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Lemma 6.3. Let \( R \) be a ring of characteristic 0 that is complete with respect to a discrete valuation \( v \), and let \( p \in \mathbb{Z} \) be a prime with \( v\left( p\right) > 0 \) . (a) Let \( f\left( T\right) \) be a power series of the form\n\n\[ f\left( T\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\... | Proof. (a) For a general term of \( f\left( x\right) \) we have\n\n\[ v\left( {{a}_{n}{x}^{n}/n}\right) \geq {nv}\left( x\right) - v\left( n\right) \;\text{ since }{a}_{n} \in R,\]\n\n\[ \geq {nv}\left( x\right) - \left( {{\log }_{p}n}\right) v\left( p\right) \]\n\nThis last expression goes to \( \infty \) as \( n \) g... | Yes |
Theorem 6.4. Let \( K \) be a field of characteristic 0 that is complete with respect to a normalized discrete valuation \( v \), i.e., \( v\left( {K}^{ * }\right) = \mathbb{Z} \), let \( R \) be the valuation ring of \( K \) , let \( \mathcal{M} \) be the maximal ideal of \( R \), and let \( p \) be a prime with \( v\... | Proof. (a) From (IV.5.2) we have an identity of power series\n\n\[ \n{\log }_{\mathcal{F}}\left( {F\left( {X, Y}\right) }\right) = {\log }_{\mathcal{F}}\left( X\right) + {\log }_{\mathcal{F}}\left( Y\right) \n\]\n\nHence \( {\log }_{\mathcal{F}} \) will be a homomorphism on \( \mathcal{M} \) provided that \( {\log }_{\... | Yes |
Proposition 7.2. Let \( \mathcal{F}/R \) and \( \mathcal{G}/R \) be formal groups, and let \( f : \mathcal{F} \rightarrow \mathcal{G} \) be a homomorphism defined over \( R \) . (a) If \( {f}^{\prime }\left( 0\right) = 0 \), then \( f\left( T\right) = {f}_{1}\left( {T}^{p}\right) \) for some \( {f}_{1} \in R\llbracket ... | Proof. (a) Let \( {\omega }_{\mathcal{F}} \) and \( {\omega }_{\mathcal{G}} \) be the normalized invariant differentials on \( \mathcal{F} \) and \( \mathcal{G} \) . Then \[ 0 = {f}^{\prime }\left( 0\right) {\omega }_{\mathcal{F}}\left( T\right) \] \[ = {\omega }_{\mathcal{G}}\left( {f\left( T\right) }\right) \;\text{f... | Yes |
Corollary 7.5. Let \( E/K \) be an elliptic curve defined over a field of positive characteristic. Then\n\n\[ \operatorname{ht}\left( \widehat{E}\right) = 1\\text{ or }2. \] | Proof. We start with two special cases.\n\nCase 1. \( \\phi \) is the \( {p}^{r} \) -power Frobenius map.\n\nThen (II.2.11) says that \( {\\deg }_{i}\\phi = {p}^{r} \), while \( f\\left( T\\right) = {T}^{{p}^{r}} \), so clearly \( \\operatorname{ht}\\left( f\\right) = r \) .\n\nCase 2. \( \\phi \) is separable.\n\nLet ... | Yes |
Theorem 1.1. (Hasse) Let \( E/{\mathbb{F}}_{q} \) be an elliptic curve defined over a finite field. Then\n\n\[ \left| {\# E\left( {\mathbb{F}}_{q}\right) - q - 1}\right| \leq 2\sqrt{q} \] | Proof. Choose a Weierstrass equation for \( E \) with coefficients in \( {\mathbb{F}}_{q} \), and let\n\n\[ \phi : E \rightarrow E,\;\left( {x, y}\right) \mapsto \left( {{x}^{q},{y}^{q}}\right) ,\]\n\nbe the \( {q}^{\text{th }} \) -power Frobenius morphism (III.4.6). Since the Galois group \( {G}_{{\mathbb{F}}_{q}/{\ma... | Yes |
