = {\left( -1\right) }^{R\left( x\right) } \]\n\nIf \( p = 2 \), then\n\n\[ \Gamma \left( x\right) \Gamma \left( {1 - x}\right) = \varepsilon \left( x\right) \]\n\nwhere\n\n\[ \varepsilon \left( x\right) = \left\{ \begin{array}... | Proof. By continuity, it suffices to prove the theorem when \( x \) is an integer \( n \geq 1 \) . Write\n\n\[ n = {a}_{0} + {a}_{1}p + {a}_{2}{p}^{2} + \cdots + {\dot{a}}_{r}{p}^{r} \]\n\nwith \( {a}_{0} \in \{ 1,\ldots, p\} \) and \( {a}_{i} \in \{ 0,\ldots, p - 1\} \) for \( i \geq 1 \) . Then\n\n\[ \left\lbrack \fr... | Yes |
Theorem 1.4 (Distribution relation). Let \( N \) be an integer \( \geq 2 \) and prime to p. Then\n\n\[ \mathop{\prod }\limits_{{i = 0}}^{{N - 1}}\Gamma \left( \frac{x + i}{N}\right) = \Gamma \left( x\right) \mathop{\prod }\limits_{{i = 1}}^{{N - 1}}\Gamma \left( \frac{i}{N}\right) {g}_{N}{\left( x\right) }^{-1}, \]\nwh... | Proof. Define \( {g}_{N}\left( x\right) \) by using the relation to be proved, so\n\n\[ {g}_{N}\left( x\right) = \mathop{\prod }\limits_{{i = 1}}^{{N - 1}}\Gamma \left( \frac{i}{N}\right) \mathop{\prod }\limits_{{i = 0}}^{{N - 1}}\Gamma {\left( \frac{x + i}{N}\right) }^{-1}\Gamma \left( x\right) . \]\n\nBy continuity, ... | Yes |
Proposition 1.5.\n\n\\[ \n\\mathop{\\prod }\\limits_{{i = 1}}^{{N - 1}}\\Gamma \\left( \\frac{i}{N}\\right) = \\left\\{ \\begin{array}{ll} \\pm 1, & \\text{ if }N\\text{ is odd } \\\\ \\pm 1, \\pm \\sqrt{-1} & \\text{ if }N\\text{ is even. } \\end{array}\\right. \\]\n | Proof. Suppose \\( N \\) is odd. We write\n\n\\[ \n\\mathop{\\prod }\\limits_{{i = 1}}^{{N - 1}}\\Gamma \\left( \\frac{i}{N}\\right) = \\mathop{\\prod }\\limits_{{i = 1}}^{{\\left( {N - 1}\\right) /2}}\\Gamma \\left( \\frac{i}{N}\\right) \\Gamma \\left( {1 - \\frac{i}{N}}\\right) \n\\]\n\nand each factor on the right i... | Yes |
Theorem 2.1. We have \( \mathrm{{AH}}\left( X\right) \in {\mathbf{Z}}_{p}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) . | Proof. The left-hand side obviously implies the right-hand side in the equivalence to be proved. So assume the right-hand side. If \( R \) is any ring, and\n\n\[ f\left( X\right) \in 1 + {XR}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \]\n\nthen a simple recursion shows that \( f\left( X\right) \) has an inf... | Yes |
(i) We have ord \( {e}_{n} \geq n\left( {p - 1}\right) /{p}^{2} \), and \( {e}_{n} \) is p-integral.\n\n(ii) If \( n \geq 2 \) then \( {\operatorname{ord}}_{\pi }{e}_{n} \geq 2 \) . | Proof. By (1), letting \( X \mapsto {\pi X} \), we find at once that\n\n\[ \text{ord}{e}_{n} \geq n\left( \frac{p - 1}{{p}^{2}}\right) \text{.}\]\n\nTo prove (ii), i.e. to prove that\n\n\[ \text{ord}{e}_{n} \geq \frac{2}{p - 1}\text{for}n \geq 2\text{,}\]\n\n it suffices to show that\n\n\[ \text{ord}{e}_{n} > \frac{1}{... | Yes |
For each element \( \pi \in {\mathbf{C}}_{p} \) such that \( {\pi }^{p - 1} = - p \), there exists a unique \( p \) -th root of unity \( {\zeta }_{\pi } \) such that\n\n\[ \n{\zeta }_{\pi } \equiv 1 + \pi \left( {\;\operatorname{mod}\;{\pi }^{2}}\right) \n\]\n\nThe correspondence \( \pi \mapsto {\zeta }_{\pi } \) estab... | Proof. If \( \zeta \) is a non-trivial \( p \) -th root of unity, then \( {\left( \zeta - 1\right) }^{p - 1} \sim p \) (where \( x \sim y \) means that \( x/y \) is a \( p \) -unit). If \( {\zeta }_{1},{\zeta }_{2} \) are two primitive \( p \) -th roots of unity which are both \( \equiv 1 + \pi \left( {\;\operatorname{... | Yes |
Theorem 3.2 (Dwork). We have \( {E}_{\pi }\left( 1\right) = {\zeta }_{\pi } \), so \( {E}_{\pi }\left( 1\right) \) is the unique \( p \) -th root of unity \( \equiv 1 + \pi {\;\operatorname{mod}\;{\pi }^{2}} \) . For any \( c \in {\mathbf{Z}}_{p} \) such that \( {c}^{p} = c \), we have\n\n\[ \n{E}_{\pi }\left( c\right)... | Proof. First we observe that \( {E}_{\pi }\left( 1\right) \) is defined by substituting 1 for \( X \) in the power series for \( {E}_{\pi }\left( X\right) \), which converges in light of the lower bound for the orders of the coefficients. Then note that\n\n\[ \n{E}_{\pi }{\left( X\right) }^{p} = \exp \left( {{p\pi X} -... | Yes |
Theorem 3.3 (Dwork). We have\n\n\[ \n{E}_{\pi, q}\left( \zeta \right) = {\psi }_{\pi, q}\left( {\zeta {\;\operatorname{mod}\;p}}\right) ,\n\]\n\nand this is the unique \( p \) -th root of unity \( \equiv 1 + T\left( \zeta \right) \pi {\;\operatorname{mod}\;{\pi }^{2}} \) . | Proof. From Lemma 2.2(ii), we know that \( {E}_{\pi }\left( \zeta \right) \equiv 1 + {\zeta \pi }{\;\operatorname{mod}\;{\pi }^{2}} \) . Hence\n\n\[ \n{E}_{\pi, q}\left( \zeta \right) = {E}_{\pi }\left( \zeta \right) {E}_{\pi }\left( {\zeta }^{2}\right) \cdots {E}_{\pi }\left( {\zeta }^{{p}^{r - 1}}\right)\n\]\n\n\[ \n... | Yes |
