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Example 2.4. Let \( E/K \) be an elliptic curve, let \( K\left( \sqrt{d}\right) \) be a quadratic extension of \( K \), and let\n\n\[ \n\chi : {G}_{\bar{K}/K} \rightarrow \{ \pm 1\} ,\;\chi \left( \sigma \right) = {\sqrt{d}}^{\sigma }/\sqrt{d},\n\]\n\nbe the quadratic character associated to \( K\left( \sqrt{d}\right) ... | We choose a Weierstrass equation for \( E/K \) of the form \( {y}^{2} = f\left( x\right) \) and we write \( \bar{K}\left( E\right) = \bar{K}\left( {x, y}\right) \) and \( \bar{K}\left( C\right) = \bar{K}{\left( x, y\right) }_{\xi } \) . Since \( \left\lbrack {-1}\right\rbrack \left( {x, y}\right) = \left( {x, - y}\righ... | Yes |
Lemma 3.1. Let \( C/K \) be a homogeneous space for \( E/K \) . Then for all \( p, q \in C \) and all \( P, Q \in E \) :\n\n(a)\n\[ \mu \left( {p, O}\right) = p\;\text{ and }\;\nu \left( {p, p}\right) = O. \]\n\n(b)\n\[ \mu \left( {p,\nu \left( {q, p}\right) }\right) = q\;\text{ and }\;\nu \left( {\mu \left( {p, P}\rig... | Proof. (a) The equality \( \mu \left( {p, O}\right) = p \) is part of the definition of homogeneous space. Next, the definition of \( \nu \) says that \( \nu \left( {p, p}\right) \) is the unique point \( P \in E \) satisfying \( \mu \left( {p, P}\right) = p \) . We know that this last equation is true for \( P = O \),... | Yes |
Proposition 3.2. Let \( E/K \) be an elliptic curve, and let \( C/K \) be a homogeneous space for \( E/K \) . Fix a point \( {p}_{0} \in C \) and define a map\n\n\[ \n\theta : E \rightarrow C,\;\theta \left( P\right) = {p}_{0} + P.\n\]\n\n(a) The map \( \theta \) is an isomorphism defined over \( K\left( {p}_{0}\right)... | Proof. (a) The action of \( E \) on \( C \) is defined over \( K \) . Hence for any \( \sigma \in {G}_{\bar{K}/K} \) satisfying \( {p}_{0}^{\sigma } = {p}_{0} \), we have\n\n\[ \n\theta {\left( P\right) }^{\sigma } = {\left( {p}_{0} + P\right) }^{\sigma } = {p}_{0}^{\sigma } + {P}^{\sigma } = {p}_{0} + {P}^{\sigma } = ... | Yes |
Proposition 3.3. Let \( C/K \) be a homogeneous space for \( E/K \) . Then \( C/K \) is in the trivial class if and only if \( C\left( K\right) \) is not the empty set. | Proof. Suppose that \( C/K \) is in the trivial class. Then there is a \( K \) -isomorphism \( \theta : E \rightarrow C \), and thus \( \theta \left( O\right) \in C\left( K\right) \). Conversely, suppose that \( {p}_{0} \in C\left( K\right) \) . Then from (X.3.2a), the map \[ \theta : E \rightarrow C,\;\theta \left( P\... | Yes |
Lemma 3.5. Let \( \theta : C/K \rightarrow {C}^{\prime }/K \) be an equivalence of homogeneous spaces for \( E/K \). Then\n\n\[ \theta \left( q\right) - \theta \left( p\right) = q - p\;\text{ for all }p, q \in C. \] | Proof. This is just a matter of grouping points so that the additions and subtractions are well-defined. Thus\n\n\[ \theta \left( q\right) - \theta \left( p\right) = \left( {\left( {\theta \left( q\right) + \left( {p - q}\right) }\right) - \theta \left( p\right) }\right) + \left( {q - p}\right) \]\n\n\[ = \left( {\thet... | Yes |
Theorem 3.8. Let \( C/K \) be a homogeneous space for an elliptic curve \( E/K \) . Choose a point \( {p}_{0} \in C \) and consider the summation map\n\n\[ \n\operatorname{sum} : {\operatorname{Div}}^{0}\left( C\right) \rightarrow E\n\]\n\n\[ \n\sum {n}_{i}\left( {p}_{i}\right) \mapsto \sum \left\lbrack {n}_{i}\right\r... | Proof. (a) Using (II.3.4), we see that we must check that the summation map is a surjective homomorphism and that its kernel is the set of principal divisors. It is clear that it is a homomorphism. Let \( P \in E \) and \( D = \left( {{p}_{0} + P}\right) - \left( {p}_{0}\right) \in {\operatorname{Div}}^{0}\left( C\righ... | Yes |
Theorem 4.2. Let \( \phi : E/K \rightarrow {E}^{\prime }/K \) be an isogeny of elliptic curves defined over \( K \). (a) There is an exact sequence \[ 0 \rightarrow {E}^{\prime }\left( K\right) /\phi \left( {E\left( K\right) }\right) \rightarrow {S}^{\left( \phi \right) }\left( {E/K}\right) \rightarrow \coprod \left( {... | Proof. (a) This is immediate from the diagram \( \left( {* * }\right) \) and the definitions of the Selmer and Shafarevich-Tate groups. | No |
Lemma 4.3. Let \( M \) be a finite (abelian) \( {G}_{\bar{K}/K} \) -module, let \( S \subset {M}_{K} \) be a finite set of places, and define\n\n\[ \n{H}^{1}\left( {{G}_{\bar{K}/K}, M;S}\right) = \left\{ {\xi \in {H}^{1}\left( {{G}_{\bar{K}/K}, M}\right) : \xi \text{ is unramified outside }S}\right\} .\n\]\n\nThen \( {... | Proof. Since \( M \) is finite and \( {G}_{\bar{K}/K} \) acts continuously on \( M \), there is a subgroup of finite index in \( {G}_{\bar{K}/K} \) that fixes every element of \( M \) . Using the inflation-restriction sequence (B.2.4), it suffices to prove the lemma with \( K \) replaced by a finite extension, so we ma... | Yes |
We reformulate the example described in (X §1) in these terms, leaving some details to the reader. Let \( E/K \) be an elliptic curve with \( E\left\lbrack m\right\rbrack \subset E\left( K\right) \) , let \( S \subset {M}_{K} \) be the usual set of places (X.4.4), and let \( K\left( {S, m}\right) \) be as in (X.1.1c). ... | \[ {H}^{1}\left( {{G}_{\bar{K}/K}, E\left\lbrack m\right\rbrack ;S}\right) \cong K\left( {S, m}\right) \times K\left( {S, m}\right) ,\] where this map uses the isomorphism \( {K}^{ * }/{\left( {K}^{ * }\right) }^{m}\overset{ \sim }{ \rightarrow }{H}^{1}\left( {{G}_{\bar{K}/K},{\mathbf{\mu }}_{m}}\right) \). | No |
Proposition 4.6. Let \( \phi : E/K \rightarrow {E}^{\prime }/K \) be a \( K \) -isogeny, let \( \xi \) be a cocycle representing an element of \( {H}^{1}\left( {{G}_{\bar{K}/K}, E\left\lbrack \phi \right\rbrack }\right) \), and let \( C/K \) be a homogeneous space representing the image of \( \xi \) in \( \mathrm{{WC}}... | Proof. (a) Let \( \sigma \in {G}_{\bar{K}/K} \) and let \( P \in C \) . Then, since \( \phi \) is defined over \( K \) and \( {\xi }_{\sigma } \in E\left\lbrack \phi \right\rbrack \), we have\n\n\[ \n{\left( \phi \circ \theta \left( P\right) \right) }^{\sigma } = \left( {\phi \circ {\theta }^{\sigma }}\right) \left( {P... | Yes |
Two-isogenies. We illustrate the general theory by completely analyzing the case of isogenies of degree 2 . Let \( \phi : E \rightarrow {E}^{\prime } \) be an isogeny of degree 2 defined over \( K \) . Then the kernel \( E\left\lbrack \phi \right\rbrack = \{ O, T\} \) is defined over \( K \), so \( T \in E\left( K\righ... | More precisely, if \( d \in K\left( {S,2}\right) \), then tracing through the above identification shows that the corresponding cocycle is\n\n\[ \sigma \mapsto \left\{ \begin{array}{ll} O & \text{ if }{\sqrt{d}}^{\sigma } = \sqrt{d} \\ T & \text{ if }{\sqrt{d}}^{\sigma } = - \sqrt{d} \end{array}\right. \]\n\nThe homoge... | Yes |