Lemma 1.2. Let \( A \) be an abelian group, and let\n\n\[ d : A \rightarrow \mathbb{Z} \]\n\nbe a positive definite quadratic form. Then\n\n\[ \left| {d\left( {\psi - \phi }\right) - d\left( \phi \right) - d\left( \psi \right) }\right| \leq 2\sqrt{d\left( \phi \right) d\left( \psi \right) }\;\text{ for all }\psi ,\phi ... | Proof. For \( \psi ,\phi \in A \), let\n\n\[ L\left( {\psi ,\phi }\right) = d\left( {\psi - \phi }\right) - d\left( \phi \right) - d\left( \psi \right) \]\n\nbe the bilinear form associated to the quadratic form \( d \) . Since \( d \) is positive definite, we have for all \( m, n \in \mathbb{Z} \),\n\n\[ 0 \leq d\left... | Yes |
Corollary 1.4. With notation as above,\n\n\\[ \n\\left| {\\mathop{\\sum }\\limits_{{x \\in {\\mathbb{F}}_{q}}}\\chi \\left( {f\\left( x\\right) }\\right) }\\right| \\leq 2\\sqrt{q} \n\\] | We note that the sum in (V.1.4) consists of \\( q \\) terms, most of which are \\( \\pm 1 \\) , so (V.1.4) says that as \\( x \\) runs through \\( {\\mathbb{F}}_{q} \\), the values of the cubic polynomial \\( f\\left( x\\right) \\) tend to be equally distributed between squares and nonsquares. Indeed, if one takes a ra... | No |
Let \( V = {\mathbb{P}}^{N} \) . Then a point of \( V\left( {\mathbb{F}}_{{q}^{n}}\right) \) is given by homogeneous coordinates \( \left\lbrack {{x}_{0},\ldots ,{x}_{N}}\right\rbrack \) with \( {x}_{i} \in {\mathbb{F}}_{{q}^{n}} \) not all zero. Two sets of coordinates give the same point if they differ by multiplicat... | \[
\# {\mathbb{P}}^{N}\left( {\mathbb{F}}_{{q}^{n}}\right) = \frac{{q}^{n\left( {N + 1}\right) } - 1}{{q}^{n} - 1} = \mathop{\sum }\limits_{{i = 0}}^{N}{q}^{ni},
\]
so
\[
\log Z\left( {{\mathbb{P}}^{n}/{\mathbb{F}}_{q};T}\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {\mathop{\sum }\limits_{{i = 0}}^{N}{q}^... | Yes |
Proposition 2.3. Let \( \psi \in \operatorname{End}\left( E\right) \) . Then\n\n\[ \det \left( {\psi }_{\ell }\right) = \deg \left( \psi \right) \;\text{ and }\;\operatorname{tr}\left( {\psi }_{\ell }\right) = 1 + \deg \left( \psi \right) - \deg \left( {1 - \psi }\right) . \]\n\nIn particular, \( \det \left( {\psi }_{\... | PROOF. We already proved this result (III.8.6). | No |
Theorem 2.3.1. Let \( E/{\mathbb{F}}_{q} \) be an elliptic curve, let\n\n\[ \phi : E \rightarrow E,\;\left( {x, y}\right) \mapsto \left( {{x}^{q},{y}^{q}}\right) ,\]\n\nbe the \( {q}^{\text{th }} \) -power Frobenius endomorphism, and let\n\n\[ a = q + 1 - \# E\left( {\mathbb{F}}_{q}\right) .\n\n(a) Let \( \alpha ,\beta... | Proof. We observed in (V§1) that (III.5.5) and (III.4.10c) imply that\n\n\[ \# E\left( {\mathbb{F}}_{q}\right) = \deg \left( {1 - \phi }\right) \]\n\nWe use (V.2.3) to compute\n\n\[ \det \left( {\phi }_{\ell }\right) = \deg \left( \phi \right) = q \]\n\n\[ \operatorname{tr}\left( {\phi }_{\ell }\right) = 1 + \deg \left... | Yes |
Theorem 2.4. Let \( E/{\mathbb{F}}_{q} \) be an elliptic curve. Then there is an \( a \in \mathbb{Z} \) such that\n\n\[ Z\left( {E/{\mathbb{F}}_{q};T}\right) = \frac{1 - {aT} + q{T}^{2}}{\left( {1 - T}\right) \left( {1 - {qT}}\right) }.\]\n\nFurther,\n\n\[ Z\left( {E/{\mathbb{F}}_{q};1/{qT}}\right) = Z\left( {E/{\mathb... | Proof. We compute\n\n\[ \log Z\left( {E/{\mathbb{F}}_{q};T}\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\# E\left( {\mathbb{F}}_{{q}^{n}}\right) {T}^{n}}{n}\;\text{by definition,}\]\n\n\[ = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\left( {1 - {\alpha }^{n} - {\beta }^{n} + {q}^{n}}\right) {T}^{n}}{n}\... | Yes |