Lemma 1.1. Let \( \alpha \in {\mathbf{Z}}_{p} \) and suppose that \( \alpha \) is not an integer \( \leq 0 \) . Let\n\n\[ D = x\frac{d}{dx} - {\pi x} + \alpha . \]\n\n(i) We have a direct sum decomposition\n\n\[ L\left( {0 + }\right) = K \oplus {DL}\left( {0 + }\right) \]\n\n(ii) If \( \delta \geq 1/\left( {p - 1}\righ... | Proof. We wish first to write an arbitrary series \( \varphi \left( x\right) \) as some constant plus \( {Dg} \) for some \( g \) . We first solve this problem for powers of \( x \) . We have for integers \( m \geq 0 \) :\n\n\[ D{x}^{m} = \left( {m + \alpha }\right) {x}^{m} - \pi {x}^{m + 1}, \]\n\nso that\n\n\[ {x}^{m... | No |
Lemma 1.2. Let \( \alpha \in {\mathbf{Z}}_{p} \), and let \( n \) be a positive integer. Then\n\n\[ \text{ord}\alpha \left( {\alpha - 1}\right) \cdots \left( {\alpha - n + 1}\right) \geq - \frac{s\left( n\right) }{p - 1} \geq - 1 - {\log }_{p}n\text{.} \] | Proof. Obvious, from\n\n\[ \frac{\alpha \left( {\alpha - 1}\right) \cdots \left( {\alpha - n + 1}\right) }{{\pi }^{n}} = \frac{n!}{{\pi }^{n}}\left( \begin{array}{l} \alpha \\ n \end{array}\right) \]\n\nfrom the fact that the binomial coefficient is \( p \) -integral, and from\n\n\[ \text{ord}n! = \frac{n - s\left( n\r... | Yes |
Lemma 1.3. The equation \( {Dg} = 1 \) has a unique solution\n\n\[ g\left( x\right) = \sum {b}_{n}{x}^{n} \in K\left\lbrack \left\lbrack x\right\rbrack \right\rbrack \]\n\nand the coefficients satisfy\n\n\[ \text{ord}{b}_{n} \leq 1 - \frac{1}{p - 1} + {\log }_{p}\left( {n + 1}\right) \text{.} \] | Proof. Write\n\n\[ 1 = \left( {x\frac{d}{dx} + \alpha - {\pi x}}\right) \left( {\mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n}{x}^{n}}\right) . \]\n\nThen\n\n\[ 1 + \sum \pi {b}_{n}{x}^{n + 1} = \sum \left( {n{b}_{n} + \alpha {b}_{n}}\right) {x}^{n} \]\n\nso that\n\n\[ 1 = \alpha {b}_{0}\text{ and }\pi {b}_{n - 1} = ... | Yes |
Theorem 1.4. We have isomorphisms for \( \delta \geq 1/\left( {p - 1}\right) \) :\n\n\[ \nL\left( \delta \right) /{D}_{j}L\left( \delta \right) \approx L\left( {0 + }\right) /{D}_{j}L\left( {0 + }\right) \approx {H}_{j,\pi }, \n\]\n\nand these spaces are 1-dimensional. | Proof. Immediate from Lemma 1.1. | No |
Lemma 2.1. Under the above conditions on \( \delta \) and \( {\delta }^{\prime } \), \[ {A}_{j, q} : L\left( \delta \right) \rightarrow L\left( \delta \right) \] maps \( L\left( \delta \right) \) into itself, and induces a homomorphism \[ {\bar{A}}_{j, q} : L\left( \delta \right) /{D}_{j}L\left( \delta \right) \rightar... | Proof. This is a special case of the preceding discussion, except that the negative power \( {x}^{-a} \) occurs inside the operator \( {A}_{j} \) . However, since \( a < q - 1 \) , we have already seen that \( {\Psi }_{q} \) annihilates such negative powers, so \( {A}_{j, q} \) maps \( L\left( \delta \right) \) into it... | Yes |
Theorem 3.1. Let \( g\\left( x\\right) \\in R\\langle x{\\rangle }_{p} \) . Let \( 1 \\leq a \\leq q - 1 \) . Then\n\n\\[ \n\\left( {q - 1}\\right) \\operatorname{tr}\\left( {{\\Psi }_{q} \\circ {x}^{-a}g\\left( x\\right) }\\right) = \\mathop{\\sum }\\limits_{\\zeta }{\\zeta }^{-a}g\\left( \\zeta \\right)\n\\]\n\nwhere... | Proof. Write\n\n\\[ \nh\\left( x\\right) = {x}^{-a}g\\left( x\\right) = \\sum {a}_{n}{x}^{n}.\n\\]\n\nThen \( {a}_{n} \\rightarrow 0 \) ( \( p \) -adically). Let\n\n\\[ \n{\\Psi }_{q}\\left( {{x}^{i}h\\left( x\\right) }\\right) = \\sum {a}_{ij}{x}^{j}\n\\]\n\nThen \( {a}_{ij} \) is the coefficient of \( {x}^{qj} \) in ... | Yes |
Theorem 3.3. Let \( 1 \leq j \leq N - 1 \), and \( a = j\left( {q - 1}\right) /N \), where \( N \) divides \( q - 1 \) . Let \( \delta \) be a rational number such that\n\n\[ \n\frac{1}{p - 1} \leq \delta < \frac{p - 1}{p}\n\]\n\nLet \( W = L\left( \delta \right) /{D}_{j}L\left( \delta \right) \) . Let \( {\lambda }_{j... | Proof. By FR 4 of \( §2 \), we have a commutative diagram with exact rows:\n\n\n\nBy the additivity of the trace (cf. Proposition 5.6 below), we have\n\n\[ \n\operatorname{tr}{A}_{j} = \operatorname{tr}q{A}_{j} + \op... | Yes |
Theorem 4.2. Let \( {\lambda }_{q} \) be the eigenvalue of Frobenius \( {\Phi }_{q}^{ * } \) on \( {H}_{j,\pi } \) . Then\n\n\[ \n{\lambda }_{q} = {\tau }_{q}\left( {{\omega }_{q}^{a},{\psi }_{\pi, q}}\right) \n\] | Proof. This follows from Theorem 3.3, the remark at the end of §2, FR 3, and the fact that\n\n\[ \n\tau \left( {\chi ,\psi }\right) \tau \left( {{\chi }^{-1},{\psi }^{-1}}\right) = q. \n\] | Yes |