Proposition 4.12. Let \( E/K \) be an elliptic curve. For any integers \( m \geq 2 \) and \( n \geq 1 \) , let \( {S}^{\left( m, n\right) }\left( {E/K}\right) \) be the image of \( {S}^{\left( {m}^{n}\right) }\left( {E/K}\right) \) in \( {S}^{\left( m\right) }\left( {E/K}\right) \) . Then there exists an exact sequence... | Proof. This is immediate from the commutative diagram given above. | No |
Proposition 5.1. The map \[ E \times \operatorname{Aut}\left( E\right) \rightarrow \operatorname{Isom}\left( E\right) ,\;\left( {P,\alpha }\right) \mapsto {\tau }_{P} \circ \alpha , \] is a bijection of sets. It identifies \( \operatorname{Isom}\left( E\right) \) with the product of \( E \) and \( \operatorname{Aut}\le... | Proof. Let \( \phi \in \operatorname{Isom}\left( E\right) \) . Then \( {\tau }_{-\phi \left( O\right) } \circ \phi \in \operatorname{Aut}\left( E\right) \), so writing \[ \phi = {\tau }_{\phi \left( O\right) } \circ \left( {{\tau }_{-\phi \left( O\right) } \circ \phi }\right) \] shows that the map is surjective. On the... | Yes |
Proposition 5.3. Let \( E/K \) be an elliptic curve.\n\n(a) The natural inclusion \( \operatorname{Aut}\left( E\right) \subset \operatorname{Isom}\left( E\right) \) induces an inclusion\n\n\[ \n{H}^{1}\left( {{G}_{\bar{K}/K},\operatorname{Aut}\left( E\right) }\right) \subset {H}^{1}\left( {{G}_{\bar{K}/K},\operatorname... | Proof. (a) Let \( i : \operatorname{Aut}\left( E\right) \rightarrow \operatorname{Isom}\left( E\right) \) be the natural inclusion. From (X.5.1), there is a homomorphism \( j : \operatorname{Isom}\left( E\right) \rightarrow \operatorname{Aut}\left( E\right) \) satisfying \( j \circ i = 1 \) . It follows that the induce... | Yes |
Proposition 5.4. Assume that \( \operatorname{char}\left( K\right) \neq 2,3 \), and let\n\n\[ n = \left\{ \begin{array}{ll} 2 & \text{ if }j\left( E\right) \neq 0,{1728} \\ 4 & \text{ if }j\left( E\right) = {1728} \\ 6 & \text{ if }j\left( E\right) = 0 \end{array}\right. \]\n\nThen \( \operatorname{Twist}\left( {\left(... | More precisely, choose a Weierstrass equation\n\n\[ E : {y}^{2} = {x}^{3} + {Ax} + B \]\n\nfor \( E/K \), and let \( D \in {K}^{ * } \) . Then the elliptic curve \( {E}_{D} \in \operatorname{Twist}\left( {\left( {E, O}\right) /K}\right) \) corresponding to \( D\left( {\;\operatorname{mod}\;{\left( {K}^{ * }\right) }^{n... | Yes |
Define an equivalence relation on the set \( K \times {K}^{ * } \) by\n\n\[ \left( {j, D}\right) \sim \left( {{j}^{\prime },{D}^{\prime }}\right) \;\text{ if }\;j = {j}^{\prime }\;\text{ and }\;D/{D}^{\prime } \in {\left( {K}^{ * }\right) }^{n\left( j\right) }, \]\n\nwhere \( n\left( j\right) = 2 \) (respectively 4, re... | Proof. From (III.10.2.) we have an isomorphism\n\n\[ \operatorname{Aut}\left( E\right) \cong {\mathbf{\mu }}_{n} \]\n\nof \( {G}_{\bar{K}/K} \) -modules. It follows from (B.2.5c) that\n\n\[ \operatorname{Twist}\left( {\left( {E, O}\right) /K}\right) = {H}^{1}\left( {{G}_{\bar{K}/K},\operatorname{Aut}\left( E\right) }\r... | No |
Proposition 6.2. Let \( p \) be an odd prime, let \( {E}_{p} \) be the elliptic curve\n\n\[ \n{E}_{p} : {y}^{2} = {x}^{3} + {px} \n\]\n\nand let \( \phi : {E}_{p} \rightarrow {E}_{p}^{\prime } \) be the isogeny of degree 2 with kernel \( {E}_{p}\left\lbrack \phi \right\rbrack = \{ O,\left( {0,0}\right) \} \).\n\n(a)\n\... | Proof. To ease notation, we let \( E = {E}_{p} \) and \( {E}^{\prime } = {E}_{p}^{\prime } \).\n\n(a) This is a special case of (X.6.1a). | No |
Corollary 6.2.1. There are infinitely many elliptic curves \( E/\mathbb{Q} \) satisfying\n\n\[ \n\operatorname{rank}E\left( \mathbb{Q}\right) = 0\;\text{ and }\;\operatorname{III}\left( {E/\mathbb{Q}}\right) \left\lbrack 2\right\rbrack = 0.\n\] | Proof. From (X.6.2), the elliptic curves \( {y}^{2} = {x}^{3} + {px} \) with \( p \equiv 7,{11}\left( {\;\operatorname{mod}\;{16}}\right) \) have this property. | Yes |
Proposition 6.5. Let \( p \equiv 1\left( {\;\operatorname{mod}\;8}\right) \) be a prime for which \( 2 \) is not a quartic residue. (a) The curves \[ {w}^{2} + 1 = {4p}{z}^{4},\;{w}^{2} + 2 = {2p}{z}^{4},\;{w}^{2} + {2p}{z}^{4} = 2, \] have points defined over every completion of \( \mathbb{Q} \), but they have no \( \... | Proof of (X.6.5). During the course of proving (X.6.2b), we showed that the Selmer group \( {S}^{\left( p\right) }\left( {{E}_{p}/\mathbb{Q}}\right) \subset {\mathbb{Q}}^{ * }/{\left( {\mathbb{Q}}^{ * }\right) }^{2} \) is given by \( \{ \pm 1, \pm 2, \pm p, \pm {2p}\} \) . Further, we showed that \( - p \) is the image... | Yes |
Proposition 6.6. Let \( p \) be a prime satisfying \( p \equiv 1\\left( {\\operatorname{mod} 8}\\right) \), and write \( p \) as a sum of two squares, \( p = {A}^{2} + {B}^{2} \). Then\n\n\\[ \n{\\left( \\frac{2}{p}\\right) }_{4} = {\\left( -1\\right) }^{{AB}/4} \n\\]\n\nIn other words, 2 is a quartic residue modulo \(... | Proof. Using the fact that \( {A}^{2} + {B}^{2} \equiv 0\\left( {\\operatorname{mod} p}\\right) \), we compute\n\n\\[ \n{\\left( A + B\\right) }^{\\left( {p - 1}\\right) /2} \equiv {\\left( 2AB\\right) }^{\\left( {p - 1}\\right) /4}\\left( {\\operatorname{mod} p}\\right) \n\\]\n\n\\[ \n\equiv {2}^{\\left( {p - 1}\\righ... | Yes |
Example 2.2. We use Pollard’s algorithm to factor \( N = {71384665949740607} \) . Using the base \( a = 2 \), we find on the \( {33}^{\text{rd }} \) iteration of the loop that | \[ {2}^{{33}!} \equiv {58248995050016779}\;\left( {\;\operatorname{mod}\;{71384665949740607}}\right) ,\] \[ \gcd \left( {{58248995050016778},{71384665949740607}}\right) = {228266501}. \] Thus \[ N = {71384665949740607} = {228266501} \cdot {312725107}, \] and one can check that both factors are prime. | Yes |
Example 2.5. We use Lenstra’s algorithm to factor \( N = {6887} \) . We randomly select \( P = \left( {{1512},{3166}}\right) \) and \( A = {14} \), and we set\n\n\[ B \equiv {3166}^{2} - {1512}^{3} - {14} \cdot {1512} \equiv {19}\left( {\;\operatorname{mod}\;{6887}}\right) ,\]\n\nso \( P \) is a mod \( N \) point on th... | We compute successively (always working modulo 6887)\n\n\[ \left\lbrack 2\right\rbrack P \equiv \left( {{3466},{2996}}\right) \]\n\n\[ \left\lbrack {3!}\right\rbrack P = \left\lbrack 3\right\rbrack \left( {\left\lbrack 2\right\rbrack P}\right) \equiv \left( {{3067},{396}}\right) ,\]\n\n\[ \left\lbrack {4!}\right\rbrack... | Yes |