Theorem 3.1. ([60]) Let \( K \) be a field of characteristic \( p \), and let \( E/K \) be an elliptic curve. For each integer \( r \geq 1 \), let\n\n\[ \n{\phi }_{r} : E \rightarrow {E}^{\left( {p}^{r}\right) }\;\text{ and }\;{\widehat{\phi }}_{r} : {E}^{\left( {p}^{r}\right) } \rightarrow E \n\]\n\nbe the \( {p}^{r} ... | Proof of V.3.1. Conditions (i)-(v) are invariant under field extension, so we may assume that \( K \) is algebraically closed, and in particular, a perfect field. For notational convenience, we let \( \phi = {\phi }_{1} \).\n\n(a) Since the Frobenius map is purely inseparable (II.2.11b), we have\n\n\[ \n{\deg }_{s}\lef... | No |
Theorem 4.1. Let \( {\mathbb{F}}_{q} \) be a finite field of characteristic \( p \geq 3 \). (a) Let \( E/{\mathbb{F}}_{q} \) be an elliptic curve given by a Weierstrass equation \( E : {y}^{2} = f\left( x\right) \) where \( f\left( x\right) \in {\mathbb{F}}_{q}\left\lbrack x\right\rbrack \) is a cubic polynomial with d... | Proof. Let \( \chi : {\mathbb{F}}_{q}^{ * } \rightarrow \{ \pm 1\} \) be the unique nontrivial character of order 2, and extend \( \chi \) to \( {\mathbb{F}}_{q} \) by setting \( \chi \left( 0\right) = 0 \). As we have seen in (V.1.3), the character \( \chi \) can be used to count the number of points of \( E \), \( \#... | Yes |
For \( p = {11} \), what is \( {H}_{11}\left( t\right) \) and its factorization modulo 11? | \[ {H}_{11}\left( t\right) = {t}^{5} + 3{t}^{4} + {t}^{3} + {t}^{2} + {3t} + 1 \] \[ \equiv \left( {{t}^{2} - t + 1}\right) \left( {t + 1}\right) \left( {t - 2}\right) \left( {t + 5}\right) \;\left( {\;\operatorname{mod}\;{11}}\right) . \] | Yes |
We compute for which primes \( p \geq 5 \) the elliptic curve \( E : {y}^{2} = {x}^{3} + 1 \) with \( j = 0 \) is supersingular. | The criterion (V.4.1a) says that we need to compute the coefficient of \( {x}^{p - 1} \) in the polynomial \( {\left( {x}^{3} + 1\right) }^{\left( {p - 1}\right) /2} \) . If \( p \equiv 2\left( {\;\operatorname{mod}\;3}\right) \), then there is no \( {x}^{p - 1} \) term, so \( E \) is supersingular. On the other hand, ... | Yes |
For which primes \( p \geq 3 \) the elliptic curve \( E : {y}^{2} = {x}^{3} + x \) with \( j = {1728} \) is supersingular? | This is determined by the coefficient of \( {x}^{\left( {p - 1}\right) /2} \) in the polynomial \( {\left( {x}^{2} + 1\right) }^{\left( {p - 1}\right) /2} \) . This coefficient is equal to 0 if \( p \equiv 3\left( {\;\operatorname{mod}\;4}\right) \) and to \( \left( \begin{matrix} \left( {p - 1}\right) /2 \\ \left( {p ... | Yes |
Proposition 2.1. A holomorphic elliptic function, i.e., an elliptic function with no poles, is constant. Similarly, an elliptic function with no zeros is constant. | Proof. Suppose that \( f\left( z\right) \in \mathbb{C}\left( \Lambda \right) \) is holomorphic. Let \( D \) be a fundamental parallelogram for \( \Lambda \) . The periodicity of \( f \) implies that\n\n\[ \mathop{\sup }\limits_{{z \in \mathbb{C}}}\left| {f\left( z\right) }\right| = \mathop{\sup }\limits_{{z \in \bar{D}... | Yes |