Theorem 4.3 (Gross-Koblitz formula). \[ {\tau }_{q}\left( {{\omega }_{q}^{a},{\psi }_{\pi, q}}\right) = {\left( -1\right) }^{r}q{\pi }^{-s\left( a\right) }\mathop{\prod }\limits_{{i = 0}}^{{r - 1}}{\Gamma }_{p}\left( {1 - \left\langle \frac{{p}^{i}a}{q - 1}\right\rangle }\right) . \] | Proof. We merely put together Theorem 4.2, and the expression for the eigenvalue obtained in Theorem 4.1. The only quantity remaining to be worked out is the power of \( \pi \) appearing on the right-hand side, and this follows from the following lemma. | No |
Lemma 4.4. Let \( a = j\left( {q - 1}\right) /N \) . Then\n\n\[ \mathop{\sum }\limits_{{i{\;\operatorname{mod}\;r}}}\frac{p{j}^{\left( i\right) } - {j}^{\left( i + 1\right) }}{N} = s\left( {j\frac{q - 1}{N}}\right) = s\left( a\right) . \] | Proof. This is essentially the easy Lemma 1 of Chapter 1, §2. Indeed,\n\n\[ \frac{{j}^{\left( i\right) }}{N} = \frac{{a}^{\left( i\right) }}{q - 1} \]\n\nwhere \( {a}^{\left( i\right) } \) is defined by\n\n\[ 1 \leq {a}^{\left( i\right) } < q - 1\text{ and }\frac{{a}^{\left( i\right) }}{q - 1} = \left\langle \frac{{p}^... | Yes |
Theorem 4.5 (Stickelberger). Let \( \gamma \left( a\right) = \prod {a}_{i} \) ! . Then \[ {\pi }^{-s\left( a\right) }\tau \left( {{\omega }_{q}^{-a},{\psi }_{\pi, q}}\right) \equiv \frac{1}{\gamma \left( a\right) }{\;\operatorname{mod}\;\mathfrak{P}}. \] | Proof. From the formula \( \tau \bar{\tau } = q \), and Theorem 4.3 we obtain (*) \[ \frac{\tau \left( {{\omega }_{q}^{-a},{\psi }_{\pi, q}^{-1}}\right) }{{\pi }^{s\left( a\right) }} = \frac{{\left( -1\right) }^{r}}{G\left( a\right) } \] where \[ G\left( a\right) = \mathop{\prod }\limits_{i}{\Gamma }_{p}\left( {1 - \le... | Yes |
Lemma 4.6.\n\n\\[ \n{\\Gamma }_{p}\\left( {1 - \\left\\langle \\frac{{p}^{r - i}a}{q - 1}\\right\\rangle }\\right) \\equiv {\\left( -1\\right) }^{1 + {a}_{i}}{a}_{i}!{\\;\\operatorname{mod}\\;p}.\n\\] \n | Proof. Note that\n\n\\[ \n{\\Gamma }_{p}\\left( {1 + {a}_{i}}\\right) = {\\left( -1\\right) }^{1 + {a}_{i}}{a}_{i}!\n\\] \n\nThis is true by definition if \\( {a}_{i} \\geq 2 \\) because \\( {a}_{i} \\leq p - 1 \\) . It is also true when \\( {a}_{i} = 1 \\) and \\( {a}_{i} = 0 \\) by the direct computations of Chapter ... | Yes |
Lemma 5.1. Let \( \mathfrak{m} \) be the maximal ideal of \( R \) . Let \( \bar{E} = {E}_{0}/\mathfrak{m}{E}_{0} \) . A family \( \left\{ {e}_{i}\right\} \) in \( E \) is a Banach basis if and only if all \( {e}_{i} \in {E}_{0} \), and their images \( {\bar{e}}_{i} \) in \( \bar{E} \) form an algebraic basis of \( \bar... | Proof. Suppose that \( \left\{ {e}_{i}\right\} \) is a Banach basis for \( E \) . Then first it is clear that every element of \( \bar{E} \) can be written as a linear combination of the \( {\bar{e}}_{i} \) with coefficients in \( k \), and such a combination is unique, as one sees by lifting back to \( {E}_{0} \) .\n\... | Yes |
Proposition 5.2. Every Banach space over \( K \) is isomorphic with a space \( C\left( {I, K}\right) \) . Equivalently, every Banach space has a Banach basis. | Proof. Immediate from the lemma, by lifting an algebraic basis from \( \bar{E} \) back to \( {E}_{0} \) . | No |
Proposition 5.3. Let \( E = C\\left( {I, K}\\right) \) and let \( \\left\\{ {e}_{i}\\right\\} \) be a Banach basis. Then we have an isomorphism\n\n\[ L\\left( {E, F}\\right) \rightarrow B\\left( {I, F}\\right) \]\n\ngiven by\n\n\[ u \mapsto {\\left( u{e}_{i}\\right) }_{i \in I}. \] | Proof. Let \( \\varphi \) be the map from \( L\\left( {E, F}\\right) \) to \( B\\left( {I, F}\\right) \) as given in the statement of the proposition. Conversely, if \( \\left( {f}_{i}\\right) \) is a bounded family in \( F \), define a map\n\n\[ \\psi : B\\left( {I, F}\\right) \rightarrow L\\left( {E, F}\\right) \]\n\... | Yes |
Proposition 5.4. Let \( F = C\left( {J, K}\right) \), with Banach basis \( \left( {f}_{j}\right) \). The map\n\n\[ u \mapsto \left( {u}_{j}\right) \]\n\n\ngives an isomorphism\n\n\[ {CC}\left( {E, F}\right) \approx C\left( {J,{E}^{ * }}\right) \]\n\n\nbetween \( {CC}\left( {E, F}\right) \) and the space of families \( ... | Proof. First suppose \( u \) has finite dimensional image, so without loss of generality, we may assume the image is one-dimensional. Then \( u\left( x\right) = v\left( x\right) f \) for some \( f \in F \), and \( v \) is a functional. Then \( {u}_{j} = {f}_{j}v \), where \( {f}_{j} \) is the \( j \) -th\ncoordinate of... | Yes |