The DLP for the additive group of a finite field \( {\mathbb{F}}_{q} \) asks for a solution \( m \) to the linear equation \( {xm} = y \) for given \( x, y \in {\mathbb{F}}_{q} \). | To solve this equation, we need only find the multiplicative inverse of \( x \) in \( {\mathbb{F}}_{q} \), which takes \( O\left( {\log q}\right) \) steps using the Euclidean algorithm. Thus the DLP in \( {\mathbb{F}}_{q}^{ + } \) is a very easy problem. | Yes |
Proposition 5.2. (Shanks’s Babystep-Giantstep Algorithm) Let \( G \) be a group, let \( x, y \in G \), and let \( n \) be the order of \( x \) . Then the following algorithm solves the DLP in \( O\left( \sqrt{n}\right) \) steps with \( O\left( \sqrt{n}\right) \) storage: | Proof. Suppose that \( y \) is equal to a power of \( x \), say \( y = {x}^{m} \) with \( 0 \leq m < n \) . We write \( m = {jN} + i \) with \( 0 \leq i < N \), so\n\n\[ 0 \leq j = \left( {m - i}\right) /N \leq N,\;\text{ since }m \leq n\text{ and }N \geq \sqrt{n}. \]\n\nIt follows that \( {x}^{i} \) is in the first li... | Yes |
Theorem 5.3. Let \( S \) be a finite set containing \( N \) elements, and let \( f : S \rightarrow S \) be a function. Starting with an initial value \( {x}_{0} \in S \), define a sequence of points \( {x}_{0},{x}_{1},{x}_{2},\ldots \) by\n\n\[ \n{x}_{i} = f\left( {x}_{i - 1}\right) = \underset{i\text{ iterations of }f... | PROOF. (a) It is clear from Figure 11.5 that for \( j > i \) we have\n\n\[ \n{x}_{j} = {x}_{i}\;\text{ if and only if }\;i \geq T\;\text{ and }\;j \equiv i\left( {\;\operatorname{mod}\;L}\right) .\n\]\n\nHence \( {x}_{2i} = {x}_{i} \) if and only if \( i \geq T \) and \( L \mid i \) . The first such \( i \) lies betwee... | Yes |
Proposition 6.1. (MOV Algorithm [168]) Let \( E/{\mathbb{F}}_{q} \) be an elliptic curve, let \( P, Q \in \) \( E\left( {\mathbb{F}}_{q}\right) \) be points of prime order \( N \), and let \( d \) be the embedding degree of \( N \) in \( {\mathbb{F}}_{q} \) . Assume that \( \gcd \left( {q - 1, N}\right) = 1 \) . Then t... | Proof. We are looking for an integer \( m \) such that \( Q = \left\lbrack m\right\rbrack P \) . We choose a point \( T \in E\left\lbrack N\right\rbrack \left( {\overline{\mathbb{F}}}_{q}\right) \) such that \( P \) and \( T \) generate \( E\left\lbrack N\right\rbrack \) . Then the value of the Weil pairing \( {e}_{N}\... | Yes |
Lemma 6.2. Let \( E/{\mathbb{F}}_{q} \) be an elliptic curve, let \( N \geq 1 \) be an integer satisfying \( \gcd \left( {q - 1, N}\right) = 1 \), let \( d \) be the embedding degree of \( N \) in \( {\mathbb{F}}_{q} \), and suppose that \( E\left( {\mathbb{F}}_{q}\right) \) contains a point of exact order \( N \) . Th... | Proof. Let \( P \in E\left( {\mathbb{F}}_{q}\right) \) be the given point of exact order \( N \) defined over \( {\mathbb{F}}_{q} \), and choose a point \( T \in E\left\lbrack N\right\rbrack \) such that \( \{ P, T\} \) is a basis for \( E\left\lbrack N\right\rbrack \) . Let \( \phi \in {G}_{{\overline{\mathbb{F}}}_{q}... | Yes |
Let \( p \geq 5 \) be prime, and let \( E/{\mathbb{F}}_{p} \) be a supersingular elliptic curve. We can compute embedding degrees for \( E \) using Exercise 5.15, which implies that\n\n\[ \n\# E\left( {\mathbb{F}}_{p}\right) = p + 1 \n\]\n\nSuppose that \( P \in E\left( {\mathbb{F}}_{p}\right) \) is a point of exact or... | This militates against using supersingular curves in most cryptographic settings. However, we will see later (XI §7) that there are cryptographic applications that make use of the Weil pairing and low embedding degrees. For these applications, supersingular curves may be used; we simply must ensure that it is computati... | Yes |
Proposition 6.5. (Semaev [228], Satoh-Araki [218], Smart [269]) Let \( p \geq 3 \) and let \( E/{\mathbb{F}}_{p} \) be an elliptic curve satisfying\n\n\[ \n\# E\left( {\mathbb{F}}_{p}\right) = p \n\]\n\n(Such curves are called anomalous.) The following algorithm solves the ECDLP in \( E\left( {\mathbb{F}}_{p}\right) \)... | Proof. Using the fact that \( \# E\left( {\mathbb{F}}_{p}\right) = p \), we have\n\n\[ \n\widetilde{\left\lbrack p\right\rbrack {P}^{\prime }} = \left\lbrack p\right\rbrack P = O\;\text{ and }\;\widetilde{\left\lbrack p\right\rbrack {Q}^{\prime }} = \left\lbrack p\right\rbrack Q = O\;\text{ in }E\left( {\mathbb{F}}_{p}... | Yes |
Example 6.7. We work over the field \( {\mathbb{F}}_{127} \) and consider the anomalous curve and points\n\n\[ E : {y}^{2} = {x}^{3} + {19x} + {112},\;P = \left( {{106},{72}}\right) \in E\left( {\mathbb{F}}_{127}\right) ,\;Q = \left( {{12},{121}}\right) \in E\left( {\mathbb{F}}_{127}\right) . \]\n\nWe take the same equ... | We use the double-and-add algorithm, working modulo \( {127}^{2} \) with \( \left( {z, w}\right) \) -coordinates, to compute\n\n\[ \left\lbrack {127}\right\rbrack {P}^{\prime } = \left( {{12319},0}\right) \in E\left( {\mathbb{Z}/{127}^{2}\mathbb{Z}}\right) \;\text{ and }\;\left\lbrack {127}\right\rbrack {Q}^{\prime } =... | Yes |
Example 7.1. Let \( E \) be the elliptic curve \( {y}^{2} = {x}^{3} + x \) having complex multiplication by \( \mathbb{Z}\left\lbrack i\right\rbrack \), and let \( \phi \) be the isogeny\n\n\[ \phi : E \rightarrow E,\;\phi \left( {x, y}\right) = \left\lbrack i\right\rbrack \left( {x, y}\right) = \left( {-x,{iy}}\right)... | To see this, let \( T \in E\left\lbrack N\right\rbrack \) be a point of exact order \( N \), and suppose that some linear combination of \( T \) and \( \phi \left( T\right) \) is zero. Then\n\n\[ \left\lbrack a\right\rbrack T + \left\lbrack b\right\rbrack \phi \left( T\right) = O\; \Leftrightarrow \;\left\lbrack {a + {... | Yes |
Theorem 7.4. The following procedure allows Alice to sign a digital document and Bob to verify that the signature is valid. | Proof. Assuming that Alice has constructed \( A \) and \( S \) as in steps (2) and (4), bilin-earity of the pairing yields\n\n\[ \langle A, D\rangle = \langle \left\lbrack a\right\rbrack T, D\rangle = \langle T, D{\rangle }^{a}\;\text{ and }\;\langle T, S\rangle = \langle T,\left\lbrack a\right\rbrack D\rangle = \langl... | "No" |
Theorem 8.1. Let \( E \) be an elliptic curve given by a Weierstrass equation\n\n\[ E : {y}^{2} + {a}_{1}{xy} + {a}_{3}y = {x}^{3} + {a}_{2}{x}^{2} + {a}_{4}x + {a}_{6}, \]\n\nand let \( P = \left( {{x}_{P},{y}_{P}}\right) \) and \( Q = \left( {{x}_{Q},{y}_{Q}}\right) \) be nonzero points on \( E \). | (a) Suppose first that \( \lambda \neq \infty \), and let \( y = {\lambda x} + \nu \) be the line through \( P \) and \( Q \), or the tangent line at \( P \) if \( P = Q \). This line intersects \( E \) at the three points \( P, Q \), and \( - P - Q \), so\n\n\[ \operatorname{div}\left( {y - {\lambda x} - \nu }\right) ... | Yes |