Proposition 5.5. Let \( u \) be a completely continuous endomorphism of the Banach space E. Let \( {\left( {e}_{i}\right) }_{i \in I} \) be a Banach basis, and let \( A = \left( {a}_{ij}\right) \) be the matrix of \( u \) with respect to this basis. Let\n\n\[ \det \left( {I - {tu}}\right) = \mathop{\sum }\limits_{{m = ... | Proof. Suppose first that \( \left| u\right| \leq 1 \) . Then the formula for \( {c}_{m} \) is valid for each \( u\left( {\pi }^{n}\right) \), and so remains valid in the limit. The general case follows by using \( c \neq 0 \) such that \( \left| {cu}\right| \leq 1 \) .\n\nFor (ii), let \( S \) be a finite subset of \(... | Yes |
Corollary 1. Let \( u, v \in {CC}\left( {E, E}\right) \) . Then\n\n\[ \det \left( {\left( {I - {tu}}\right) \left( {I - {tv}}\right) }\right) = \det \left( {I - {tu}}\right) \det \left( {I - {tv}}\right) . \] | Proof. After multiplying \( u, v \) by appropriate scalars, we may assume without loss of generality that \( \left| u\right| \leq 1 \) and \( \left| v\right| \leq 1 \) . In that case, we view \( u, v \) as acting on \( {E}_{0} \), and reduce \( {\;\operatorname{mod}\;{\pi }^{n}} \), in which case the formula is true wh... | Yes |
Corollary 2. Let \( u \in {CC}\left( {E, F}\right) \) and \( v \in L\left( {F, E}\right) \) . Then\n\n\[ \det \left( {I - {tu} \circ v}\right) = \det \left( {I - {tv} \circ u}\right) . \] | Proof. By (iii) and (iv) we may assume that \( u, v \) have finite rank, in which case the assertion is standard by the linear algebra of finite dimensional spaces. | No |
Corollary 3. Assume that \( K \) has characteristic 0 . Then\n\n\[ \det \left( {I - {tu}}\right) = \exp \left( {-\mathop{\sum }\limits_{{m = 1}}^{\infty }\operatorname{tr}\left( {u}^{m}\right) \frac{{t}^{m}}{m!}}\right) . \] | Proof. The assertion is true if \( u \) has finite rank by ordinary linear algebra. Let \( \left\{ {u}_{n}\right\} \) be a sequence of such endomorphisms converging to \( u \) . Then both the right-hand side and left-hand side with \( u \) replaced by \( {u}_{n} \) converge to the corresponding expressions of the corol... | No |
Proposition 5.6. Let\n\n\\[ \n0 \rightarrow {E}^{\prime } \rightarrow E \rightarrow {E}^{\prime \prime } \rightarrow 0 \n\\]\n\nbe an exact sequence of Banach spaces, and let \( {u}^{\prime }, u,{u}^{\prime \prime } \) be continuous linear maps making the diagram commutative:\n\n![c5b6c766-fed7-488a-9a60-a8beceaf977a_1... | Proof. We may view \( {E}^{\prime } \) as a subspace of \( E \), and \( {u}^{\prime } \) as the restriction of \( u \) to \( {E}^{\prime } \) . Thus \( {u}^{\prime \prime } \) is the map induced on \( {E}^{\prime \prime } \) as factor space \( E/{E}^{\prime } \) . By the Corollary of Proposition 5.2 there exists a Bana... | Yes |
Lemma 1.1. The ring \( R\langle x{\rangle }_{p} \) is integrally closed. | Proof. An element \( \varphi \left( x\right) \) of \( R\langle x{\rangle }_{p} \) not divisible by \( \pi \) can be written\n\n\[ \varphi \left( x\right) = {b}_{0} + \cdots + {b}_{d}{x}^{d} + \cdots \]\n\nsuch that \( {b}_{d} \) is a unit, and \( {b}_{n} \equiv 0{\;\operatorname{mod}\;\pi } \) for \( n \geq d + 1 \) . ... | Yes |
Lemma 1.2. The ring \( R\langle x\rangle \) is algebraically closed in \( R\langle x{\rangle }_{p} \), and therefore it is integrally closed (in its quotient field). | Proof. As pointed out to me by Dwork, this lemma can be viewed as a special case of a result in Dwork-Robba [Dw-Ro], Theorem 3.1.6. The proof given below was derived in collaboration with Dwork.\n\nLet \( A = R\langle x\rangle \) and \( {A}_{p} = R\langle x{\rangle }_{p} \) . Let \( y \in {A}_{p} \) be algebraic over \... | Yes |
Lemma 1.2.1. For any rational function \( H \) holomorphic on \( B \), we have\n\n\[ \parallel H\parallel \leq \parallel H{\parallel }_{B} \] | Proof. We factor the rational function into a product of a constant factor and linear factors of type\n\n\[ x - a\text{ and }\frac{1}{x - a},\text{ with }a \in {\mathbf{C}}_{p}. \]\n\nThe Gauss norm of \( x - a \) is \( \max \left( {1,\left| a\right| }\right) \) . We pick \( x \in B \) to be a unit which is not congrue... | No |
Lemma 1.2.2. Given a rational function \( H\left( x\right) \), with \( \parallel H\parallel < 1 \), there exists a set \( B\left( {t;\alpha, r}\right) \) with \( t > 1 \) and \( {r}_{j} < 1 \), such that\n\n\[ \parallel H{\parallel }_{B} < 1\text{.} \] | Proof. Factor\n\n\[ H\left( x\right) = c\prod {\left( 1 - {a}_{i}x\right) }^{{m}_{i}}\prod {\left( x - {b}_{j}\right) }^{{n}_{j}} \]\n\nwhere \( \left| {a}_{i}\right| < 1 \) and \( \left| {b}_{j}\right| \leq 1 \) . Then \( \left| c\right| < 1 \) because \( \parallel H\parallel < 1 \) . Let \( N \) be the number of line... | Yes |