Proposition 9.1. The Tate-Lichtenbaum pairing is a well-defined bilinear pairing. | Proof. Let \( \xi \left( \sigma \right) = {e}_{N}\left( {{Q}^{\sigma } - Q, T}\right) \) be the given map \( \xi : {G}_{\bar{K}/K} \rightarrow {\mathbf{\mu }}_{N} \) . We use basic properties of the Weil pairing (III.8.1ad) and the assumption that \( T \in E\left( K\right) \left\lbrack N\right\rbrack \) to verify that ... | Yes |
Proposition 9.2. Let \( T \in E\left( K\right) \left\lbrack N\right\rbrack \) and choose a function \( f \in K\left( E\right) \) satisfying\n\n\[ \operatorname{div}\left( f\right) = N\left( T\right) - N\left( O\right) \;\text{ and }\;f \circ \left\lbrack N\right\rbrack \in {\left( K{\left( E\right) }^{ * }\right) }^{N}... | Proof of (XI.9.2). As explained in the construction of the Weil \( {e}_{N} \) -pairing (III \( §8 \) ), there are functions \( f, g \in \bar{K}\left( E\right) \) satisfying\n\n\[ \operatorname{div}\left( f\right) = N\left( T\right) - N\left( O\right) \;\text{ and }\;f \circ \left\lbrack N\right\rbrack = {g}^{N}, \]\n\n... | Yes |
Proposition 1.1. Let \( E/K \) be a curve given by a Weierstrass equation. Then, under the boxed assumptions, there is a substitution\n\n\[ x = {u}^{2}{x}^{\prime } + r,\;y = {u}^{3}{y}^{\prime } + {u}^{2}s{x}^{\prime } + t,\;\text{ with }u \in {K}^{ * }\text{ and }r, s, t \in K, \]\n\nthat transforms the given Weierst... | Proof. (a) See (III §1). | No |
Proposition 1.2. (a) A curve given by a Weierstrass equation is nonsingular if and only if the discriminant of the equation is nonzero. | Proof. (a) We already proved most of this result in (III.1.4a). All that remains is to show that if \( \operatorname{char}\left( K\right) = 2 \) and \( \Delta = 0 \), then the curve is singular. But this is immediate from the normal forms given in (A.1.1c). | No |
Proposition 1.3. (Deuring Normal Form) Let \( E/K \) be an elliptic curve over a field with \( \operatorname{char}K \neq 3 \) . Then \( E \) has a Weierstrass equation over \( \bar{K} \) of the form\n\n\[ \n{E}_{\alpha } : {y}^{2} + {\alpha xy} + y = {x}^{3},\;\alpha \in \bar{K},{\alpha }^{3} \neq {27}.\n\]\n\nThis Wei... | Proof. The computation of \( \Delta \) and \( j \) for \( {E}_{\alpha } \) is an exercise. In order to show that \( E \) has an equation of the form \( {E}_{\alpha } \), we could find appropriate substitutions. However, using (A.1.2b), a quicker route is available. Given an elliptic curve \( E/K \), we let \( \alpha \i... | No |
Corollary 1.4. Let \( E/K \) be an elliptic curve defined over a local field, i.e., the field \( K \) comes equipped with a discrete valuation.\n\n(a) There exists a finite extension \( {K}^{\prime }/K \) such that \( E \) has either good or split multiplicative reduction over \( {K}^{\prime } \) . | Proof. Let \( R \) be the ring of integers of \( K \), let \( \mathcal{M} \) be its maximal ideal, and let \( k = R/\mathcal{M} \) be its residue field. From the proofs of (VII.5.4c) and (VII.5.5), we are left to deal with the case that \( \operatorname{char}k = 2 \) . In particular, we may assume that \( \operatorname... | Yes |
Proposition 1.2. Let\n\n\\[ \n0 \rightarrow P\\xrightarrow[]{\\phi }M\\xrightarrow[]{\\psi }N \rightarrow 0 \n\\]\n\nbe an exact sequence of \\( G \\) -modules. Then there is a long exact sequence  where the connecti... | Proof. Everything follows from a straightforward, but tedious, diagram chase that we leave to the reader (Exercise B.1). Or see any of the references listed at the beginning of this appendix. | No |
Proposition 1.3. (Inflation-Restriction Sequence) Let \( M \) be a \( G \) -module and let \( H \) be a normal subgroup of \( G \) . Then the following sequence is exact:\n\n\[ 0 \rightarrow {H}^{1}\left( {G/H,{M}^{H}}\right) \overset{\text{ Inf }}{ \rightarrow }{H}^{1}\left( {G, M}\right) \overset{\text{ Res }}{ \righ... | Proof. From the definitions it is clear that \( \operatorname{Res} \circ \operatorname{Inf} = 0 \) .\n\nNext let \( \xi : G/H \rightarrow {M}^{H} \) be a 1-cocycle with \( \operatorname{Inf}\{ \xi \} = 0 \), where we use braces \( \{ \cdot \} \) to indicate the cohomology class of a cocycle. Thus there is an \( m \in M... | Yes |
Both \( {\bar{K}}^{ + } \) and \( {\bar{K}}^{ * } \) are \( {G}_{\bar{K}/K} \) -modules under the natural action of \( {G}_{\bar{K}/K} \). | This is true, because for any \( x \in \bar{K} \), the extension \( K\left( x\right) /K \) is finite, so the stabilizer of \( x \) has finite index in \( {G}_{\bar{K}/K} \). | Yes |
Proposition 2.4. (Inflation-Restriction Sequence) With notation as above, there is an exact sequence\n\n\[ 0 \rightarrow {H}^{1}\left( {{G}_{L/K},{M}^{{G}_{\bar{K}/L}}}\right) \overset{\text{ Inf }}{ \rightarrow }{H}^{1}\left( {{G}_{\bar{K}/K}, M}\right) \overset{\text{ Res }}{ \rightarrow }{H}^{1}\left( {{G}_{\bar{K}/... | Proof. Virtually identical to the proof of (B.1.3). | No |
Proposition 2.5. Let \( K \) be a field.\n\n(a) \( {H}^{1}\left( {{G}_{\bar{K}/K},{\bar{K}}^{ + }}\right) = 0 \) . | Proof. (a) [233, Chapter X, Proposition 1]. | No |
If \( \mathcal{D}/K \) is any algebraic group, then there is a natural action of \( {G}_{\bar{K}/K} \) on \( \mathcal{D} = \mathcal{D}\left( \bar{K}\right) \), and as explained earlier (B.2.1.2), this action is discrete. | \[ {H}^{0}\left( {{G}_{\bar{K}/K},\mathcal{D}}\right) = \mathcal{D}\left( K\right) \] is the subgroup of \( K \) -rational points of \( \mathcal{D} \). | Yes |
Proposition 3.2. For all integers \( n \geq 1 \) ,\n\n\[ \n{H}^{1}\left( {{G}_{\bar{K}/K},{\mathrm{{GL}}}_{n}\left( \bar{K}\right) }\right) = \{ 1\} .\n\] | Proof. [233, Chapter X, Proposition 3]. | No |
Corollary 11.1.1. (a) There are only finitely many isomorphism classes of elliptic curves \( E/\mathbb{C} \) with \( \operatorname{End}\left( E\right) \cong \mathcal{R} \) . | Proof. (a) Clear from (C.11.1), since \( \mathcal{C}\left( \mathcal{R}\right) \) is finite. | No |
Theorem 11.2. (Weber, Fueter) Let \( \{ \Lambda \} \in \mathcal{C}\left( \mathcal{R}\right) \) . (a) \( j\left( \Lambda \right) \) is an algebraic integer. | Proof. (a) The original proof of the integrality of \( j\left( \Lambda \right) \) uses the theory of modular functions; see, for example, [140, Chapter 5, Theorem 4], [249, §4.6], or [266, II §6]. An algebraic proof that generalizes to higher dimensions can be based on the criterion of Néron-Ogg-Shafarevich; see [239, ... | No |