Lemma 1.3. Let \( w \in \mathcal{A} \) and assume \( w \equiv 0{\;\operatorname{mod}\;\pi } \) . Then the geometric series\n\n\[1 + w + {w}^{2} + \cdots\]\n\nconverges to an inverse of \( 1 - w \) in \( \mathcal{A} \) . | Proof. For convenience, assume that the lifted polynomial \( f\left( Y\right) \) has coefficients in \( R\left\lbrack x\right\rbrack \) . This is all that we need in the applications, and the proof is slightly easier in this case. We write\n\n\[w = \pi \mathop{\sum }\limits_{{i = 0}}^{{d - 1}}{g}_{i}\left( x\right) {y}... | Yes |
Lemma 1.4. Assume that \( {f}_{0} \) is special. Then \( \mathcal{A} \) is integrally closed in \( R\langle x{\rangle }_{p}\left\lbrack y\right\rbrack \), and also in the quotient field \( K \) of \( \mathcal{A} \) . | Proof. The powers \( {y}^{j}\left( {j = 0,\ldots, d - 1}\right) \) form a basis of \( K \) over the quotient field of \( R\langle x\rangle \) . The dual basis with respect to the trace is contained in \( {f}^{\prime }{\left( y\right) }^{-1}\mathcal{A} \), and hence in \( \mathcal{A} \) . If an element \( z \in R\langle... | Yes |
To each root \( \bar{z} \) of \( {f}_{0} \) in \( k\left\lbrack {x,{y}_{0}}\right\rbrack \) there exists a unique root \( z \) of \( f \) in \( \mathcal{A} \) such that \( z{\;\operatorname{mod}\;\pi } = \bar{z} \). If in addition \( k\left( {x,{y}_{0}}\right) \) is Galois over \( k\left( x\right) \) with group \( {G}_... | Proof. Let \( {z}_{1} \in \mathcal{A} \) reduce to \( \bar{z}{\;\operatorname{mod}\;\pi } \) . Such \( {z}_{1} \) exists because reduction \( {\;\operatorname{mod}\;\pi } \) gives a surjective homomorphism of \( \mathcal{A} \) onto \( k\left\lbrack {x,{y}_{0}}\right\rbrack \) . We can find \( w \in \mathcal{A} \) such ... | Yes |
Lemma 1.6. Let \( {W}_{0},{V}_{0} \) be two special affine curves, defined over \( k \) by polynomials \( {g}_{0},{f}_{0} \) respectively. Let \( W, V \) be liftings, defined by \( g, f \) . Let\n\n\[ \n{\varphi }_{0} : {\mathcal{A}}_{0}\left( {V}_{0}\right) \rightarrow {\mathcal{A}}_{0}\left( {W}_{0}\right)\n\]\n\nbe ... | Proof. Suppose \( \varphi \left( x\right) \) lifts \( {\varphi }_{0}\left( x\right) \) . Then we can define a unique \( p \) -adically continuous homomorphism \( R\langle x\rangle \rightarrow R\langle x\rangle \) by\n\n\[ \n\sum {a}_{n}{x}^{n} \mapsto \sum {a}_{n}\varphi {\left( x\right) }^{n}\n\]\n\nLet \( {\varphi f}... | Yes |
Lemma 2.1. We have \( \mathcal{A}\left( {1,{\chi }_{j}}\right) = {x}^{j}R\left\langle {x}^{N}\right\rangle \), for \( 0 \leq j \leq N - 1 \) . | Proof. Obvious. | No |
Lemma 2.2. The element \( {E}_{\pi }\left( y\right) \) is an eigenvector of \( G\left( p\right) \) with eigencharacter \( {\psi }_{\pi } \) . In other words, for \( \alpha \in \mathbf{Z}\left( p\right) \) , \[ {\sigma }_{\alpha }\left( {{E}_{\pi }\left( y\right) }\right) = {\psi }_{\pi }\left( \alpha \right) {E}_{\pi }... | Proof. For \( p \) odd, the inverse of \( {E}_{\pi }\left( y\right) \) is \( {E}_{\pi }\left( {-y}\right) \), which is thus also in \( R\langle y\rangle \) . If \( p = 2 \), one has to apply Lemma 1.3. Next we verify that \( {\sigma }_{\alpha }{E}_{\pi }\left( y\right) \) and \( {E}_{\pi }\left( y\right) \) differ by a... | Yes |
Theorem 2.3. We have the eigenspace decomposition\n\n\[ \n{\mathcal{A}}_{K} = {\bigoplus }_{\alpha \in \mathbf{Z}\left( p\right) }{E}_{\alpha } \cdot R\langle x{\rangle }_{K} \n\]\n\nor also\n\n\[ \n{\mathcal{A}}_{K} = R\langle x{\rangle }_{K} \oplus {\bigoplus }_{i = 1}^{p - 1}{E}_{\pi }{\left( y\right) }^{i}R\langle ... | Proof. Note that \( {\mathcal{A}}_{K} \) is free of dimension \( p \) over \( R\langle \langle x{\rangle }_{K} \) . Each eigenvector provides for a one-dimensional subspace, and hence their sum (necessarily direct) is the whole space \( {\mathcal{A}}_{K} \) .\n\nThe fact that \( R\langle x\rangle \) is the fixed subrin... | Yes |
Theorem 2.4. Let \( 0 \leq i \leq p - 1 \) and \( 0 \leq j \leq N - 1 \) . Then\n\n\[{\mathcal{A}}_{K}\left( {{\psi }_{\pi }^{i},{\chi }_{j}}\right) = {x}^{j}{E}_{\pi }{\left( y\right) }^{i}R{\left\langle {x}^{N}\right\rangle }_{K}.\] | Proof. Obvious from Lemma 2.1 and Lemma 2.2, and the fact that \( {E}_{\pi }\left( y\right) \) is fixed under \( G\left( N\right) \) . | No |