Suppose that \( E/\mathbb{Q} \) is an elliptic curve with complex multiplication, and suppose that \( \operatorname{End}\left( E\right) \) is the full ring of integers \( \mathcal{R} \) in the field \( \mathcal{K} = \operatorname{End}\left( E\right) \otimes \mathbb{Q} \). (Note that (VI.5.5) tells us that \( \mathcal{K... | \[ \mathcal{H} = \mathcal{K}\left( {j\left( E\right) }\right) = \mathcal{K} \] and thus that \( \mathcal{K} \) has class number one. | No |
If we relax the requirement that \( \operatorname{End}\left( E\right) \) be the full ring of integers of \( \mathcal{K} \) and allow \( \operatorname{End}\left( E\right) \) to be an arbitrary order in \( \mathcal{K} \), then \( \operatorname{End}\left( E\right) \) has the form \( \mathbb{Z} + f\mathcal{R} \) for some i... | \[ \left\lbrack {\mathcal{K}\left( {j\left( E\right) }\right) : \mathcal{K}}\right\rbrack = \# \mathcal{C}\left( {\mathbb{Z} + f\mathcal{R}}\right) \] where \( \mathcal{C}\left( {\mathbb{Z} + f\mathcal{R}}\right) \) is the group of rank-1 projective \( \left( {\mathbb{Z} + f\mathcal{R}}\right) \) -modules. In particula... | Yes |
Theorem 11.4. (Hasse) Let \( \{ \Lambda \} \in \mathcal{C}\left( \mathcal{R}\right) \), and let \( \mathcal{H} = \mathcal{K}\left( {j\left( \Lambda \right) }\right) \) be as in (C.11.2). For each prime ideal \( \mathfrak{p} \) of \( \mathcal{R} \), let \( \operatorname{Frob}\left( \mathfrak{p}\right) \in {G}_{\mathcal{... | Proof. See [140, Chapter 10, Theorem 1], [230], [249, Theorem 5.7], or [266, II.4.3]. | No |
Theorem 11.5. Let \( \mathcal{K} \) be an imaginary quadratic field, let \( \mathcal{R} \subset \mathcal{K} \) be its ring of integers, and let \( E/\mathbb{C} \) be an elliptic curve with \( \operatorname{End}\left( E\right) \cong \mathcal{R} \) .\n\n(a) The maximal unramified abelian extension of \( \mathcal{K} \) is... | Proof. (a) This is a restatement of (C.11.2c).\n\n(b) See [140, Chapter 10, Theorem 2], [230], [249, Corollary 5.6], or [266, II.5.7]. | No |
Proposition 12.1. (a) The group \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) acts properly discontinuously on \( \mathbb{H} \) . (b) The region\n\n\[ \mathcal{F} = \left\{ {\tau \in \mathbb{H} : \left| {\operatorname{Re}\left( \tau \right) }\right| \leq \frac{1}{2}\text{ and }\left| \tau \right| \geq 1}\right\} \]... | Proof. See [5, Theorems 2.1 and 2.3], [232, VII §1], or [266, I.1.5, I.1.6]. | No |
Proposition 12.4. (a) The \( j \) -function \( j\left( \tau \right) \) is a modular function of weight 0 that is holomorphic on \( \mathbb{H} \) . Its Fourier series has the form\n\n\[ j\left( \tau \right) = \frac{1}{q} + {744} + \mathop{\sum }\limits_{{n = 1}}^{\infty }c\left( n\right) {q}^{n}\;\text{ with }c\left( n\... | Proof. See [5, Theorems 1.18, 1.19, 1.20], [232, VII Propositions 4, 5, 8], or [266, I.7.1, I.7.4]. | No |
Theorem 12.5. (a) (Jacobi)\n\n\[ \n\Delta \left( \tau \right) = {\left( 2\pi \right) }^{12}q\mathop{\prod }\limits_{{n = 1}}^{\infty }{\left( 1 - {q}^{n}\right) }^{24}. \n\] | Proof. See [5, Theorems 3.1, 3.3] or [232, VII Theorem 6]. | No |
Proposition 12.6. Let \( q = {e}^{2\pi i\tau } \) and \( u = {e}^{2\pi iz} \) . Then\n\n\[ \n{\left( 2\pi i\right) }^{-2}\wp \left( {z;\tau }\right) = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }\frac{{q}^{n}u}{{\left( 1 - {q}^{n}u\right) }^{2}} + \frac{1}{12} - 2\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{q}^... | Proof. See [140, Chapter 18, §2], [210, II §5], or [266, I.6.2, I.6.4]. | No |
Theorem 12.8. (a) \( j\left( \tau \right) \) is a modular function of weight 0 . Every modular function of weight 0 is a rational function of \( j\left( \tau \right) \) . (b) The map\n\n\[ \n\mathbb{C}\left\lbrack {X, Y}\right\rbrack \rightarrow M,\;P\left( {X, Y}\right) \mapsto P\left( {{G}_{4},{G}_{6}}\right) ,\n\]\n... | Proof. See [5, Theorem 2.8, §§6.4, 6.5], [232, VII, §§3.2, 3.3], or [266, I.3.10, I.4.2, and exercise 1.10]. | No |
Proposition 12.9. (a) If \( f \) is a modular form (respectively a cusp form) of weight \( {2k} \) , then \( T\left( n\right) f \) is also a modular form (respectively a cusp form) of weight \( {2k} \) . In other words, \( T\left( n\right) \) induces linear maps\n\n\[ T\left( n\right) : {M}_{k} \rightarrow {M}_{k}\;\te... | Proof. See [5, Theorems 6.11, 6.13], [232, VII §§5.1, 5.3], or [266, I.10.2, I.10.6]. | No |
Consider the vector space \( {M}_{6,0} \) of cusp forms of weight 12. We see from (C.12.8c) and (C.12.4c) that \( {M}_{6,0} \) has dimension one and is generated by the discriminant function\n\n\[ \Delta = {\left( 2\pi \right) }^{12}q\mathop{\prod }\limits_{{n = 1}}^{\infty }{\left( 1 - {q}^{n}\right) }^{24} = {\left( ... | Since \( T\left( n\right) \Delta \) is also in \( {M}_{6,0} \), it follows that \( T\left( n\right) \Delta \) is a multiple of \( \Delta \) . Using (C.12.10), we conclude that\n\n\[ T\left( n\right) \Delta = \tau \left( n\right) \Delta \;\text{ for all }n = 1,2,\ldots \] | Yes |
Proposition 12.11. The map\n\n\[ j : \mathbb{H}/{\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \rightarrow \mathbb{C} \]\n\nis a complex analytic isomorphism of (open) Riemann surfaces. | Proof. See [232, VII Proposition 5] or [266, I.4.1]. | No |
Corollary 12.11.1. (Uniformization Theorem) Let \( E/\mathbb{C} \) be an elliptic curve. Then there exist a lattice \( \Lambda \subset \mathbb{C} \) and a complex analytic isomorphism \( \mathbb{C}/\Lambda \rightarrow E\left( \mathbb{C}\right) \) . | Proof. Let \( J \) be the \( j \) -invariant of \( E \) . From (C.12.11), there is a \( \tau \in \mathbb{H} \) such that \( j\left( \tau \right) = J \) . Then the elliptic curve\n\n\[ \n{E}_{\tau } : {y}^{2} = 4{x}^{3} - {g}_{2}\left( \tau \right) x - {g}_{3}\left( \tau \right)\n\]\n\nhas \( j \) -invariant \( J \), so... | Yes |
Proposition 12.14. Let \( \Gamma \) be a congruence subgroup of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) . There is a natural isomorphism between the space of weight-2 cusp forms for \( \Gamma \) and the space of holomorphic 1 -forms on the Riemann surface \( {\mathbb{H}}^{ * }/\Gamma \) . | Proof. See [249, §2.4]. | No |
Proposition 12.16. Let \( \Gamma \) be a congruence subgroup of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \), say \( \Gamma \supset \Gamma \left( N\right) \), and let \( f\left( \tau \right) \) be a modular form of weight \( {2k} \) for \( \Gamma \). Then for each integer \( n \geq 1 \) satisfying \( \gcd \left( {... | Proof. See [249, Proposition 3.37]. | No |