Lemma 3.1. We have:\n\n(i) \( {\Omega }_{K}^{G\left( p\right) } = R\langle x{\rangle }_{K}{dx} = {\mathcal{A}}_{K}^{G\left( p\right) }{dx} \)\n\n(ii) \( {\Omega }_{K}^{G\left( N\right) } = {\mathcal{A}}_{K}^{G\left( N\right) }\frac{dx}{x} \cap {\Omega }_{K} \) ;\n\n(iii) for non-trivial \( \psi ,\chi \) we have\n\n\[ {... | Proof. Write a differential form as\n\n\[ \mathop{\sum }\limits_{{i = 0}}^{{p - 1}}{g}_{i}\left( x\right) {y}^{i}{dx} \]\n\nInvariance under \( G\left( p\right) \) implies that \( {g}_{i} = 0 \) if \( i \geq 1 \), and conversely, so (i) is clear. If \( \zeta \) is a primitive \( N \) -th root of unity, then \( d\left( ... | Yes |
Lemma 3.2. The differential operator \( d \) maps:\n\n\[ d : {\mathcal{A}}_{K}^{G\left( p\right) } \rightarrow {\Omega }_{K}^{G\left( p\right) } \]\n\n\[ d : {\mathcal{A}}_{K}^{G\left( N\right) } \rightarrow {\Omega }_{K}^{G\left( N\right) } \]\n\n\[ d : {\mathcal{A}}_{K}\left( {\psi ,\chi }\right) \rightarrow {\mathca... | Proof. The first inclusion is clear. For the other two, we have\n\n\[ \frac{d{E}_{\pi }\left( y\right) }{{E}_{\pi }\left( y\right) } = d\left( {{\pi y} - \pi {y}^{p}}\right) = d\left( {-\pi {x}^{N}}\right) = - {\pi N}{x}^{N - 1}{dx}. \]\n\nSince \( \varphi \mapsto {d\varphi }/\varphi \) is homomorphic, we get\n\n(1)\n\... | Yes |
Theorem 3.3. (i) For \( \psi \neq 1 \) and \( \chi \neq 1 \) we have\n\n\[ \n{H}_{K}^{1}\left( \mathcal{A}\right) \left( {\psi ,\chi }\right) = {\mathcal{A}}_{K}\left( {\psi ,\chi }\right) \frac{dx}{x}/d\left( {{\mathcal{A}}_{K}\left( {\psi ,\chi }\right) }\right) .\n\]\n\n(ii) For \( 1 \leq j \leq N - 1 \), we have an... | Proof. The first assertion is clear from Lemma 3.2. The second comes from the above diagram and Lemma 3.2. | No |
Theorem 3.4. We have an isomorphism\n\n\[ \n{H}^{1}\left( {\mathcal{A}}_{K}\right) \left( {\psi ,{\chi }_{j}}\right) \overset{ \approx }{ \rightarrow }{H}_{j,\pi } \n\]\n\nwhere \( {H}_{j,\pi } \) is the representation space of the last chapter. | Proof. Clear from Theorem 3.3. | No |
Theorem 1.1. Let \( g \) be a distribution as above. Let \( K \) be a field of characteristic 0 . Assume that the distribution obtained by following \( g \) with the natural homomorphism \[ A \rightarrow A \otimes K \] has \( K \) -rank \( \phi \left( N\right) \), in the sense that the dimension of the vector space gen... | Proof. The rank of the image is at most \( \phi \left( N\right) \) . If the vector space generated by the image has that rank, then the Kubert generators must remain free under \( g \) and the tensor product, so they must be linearly independent over \( \mathbf{Z} \) in the abelian group generated by \( g\left( {\left(... | Yes |
Theorem 1.2. Let \( h \) be a distribution with values in \( K \) . Let \( V \) be the vector space generated by the image of the Stickelberger distribution. Then \( {e}_{\chi }V \) is generated by the single element \( S\left( {\bar{\chi }, h}\right) \), and in particular has dimension 0 or 1 according as that element... | Proof. The proof of Theorem 8.2 in Chapter 2 in fact proves the statement as given here, although we stated previously only the corresponding dimension property. | No |
Lemma 2.1. Let \( \{ m\left( x\right) \} \) be a family of integers and let\n\n\[ \alpha = \mathop{\prod }\limits_{x}g{\left( x\right) }^{m\left( x\right) } \]\n\nThen \( \alpha \) is pure if and only if \( \operatorname{div}\alpha = 0 \), and in that case \( \alpha \) is a root of unity. | Proof. The absolute value of \( g\left( x\right) \) in the complex numbers is 1 . Hence \( \left| \alpha \right| = 1 \) . If \( \alpha \) is pure, this implies that \( \alpha \) is a root of unity. Conversely, assume that \( \operatorname{div}\alpha = 0 \), so \( \alpha \) is a unit. The conjugates of \( g\left( x\righ... | Yes |
Lemma 2.3. There exists a family of integers \( \left\{ {m\left( {d}^{\prime }\right) }\right\} \), for divisors \( {d}^{\prime } \) of \( d,{d}^{\prime } \neq d \), having the following property. Let\n\n\[ \xi = {\mathrm{{St}}}_{h}\left( \frac{1}{d}\right) + \mathop{\sum }\limits_{{d}^{\prime }}m\left( {d}^{\prime }\r... | Proof. As in Theorem 1.2 we look at the \( \psi \) -eigenspace for odd characters \( \psi \) such that \( \psi \left( p\right) = 1 \) and cond \( \psi \) divides \( d \) . It suffices to prove that we can choose the family \( \left\{ {m\left( {d}^{\prime }\right) }\right\} \) such that\n\n\[ \xi {e}_{\psi } \neq 0\text... | Yes |
Theorem 3.1. Let \( \chi \) be an even character such that \( {\chi }_{1} \) has conductor \( d \). Then \[ {L}_{p}^{\prime }\left( {0,\chi }\right) = \mathop{\sum }\limits_{{c = 1}}^{d}{\chi }_{1}\left( c\right) {\log }_{p}{\Gamma }_{p}\left( \frac{c}{d}\right) + \left( {1 - {\chi }_{1}\left( p\right) }\right) {B}_{1,... | We shall prove this formula in \( §4 \) . | No |