Theorem 13.1. Let \( N \geq 1 \) be an integer:\n\n(a) There exist a smooth projective curve \( {X}_{0}\left( N\right) /\mathbb{Q} \) and a complex analytic isomorphism\n\n\[ {j}_{N,0} : {\mathbb{H}}^{ * }/{\Gamma }_{0}\left( N\right) \rightarrow {X}_{0}\left( N\right) \left( \mathbb{C}\right) \]\n\n such that the foll... | Proof. See [249, §6.7]. | No |
The curve \( {X}_{1}\left( 7\right) \) is isomorphic to \( {\mathbb{P}}^{1} \). | To make this precise, we associate to each point \( \left\lbrack {t,1}\right\rbrack \in {\mathbb{P}}^{1} \) the pair \( \left( {{E}_{t},{P}_{t}}\right) \), where \( {E}_{t} \) is the curve (defined over \( \mathbb{Q}\left( t\right) \) ) given by the equation\n\n\[ \n{E}_{t} : {y}^{2} + \left( {1 + t - {t}^{2}}\right) {... | Yes |
Theorem 13.6. (Modularity Theorem, Wiles et al. [28, 291, 311]) Every elliptic curve defined over \( \mathbb{Q} \) is modular, i.e., if \( E/\mathbb{Q} \) is an elliptic curve, then there exist an integer \( N \) and a surjective morphism \( \phi : {X}_{0}\left( N\right) \rightarrow E \) defined over \( \mathbb{Q} \) .... | We have credited the modularity theorem to Wiles et al., since the history of the proof is somewhat complicated and involves a number of people, although certainly the most important new ideas were due to Wiles. The original announcement of a proof of (C.13.6) was made by Wiles in 1993, but a detailed examination of th... | Yes |
Theorem 14.1. (Tate) Let \( K \) be a field that is complete with respect to a discrete valuation \( v \). (a) For every \( q \in {K}^{ * } \) with \( {\left| q\right| }_{v} < 1 \), the map \[ \phi : {\bar{K}}^{ * }/{q}^{\mathbb{Z}} \rightarrow {E}_{q}\left( \bar{K}\right) \] as described above is an isomorphism of \( ... | Proof. This result was originally proven by Tate in 1959, although not published until many years later [285]. Other accounts may be found in [210, II §5], [212], and [266, V.5.3, V.5.4]. | No |
Theorem 15.1. (Kodaira, Néron) Let \( E/K \) be as above.\n\n(a) There is a regular projective two-dimensional scheme \( \mathcal{C}/\operatorname{Spec}\left( R\right) \) whose generic fiber \( \mathcal{C}{ \times }_{\operatorname{Spec}\left( R\right) }\operatorname{Spec}\left( K\right) \) is isomorphic over \( K \) to... | Proof. See [193] or [266, §§IV.5, IV.6]. | No |
Theorem 15.2. (Kodaira, Néron) With notation as in (C.15.1), all of the possibilities for the special fiber \( \widetilde{\mathcal{C}} \) and the group of components \( \widetilde{\mathcal{E}}\left( k\right) /{\widetilde{\mathcal{E}}}^{0}\left( k\right) \) are given in Table 15.1. (Some of the components of the special... | Proof. See [193] or [266, §§IV.8, IV.9]. | No |
Corollary 15.2.1. The group \( E\left( K\right) /{E}_{0}\left( K\right) \) is finite. If \( E \) has split multiplicative reduction, then it is cyclic of order \( - {\operatorname{ord}}_{v}\left( {j\left( E\right) }\right) \) . In all other cases it has order at most 4. | Proof. The first statement follows from (C.15.1d). For the second, if \( E \) has split multiplicative reduction, then (C.14.1ac) implies that\n\n\[ E\left( K\right) = {K}^{ * }/{q}^{\mathbb{Z}}\;\text{ and }\;E\left( K\right) /{E}_{0}\left( K\right) \cong \mathbb{Z}/{\operatorname{ord}}_{v}\left( q\right) \mathbb{Z}. ... | Yes |
Proposition 16.2. (Ogg-Saito formula) Let \( {m}_{v} \) be the number of irreducible components (ignoring multiplicities) on the special fiber of the minimal (complete) Néron model of \( E \) at \( v\left( {\mathrm{C}§{15}}\right) \), and let \( {\mathcal{D}}_{E/K} \) be the minimal discriminant of \( E/K \) . Then\n\n... | Proof. This was proven by Ogg [200] except in the case that \( \operatorname{char}\left( {K}_{v}\right) = 0 \) and \( \operatorname{char}\left( {k}_{v}\right) = 2 \) . A proof in all characteristics for curves of arbitrary genus was given by Saito [217]. | Yes |
Theorem 17.1. (Tate [281],[286]) Let \( v \in {M}_{K} \) be a nonarchimedean absolute value, and let \( E/{K}_{v} \) be an elliptic curve. There exists a nondegenerate bilinear pairing\n\n\[ \langle \cdot , \cdot \rangle : E\left( {K}_{v}\right) \times \mathrm{{WC}}\left( {E/{K}_{v}}\right) \rightarrow \mathbb{Q}/\math... | See Exercise 10.24 for a construction of the Tate pairing. | No |
Theorem 20.1. (Néron [192]) Let \( K \) be a number field, and let \( E \) be an elliptic curve defined over the function field \( K\left( {\mathbb{P}}^{n}\right) \) . Then there are infinitely many points \( t \in {\mathbb{P}}^{n}\left( K\right) \) such that the specialization homomorphism\n\n\[{\sigma }_{t} : E\left(... | Proof. More precisely, Néron proves that the set of \( t \) for which \( {\sigma }_{t} \) is noninjec-tive forms a \ | No |
Corollary 20.1.1. There exist infinitely many elliptic curves \( E/\mathbb{Q} \) such that \( E\left( \mathbb{Q}\right) \) has rank at least 10 . | Proof. Néron [192] originally used (C.20.1) to find families of curves \( E/\mathbb{Q} \) of rank 9 and 10. Subsequently, others have constructed families of rank up to 19; see for example \( \left\lbrack {{76},{85},{188}}\right\rbrack \) . | Yes |
Theorem 20.3. (Silverman [255]) Let \( K \) be a number field, let \( C/K \) be a curve, and let \( E \) be an elliptic curve defined over the function field \( K\left( C\right) \) . Assume that \( E \) is nonconstant, i.e., \( j\left( E\right) \notin K \) . Then the specialization map\n\n\[ \n{\sigma }_{t} : E\left( {... | Proof. See [139, Chapter 12], [256], or [266, III.11.4]. | No |
Theorem 21.2. Let \( E/\mathbb{Q} \) be an elliptic curve with nonintegral \( j \) -invariant. Then the Sato-Tate conjecture (C.21.1) is true for \( E \) . | Proof. [290]; see also [47] and [110]. | No |
Theorem 3.1 Let \( X \) be pseudo-BCI algebra, \( F \) an associative pseudo-BCI filter of \( X \) . Then \( F \) satisfies:\n\n(1) \( \forall x \in X, x \rightarrow \left( {x \rightsquigarrow 1}\right) \in F;x \rightsquigarrow \left( {x \rightarrow 1}\right) \in F \) ;\n\n(2) \( \forall x, y \in X,\left( {x \rightarro... | Proof (1) \( \forall x \in X \), by Definition 2.1 and Theorem 2.1 we have\n\n\[ \left( {x \rightsquigarrow 1}\right) \rightsquigarrow \{ \left( {x \rightsquigarrow 1}\right) \rightarrow \left\lbrack {x \rightarrow \left( {x \rightsquigarrow 1}\right) }\right\rbrack \}\]\n\n\[ = \left( {x \rightsquigarrow 1}\right) \ri... | Yes |