Theorem 3.2. Let \( \chi \) be an even character such that \( {\chi }_{1} \) has conductor \( d \) and such that \( {\chi }_{1}\left( p\right) = 1 \) . Let \( {D}_{p} \) be the subgroup of \( \mathbf{Z}{\left( d\right) }^{ * } \) generated by the powers of p. Then\n\n\[ \n{L}_{p}^{\prime }\left( {0,\chi }\right) = \mat... | Proof. By assumption the formula of Theorem 3.1 simplifies to\n\n\[ \n{L}_{p}^{\prime }\left( {0,\chi }\right) = \mathop{\sum }\limits_{{c = 1}}^{d}{\chi }_{1}\left( c\right) {\log }_{p}{\Gamma }_{p}\left( \frac{c}{d}\right) .\n\]\n\nLet \( d \mid \left( {q - 1}\right), q = {p}^{r} \) where \( r \) is the period of \( ... | Yes |
Theorem 4.1. Let \( \chi \) be a Dirichlet character, and let \( N \) be any multiple of the conductor of \( \chi \) such that \( N \) is divisible by \( p \) . Then\n\n\[ \n{L}_{p}\left( {s,\chi }\right) = \mathop{\sum }\limits_{\substack{{a = 1} \\ {p \nmid a} }}^{{N - 1}}\chi \left( a\right) H\left( {s;a, N}\right) ... | Proof. The left-hand side and the right-hand side have the same values at the negative integers, which are dense in \( {\mathbf{Z}}_{p} \), and they are both holomorphic, hence they coincide. | No |
Theorem 4.2. Let \( p \neq 2 \) . Let \( N \) be a positive integer divisible by \( p \) and let \( f \) be a function on \( \mathbf{Z}\left( N\right) \) . Then\n\n\[ \n{L}_{p}^{\prime }\left( {0, f}\right) = \mathop{\sum }\limits_{\substack{{a = 1} \\ {p \nmid a} }}^{{N - 1}}{f}_{1}\left( a\right) {G}_{p}\left( \frac{... | Proof. The desired result is an immediate consequence of the next lemma. | No |
Lemma 4.3. Let \( \\left| a\\right| > 1/p \) and let \( N \) be a positive integer divisible by \( p \) . Then the coefficient of \( s \) in \( H\\left( {s;a, N}\\right) \) is equal to | Proof. We have the expansions:\n\n\[ \n\\frac{1}{1 - s} = 1 + s + \\cdots \n\]\n\n\[ \n\\langle a\\rangle ^{1 - s} = \\langle a\\rangle \\left( {1 - s\\log _{p}\\langle a\\rangle + \\cdots }\\right) \n\]\n\nIf \( j \\geq 2,\\;\\left( \\begin{matrix} 1 - s \\\\ j \\end{matrix}\\right) = \\frac{\\left( -1\\right) ^{j - 1... | Yes |
Theorem 4.4. Extend \( {G}_{p}\left( x\right) \) to \( {\mathbf{Q}}_{p} \) by putting \( {G}_{p}\left( x\right) = 0 \) if \( x \in {\mathbf{Z}}_{p} \) . Then for all \( x \in {\mathbf{Z}}_{p} \) we have\n\n\[ \mathop{\sum }\limits_{{b = 0}}^{{p - 1}}{G}_{p}\left( \frac{x + b}{p}\right) = {\log }_{p}{\Gamma }_{p}\left( ... | Proof. Both sides are continuous and satisfy the functional equation\n\n\[ f\left( {x + 1}\right) = f\left( x\right) + \delta \left( x\right) {\log }_{p}x \]\n\nwhere \( \delta \left( x\right) = 0 \) if \( x \equiv 0{\;\operatorname{mod}\;p} \), and \( \delta \left( x\right) = 1 \) otherwise. This is true for \( {\log ... | Yes |
Theorem 4.5. Let \( \chi \) be a Dirichlet character such that the conductor \( d \) of \( {\chi }_{1} \) is not divisible by \( p \) . Then\n\n\[ \n{L}_{p}^{\prime }\left( {0,\chi }\right) = \mathop{\sum }\limits_{{c = 1}}^{{d - 1}}{\chi }_{1}\left( c\right) {\log }_{p}{\Gamma }_{p}\left( \frac{c}{d}\right) + \left( {... | Proof. Let \( N = {pd} \) . In Theorem 4.2, write\n\n\[ \na = c + {bd}\n\]\n\nwith \( 1 \leq c \leq d - 1 \) and \( 0 \leq b \leq p - 1 \) . Then \( {\chi }_{1}\left( a\right) = {\chi }_{1}\left( c\right) \) . If \( a \) is divisible by \( p \) then \( a/N \in {\mathbf{Z}}_{p} \), so that \( {G}_{p}\left( {a/N}\right) ... | Yes |
Theorem 3 (Chevalley-Warning).-Let \( {f}_{\alpha } \in K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) be polynomials in \( n \) variables such that \( \mathop{\sum }\limits_{a}\deg {f}_{a} < n \), and let \( V \) be the set of their common zeros in \( {K}^{n} \) . One has \[ \operatorname{Card}\left( V\right)... | Put \( P = \mathop{\prod }\limits_{\alpha }\left( {1 - {f}_{\alpha }^{q - 1}}\right) \) and let \( x \in {K}^{n} \) . If \( x \in V \), all the \( {f}_{\alpha }\left( x\right) \) are zero and \( P\left( x\right) = 1 \) ; if \( x \notin V \), one of the \( {f}_{a}\left( x\right) \) is nonzero and \( {f}_{a}{\left( x\rig... | Yes |
Theorem 6 (Gauss).- \( \left( \begin{array}{l} l \\ p \end{array}\right) = \left( \begin{array}{l} p \\ l \end{array}\right) {\left( -1\right) }^{\varepsilon \left( l\right) \varepsilon \left( p\right) } \) . | Let \( \Omega \) be an algebraic closure of \( {\mathbf{F}}_{p} \), and let \( w \in \Omega \) be a primitive \( l \) -th root of unity. If \( x \in {\mathbf{F}}_{l} \), the element \( {w}^{x} \) is well defined since \( {w}^{l} = 1 \) . Thus we are able to form the \ | No |
Lemma 2. \( - {y}^{p - 1} = \left( \begin{array}{l} p \\ l \end{array}\right) \) | Since \( \Omega \) is of characteristic \( p \), we have\n\n\[ \n{y}^{p} = \mathop{\sum }\limits_{{x \in {\mathbf{F}}_{l}}}\left( \frac{x}{p}\right) {w}^{xp} = \mathop{\sum }\limits_{{z \in {\mathbf{F}}_{l}}}\left( \frac{z{p}^{-1}}{l}\right) {w}^{z} = \left( \frac{{p}^{-1}}{l}\right) y = \left( \frac{p}{l}\right) y \n\... | Yes |
Theorem 3 (Hilbert).-If \( a, b \in {\mathbf{Q}}^{ * } \), we have \( {\left( a, b\right) }_{v} = 1 \) for almost all\n\n\( v \in V \) and\n\[ \mathop{\prod }\limits_{{v \in V}}{\left( a, b\right) }_{v} = 1 \] | Since the Hilbert symbols are bilinear, it suffices to prove the theorem when \( a \) or \( b \) are equal to -1 or to a prime number. In each case, theorem 1 gives the value of \( {\left( a, b\right) }_{v} \) . We find\n\n1) \( a = - 1, b = - 1 \) . One has \( {\left( -1, - 1\right) }_{\infty } = {\left( -1, - 1\right... | Yes |
Theorem 3 (Witt).-If \( \\left( {V, Q}\\right) \) and \( \\left( {{V}^{\\prime },{Q}^{\\prime }}\\right) \) are isomorphic and nondegenerate, every injective metric morphism\n\n\[ s : U \\rightarrow {V}^{\\prime } \]\nof a subvector space \( U \) of \( V \) can be extended to a metric isomorphism of \( V \) onto \( {V}... | Since \( V \) and \( {V}^{\\prime } \) are isomorphic, we can suppose that \( V = {V}^{\\prime } \). Moreover, by applying the above lemma, we are reduced to the case where \( U \) is nondegenerate. We argue then by induction on dim \( U \).\n\nIf \( \\dim U = 1, U \) is generated by a nonisotropic element \( x \) ; if... | Yes |
In order that \( f \) represent 0, it is necessary and sufficient that, for all \( v \in V \), the form \( {f}_{v} \) represent 0. | (In other words: \( f \) has a \ | No |
Corollary 2 (Meyer).-A quadratic form of rank \( \geqq 5 \) represents 0 if and only if it is indefinite (i.e. if it represents 0 in \( \mathbf{R} \) ). | Indeed, by th. 6, such a form represents 0 in each of the \( {\mathbf{Q}}_{p} \) . | No |
Corollary 1 (Lagrange).—Every positive integer is a sum of four squares. | Let \( n \) be an integer \( > 0 \) . We write \( n \) in the form \( {4}^{a}m \), where \( m \) is not divisible by 4 . If \( m \equiv 1,2,3,5,6\left( {\;\operatorname{mod}\;8}\right), m \) is a sum of three squares, and the same holds for \( n \) . Otherwise \( m \equiv - 1\left( {\;\operatorname{mod}\;8}\right) \) a... | No |
Corollary 2 (Gauss).-Every positive integer is a sum of three triangular numbers. | Let \( n \) be a positive integer. By applying the theorem to \( {8n} + 3 \), we see that there exist integers \( {x}_{1},{x}_{2},{x}_{3} \) such that\n\n\[ \n{x}_{1}^{2} + {x}_{2}^{2} + {x}_{3}^{2} = {8n} + 3.\n\]\n\nOne has\n\n\[ \n{x}_{1}^{2} + {x}_{2}^{2} + {x}_{3}^{2} \equiv 3\left( {\;\operatorname{mod}\;8}\right... | Yes |
Lemma 2 (Abel’s lemma).-Let \( \\left( {a}_{n}\\right) \) and \( \\left( {b}_{n}\\right) \) be two sequences. Put:\n\n\[ \n{A}_{m, p} = \\mathop{\\sum }\\limits_{{n = m}}^{{n = p}}{a}_{n}\\;\\text{ and }\\;{S}_{m,{m}^{\\prime }} = \\mathop{\\sum }\\limits_{{n = m}}^{{n = {m}^{\\prime }}}{a}_{n}{b}_{n}.\n\]\n\nThen one ... | One replaces \( {a}_{n} \) by \( {A}_{m, n} - {A}_{m, n - 1} \) and regroups the terms. | Yes |
Corollary 2. When \( s \rightarrow 1 \), one has \( \mathop{\sum }\limits_{p}{p}^{-s} \sim \log 1/\left( {s - 1}\right) \), and \( \mathop{\sum }\limits_{{p, k \geqq 2}}1/{p}^{ks} \) remains bounded. | One has:\n\n\[ \log \zeta \left( s\right) = \mathop{\sum }\limits_{{p \in P, k \geqq 1}}\frac{1}{k \cdot {p}^{ks}} = \mathop{\sum }\limits_{{p \in P}}1/{p}^{s} + \psi \left( s\right) ,\] \n\nwith \( \psi \left( s\right) = \mathop{\sum }\limits_{{p \in P}}\mathop{\sum }\limits_{{k \geqq 2}}\left( {1/k \cdot {p}^{ks}}\ri... | Yes |
Theorem 5 (Hecke).-Iff is a cusp form of weight \( {2k} \), then\n\n(39)\n\n\[ \n{a}_{n} = O\left( {n}^{k}\right) \n\]\n\n(In other words, the quotient \( \frac{\left| {a}_{n}\right| }{{n}^{k}} \) remains bounded when \( n \rightarrow \infty \) .) | Because \( f \) is a cusp form, we have \( {a}_{0} = 0 \) and can factor \( q \) out of the expansion (37) of \( f \) . Hence:\n\n(40)\n\n\[ \n\left| {f\left( z\right) }\right| = O\left( q\right) = O\left( {e}^{-{2\pi y}}\right) \;\text{ with }y = \operatorname{Im}\left( z\right) ,\;\text{ when }q\text{ tends to }0. \n... | Yes |
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