Theorem 3.2 A nonempty subset \( F \) of pseudo-BCI algebra \( X \) is a pseudo-a filter if and only if it satisfies \( \left( {\forall x, y \in X}\right) \) : (1) \( 1 \in F \) , (2) \( \left( {x \rightarrow 1}\right) \rightsquigarrow y \in F \Rightarrow y \rightsquigarrow x \in F \) , (3) \( \left( {x \rightsquigarro... | Proof If \( F \) is a pseudo-a filter of \( X \), by Theorem 2.8 we know that (1) \( \sim \) (3) hold for \( F \) . Assume non-empty subset \( F \) of \( X \) satisfies \( \left( 1\right) \sim \left( 3\right) .\forall x, y, z \in X \), if \( \left( {x \rightarrow 1}\right) \sim \left( {y \rightarrow z}\right) \in F \) ... | Yes |
Theorem 3.3 Let \( X \) be pseudo-BCI algebra, \( F \) an associative pseudo-BCI filter of \( X \) . Then \( F \) is a pseudo-a filter of \( X \) . | Proof \( \forall x, y \in X \), assume \( \left( {x \rightarrow 1}\right) \rightsquigarrow y \in F \) . Since \( \left( {x \rightarrow 1}\right) \rightsquigarrow y \leq \left( {y \rightsquigarrow x}\right) \rightarrow \left\lbrack {\left( {x \rightarrow 1}\right) \rightarrow x}\right\rbrack \) . By Lemma 3.1, we have \... | Yes |
Theorem 4.1 Let \( X \) be pseudo-BCI algebra, \( F \) a pseudo-a filter of \( X \) . Then \( F \) satisfies:\n\n(1) \( \forall x \in X, x \in F \Rightarrow x \rightarrow 1 \in F, x \rightsquigarrow 1 \in F \) ;\n\n(2) \( \forall x \in X,\left( {x \rightarrow 1}\right) \rightsquigarrow x \in F;\left( {x \rightsquigarro... | Proof (1) \( \forall x \in X \), assume \( x \in F \) . Since \( \left( {1 \rightsquigarrow 1}\right) \rightarrow x = 1 \rightarrow x = x \in F \) . Using Theorem 2.8(2), we get \( x \rightarrow 1 \in F \) . By Theorem 2.1(12), we have \( x \rightsquigarrow 1 = x \rightarrow 1 \in F \) .\n\n(2) \( \forall x \in X \), s... | Yes |
Theorem 4.2 Let \( X \) be pseudo-BCI algebra, \( F \) a pseudo-BCI filter of \( X \) . Then \( F \) is a pseudo-a filter of \( X \) if and only if \( F \) is an associative pseudo-BCI filter of \( X \) . | Proof By Theorem 3.4, we only prove that every pseudo-a filter is associative.\n\nAssume that \( F \) is a pseudo-a filter of \( X.\forall x, y \in X \), if \( x \rightarrow \left( {x \rightsquigarrow y}\right) \in F \) . Using Theorem 4.1(8), \( x \) \( \rightarrow \left( {x \rightsquigarrow 1}\right) \in F \) . Since... | Yes |
Theorem 4.3 Let \( \\left( {X; \\leq , \\rightarrow , \\rightsquigarrow ,1}\\right) \) be a pseudo-BCI algebra. If \( X \) is anti-grouped and T-type, then \( \\forall x, y \\in X, x \\rightarrow y = x \\rightsquigarrow y \), and \( X \) is an associative BCI-algebra. | Proof By Theorem 2.12 and Definition 2.15, we have \( x \\rightarrow 1 = \\left( {x \\rightarrow 1}\\right) \\rightarrow 1 = x,\\forall x \\in X \) . Applying Theorem 2.2(11), \( \\forall x, y \\in X \) ,\n\n\[ \nx \\rightarrow y = \\left( {x \\rightarrow y}\\right) \\rightarrow 1 = \\left( {x \\rightarrow 1}\\right) \... | Yes |
Theorem 4.4 Let \( \left( {X; \leq , \rightarrow , \rightsquigarrow ,1}\right) \) be a pseudo-BCI algebra. Then the following statements are equivalent:\n\n(1) \( \forall x, y \in X, x \rightarrow y = x \rightsquigarrow y \), and \( X \) is an associative BCI-algebra;\n\n(2) Every pseudo-BCI filter of \( X \) is a pseu... | Proof (1) \( \Rightarrow \) (2): By Theorem 2.23, Theorem 4.2 and Definition 2.4.\n\n\( \left( 2\right) \Rightarrow \left( 3\right) \) : It is Obvious.\n\n\( \left( 3\right) \Rightarrow \left( 1\right) \) : If \( \{ 1\} \) is a pseudo-a filter and associative pseudo-filter. By Theorem 4.2, Theorem 3.2 (4) and (5), \( \... | Yes |
Theorem 5.1 Let \( X \) be pseudo-BCI algebra, \( F \) a pseudo-BCI filter of \( X \) . Then \( F \) is a pseudo-a filter of \( X \) if and only if \( F \) is anti-grouped and pseudo-q filter of \( X \) . | Proof If \( F \) is a pseudo-a filter of \( X \), by Theorem 4.1 (3) and (7), \( F \) is anti-grouped and pseudo-q filter of \( X \) .\n\nAssume that \( F \) is anti-grouped and pseudo-q filter of \( X.\forall x, y \in X \), suppose \( \left( {x \rightarrow 1}\right) \rightsquigarrow y \in F \) . By Theorem 2.2(11),(9)... | Yes |
Theorem 5.2 Let \( X \) be pseudo-BCI algebra, \( F \) a pseudo-BCI filter of \( X \) . Then \( F \) is an associative if and only if \( F \) is anti-grouped and T-type pseudo-BCI filter of \( X \) . | Proof If \( F \) is an associative pseudo-BCI filter of \( X \), by Theorem 3.2(4) and (5), \( F \) is anti-grouped and T-type pseudo-BCI filter of \( X \) .\n\nAssume that \( F \) is anti-grouped and T-type. \( \forall x, y \in X \), suppose \( x \rightsquigarrow \left( {x \rightarrow y}\right) \in F \) . By Theorem 2... | Yes |
Theorem 3.2 Let \( X \) be a pseudo-BCI algebra, \( + \) and \( \oplus \) the derived addition of \( X \) . Then\n\n(1) \( x \oplus y = y + x,\forall x, y \in X \) . | Proof (1) \( \forall x, y \in X \), by Defintition 3.1 and Theorem 2.2 (4),(12), we have\n\n\[ x \oplus y = \left\lbrack {x \sim ,\left( {y \rightarrow 1}\right) }\right\rbrack \rightarrow 1 = \left\lbrack {y \rightarrow \left( {x \sim ,1}\right) }\right\rbrack \rightarrow 1 = \left\lbrack {y \rightarrow \left( {x \rig... | Yes |
Theorem 3.4 Let \( \left( {X; \leq , \rightarrow , \sim ,1}\right) \) be a pseudo-BCI algebra, \( + \) and \( \oplus \) derived addition of \( X \) . Then (1) \( X \) is anti-grouped if and only if \( \left( {X; + }\right) \) is a group with unit 1 . | Proof We only prove (1). If \( \mathrm{X} \) is an anti-grouped pseudo-BCI algebra, then for any \( x, y \in X \), by Theorem 3.3 and Theorem 2.7, \[ x + 1 = 1 + x = \left( {x \rightarrow 1}\right) \rightarrow 1 = x;x + \left( {x \rightarrow 1}\right) = \left( {x \rightarrow 1}\right) + x = 1 \] From this and Theorem 3... | Yes |
Theorem 3.5 Let \( \left( {X; \leq , \rightarrow , \sim ,1}\right) \) be a T-type pseudo-BCI algebra, \( {AG}\left( X\right) \) the anti-grouped part of \( X, + \) and \( \oplus \) derived addition of \( X \) . Then\n\n(1) \( \left( {{AG}\left( X\right) ; + }\right) \) is an involutive group with unit 1 . | Proof We only prove (1). If \( X \) is a T-type pseudo-BCI algebra, then for any \( x \in X, x + x = \lbrack x \) \( \rightarrow \left( {x \rightarrow 1}\right) \rbrack \rightarrow 1 = 1 \rightarrow 1 = 1 \) (by Theorem 2.11).\n\nIt is easy to prove that \( \forall x, y \in {AG}\left( X\right), x + y \in {AG}\left( X\r... | Yes |
Theorem 4.2 Let \( X \) be a pseudo-BCI algebra, \( {AG}\left( X\right) \) the anti-grouped part of \( X, T\left( X\right) \) the T-part of \( X \) . Then \( T\left( X\right) \subseteq {AG}\left( X\right) \) . | Proof If \( a \in T\left( X\right) \), then \( a = x \rightarrow \left( {x \rightarrow 1}\right) \) for some \( x \in X \) . By Theorem 2.2(9),(11) and (12) we have\n\n\[x \rightarrow \left( {x \rightarrow 1}\right) \]\n\n\[ = x \rightarrow \{ \left\lbrack {\left( {x \rightarrow 1}\right) \rightarrow 1}\right\rbrack \r... | Yes |
Theorem 4.3 Let \( X \) be a strong pseudo-BCI algebra, \( {AG}\left( X\right) \) the anti-grouped part of \( X, T\left( X\right) \) the T-part of \( X \) . Then \( T\left( X\right) \) is a pseudo-BCI filter of \( {AG}\left( X\right) \) . | Proof By Theorem 2.23 we know that \( {AG}\left( X\right) \) is a pseudo-BCI algebra.\n\nAssume \( x \rightarrow y \in T\left( X\right), x \in T\left( X\right) \) and \( y \in {AG}\left( X\right) \) . By Theorem 4.3 and Theorem 4.4, \( x \in T\left( X\right) \) \( \subseteq {AG}\left( X\right) \) and \( \left\lbrack {\... | Yes |
Theorem 4.4 Let \( X \) be a strong pseudo-BCI algebra, \( K\left( X\right) \) the pseudo-BCK part of \( X, T\left( X\right) \) the T-part of \( X \) . Then the following statements are equivalent:\n\n(1) \( T\left( X\right) \) is a pseudo-BCI filter of \( X \) ;\n\n(2) \( a \rightarrow x = a \rightarrow y \Rightarrow ... | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \) : Let \( T\left( X\right) \) be a pseudo-BCI filter of \( X \) and \( a \rightarrow x = a \rightarrow y, a \in T\left( X\right), x, y \in \) \( K\left( X\right) \) . Then\n\n\[ a \rightarrow \left( {x \sim \rightarrow y}\right) = x \sim \rightarrow \left( {a \righ... | Yes |
Theorem 4.5 Let \( F \) be a pseudo-BCI filter of pseudo-BCI algebra \( X, T\left( X\right) \) the T-part of \( X \) . Then \( F \) is a T-type pseudo-BCI filter of \( X \) if and only if \( T\left( X\right) \subseteq F \) . | Proof If \( F \) is a T-type pseudo-BCI filter of \( X \) . By Theorem 2.13, since \( 1 \in F \), so \( x \rightarrow \left( {x \rightarrow 1}\right) \in \) \( F,\forall x \in X \) . That is, \( T\left( X\right) \subseteq F \) .\n\nConversely, assume \( T\left( X\right) \subseteq F \) . Then \( x \rightarrow \left( {x ... | Yes |
Theorem 1.2. A function \( f \in {C}^{1}\left( D\right) \) satisfies the Cauchy-Riemann equations at the point \( a \in D \) if and only if its differential \( d{f}_{a} \) at \( a \) is \( \mathbb{C} \) -linear. In particular, \( f \in \mathcal{O}\left( D\right) \) if and only if \( {df} \) is \( \mathbb{C} \) -linear ... | Proof. Since \( \partial {f}_{a} \) is obviously \( \mathbb{C} \) -linear for any \( a \in D \), one implication is trivial. For the other implication, suppose \( {\beta }_{k} = \partial f/\partial {\bar{z}}_{k}\left( a\right) \neq 0 \) for some \( k \) . Let \( {\alpha }_{k} = \partial f/\partial {z}_{k}\left( a\right... | Yes |
Theorem 1.3. Let \( P = P\left( {a, r}\right) \) be a polydisc in \( {\mathbb{C}}^{n} \) with multiradius \( r = \left( {{r}_{1},\ldots ,{r}_{n}}\right) \) . Suppose \( f \in C\left( \bar{P}\right) \), and \( f \) is holomorphic in each variable separately, i.e., for each \( z \in \bar{P} \) and \( 1 \leq j \leq n \), ... | Proof. We use induction over the number of variables \( n \) . For \( n = 1 \) one has the classical Cauchy integral formula, which we assume as known. Suppose \( n > 1 \), and that the theorem has been proved for \( n - 1 \) variables. For \( z \in P \) fixed, apply the inductive hypothesis with respect to \( \left( {... | Yes |
Corollary 1.5. Suppose \( f \in C\left( D\right) \) (or just bounded on \( D \) ) is holomorphic in each variable separately. Then \( f \in {C}^{\infty }\left( D\right) \) and, in particular, \( f \in \mathcal{O}\left( D\right) \) . For any \( \alpha \in {\mathbb{N}}^{n} \) , \( {D}^{\alpha }f \in \mathcal{O}\left( D\r... | Proof. Apply Theorem 1.3 to a polydisc \( P\left( {a, r}\right) \subset \subset D \) ; in (1.15) it is legitimate to differentiate under the integral sign as often as needed. | Yes |
Theorem 1.6 (Cauchy estimates). Let \( f \in \mathcal{O}\left( {P\left( {a, r}\right) }\right) \) . Then, for all \( \alpha \in {\mathbb{N}}^{n} \) , (1.19) \[ \left| {{D}^{\alpha }f\left( a\right) }\right| \leq \frac{\alpha !}{{r}^{\alpha }}{\left| f\right| }_{P\left( {a, r}\right) } \] (1.20) \[ \left| {{D}^{\alpha }... | Proof. Fix \( 0 < \rho < r \) . Apply Theorem 1.3 to \( P\left( {a,\rho }\right) \subset \subset P\left( {a, r}\right) \) and differentiate under the integral sign, obtaining (1.21) \[ {D}^{\alpha }f\left( a\right) = \frac{\alpha !}{{\left( 2\pi i\right) }^{n}}{\int }_{{b}_{o}P\left( {a,\rho }\right) }\frac{f\left( \ze... | Yes |
Corollary 1.7. For each \( \\alpha \\in {\\mathbb{N}}^{n},1 \\leq p \\leq \\infty \\), and \( \\Omega \\subset \\subset D \) there is a constant \( C = C\\left( {\\alpha, p,\\Omega, D}\\right) \) such that\n\n(1.23)\n\n\[ \n{\\left| {D}^{\\alpha }f\\right| }_{\\Omega } \\leq C\\parallel f{\\parallel }_{{L}^{p}\\left( D... | Proof. Fix \( 0 < \\delta < \\operatorname{dist}\\left( {\\Omega ,{bD}}\\right) \) and let \( r = \\delta /\\sqrt{n} \) . Then (1.20) holds for each \( a \\in \\Omega \), and since \( \\parallel f{\\parallel }_{{L}^{1}\\left( {P\\left( {a, r}\\right) }\\right) } \\leq \) constant \\cdot \\parallel f{\\parallel }_{{L}^{... | Yes |
Theorem 1.8. For \( P = P\left( {a, r}\right) \) and \( z \in \bar{P} \) one has\n\n(1.24)\n\n\[ \left| {f\left( z\right) }\right| \leq {\left| f\right| }_{{b}_{o}P}\text{ for all }f \in C\left( \bar{P}\right) \cap \mathcal{O}\left( P\right) . \] | Proof. It is enough to prove (1.24) for \( z \in P \) . From (1.15) it follows by an obvious estimate that there is a constant \( {C}_{z} \) such that \( \left| {f\left( z\right) }\right| \leq {C}_{z}{\left| f\right| }_{{b}_{o}P} \) for all \( f \in A\left( P\right) \) . Hence, for \( k = 1,2,\ldots \), since \( {f}^{k... | Yes |
Theorem 1.9. Suppose \( \left\{ {{f}_{j} : j = 1,2,\ldots }\right\} \subset \mathcal{O}\left( D\right) \) converges compactly in \( D \) to the function \( f : D \rightarrow \mathbb{C} \) . Then \( f \in \mathcal{O}\left( D\right) \), and for each \( \alpha \in {\mathbb{N}}^{n} \) ,\n\n\[ \mathop{\lim }\limits_{{j \rig... | The proof of Theorem 1.9 is the same as in the classical case \( n = 1 \) and will be omitted. | No |
Lemma 1.11. The function \( \delta \) defined by (1.26) is a metric on \( C\left( D\right) \) . A sequence \( \left\{ {f}_{j}\right\} \subset C\left( D\right) \) converges compactly to \( f \) if and only if \( \mathop{\lim }\limits_{{j \rightarrow \infty }}\delta \left( {{f}_{j}, f}\right) = 0 \) . The topology on \( ... | The proof is left to the reader. | No |
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