Theorem 2.2. Let \( f \in \mathbb{C}\left( \Lambda \right) \) be an elliptic function relative to \( \Lambda \) .\n\n(a) \( \mathop{\sum }\limits_{{w \in \mathbb{C}/\Lambda }}{\operatorname{res}}_{w}\left( f\right) = 0 \) . | Proof. Let \( D \) be a fundamental parallelogram for \( \Lambda \) such that \( f\left( z\right) \) has no zeros or poles on the boundary \( \partial D \) of \( D \) . All three parts of the theorem are simple applications of the residue theorem [3, Chapter 4, Theorem 19] applied to appropriately chosen functions on \... | Yes |
Corollary 2.3. A nonconstant elliptic function has order at least 2. | Proof. If \( f\left( z\right) \) has a single simple pole, then (VI.2.2a) tells us that the residue at that pole is 0, so \( f\left( z\right) \) is actually holomorphic. Now apply (VI.2.1). | No |
Theorem 2.4. The following is an exact sequence:\n\n\[ 1 \rightarrow {\mathbb{C}}^{ * } \rightarrow \mathbb{C}{\left( \Lambda \right) }^{ * }\xrightarrow[]{\;\text{ div }\;}{\operatorname{Div}}^{0}\left( {\mathbb{C}/\Lambda }\right) \xrightarrow[]{\;\text{ sum }\;}\mathbb{C}/\Lambda \rightarrow 0. \] | Proof. Exactness on the left is clear, and exactness on the right follows from \( \operatorname{sum}\left( {\left( w\right) - \left( 0\right) }\right) = w \) . Exactness at \( \mathbb{C}{\left( \Lambda \right) }^{ * } \) is (VI.2.1), and exactness at \( {\operatorname{Div}}^{0}\left( {\mathbb{C}/\Lambda }\right) \) is ... | No |
Theorem 3.1. Let \( \Lambda \subset \mathbb{C} \) be a lattice.\n\n(a) The Eisenstein series \( {G}_{2k}\left( \Lambda \right) \) is absolutely convergent for all \( k > 1 \) . | Proof. Since \( \Lambda \) is discrete in \( \mathbb{C} \), it is not hard to see that there is a constant \( c = c\left( \Lambda \right) \) such that for all \( N \geq 1 \), the number of points in an annulus satisfies\n\n\[ \n\# \{ \omega \in \Lambda : N \leq \left| \omega \right| < N + 1\} < {cN}.\n\]\n\n(See Exerci... | No |
Theorem 3.2. Let \( \Lambda \subset \mathbb{C} \) be a lattice. Then\n\n\[ \mathbb{C}\left( \Lambda \right) = \mathbb{C}\left( {\wp \left( z\right) ,{\wp }^{\prime }\left( z\right) }\right) \]\n\n i.e., every elliptic function is a rational combination of \( \wp \) and \( {\wp }^{\prime } \) . | Proof. Let \( f\left( z\right) \in \mathbb{C}\left( \Lambda \right) \) . Writing\n\n\[ f\left( z\right) = \frac{f\left( z\right) + f\left( {-z}\right) }{2} + \frac{f\left( z\right) - f\left( {-z}\right) }{2}, \]\n\nwe see that it suffices to prove the theorem for functions that are either odd or even. Further, if \( f\... | Yes |
Lemma 3.3. (a) The infinite product for \( \sigma \left( z\right) \) defines a holomorphic function on all of \( \mathbb{C} \) . It has simple zeros at each \( z \in \Lambda \) and no other zeros. | Proof. (a) The absolute and uniform convergence of the infinite product on \( \mathbb{C} \) follows from (VI.3.1a) and standard facts about convergence of infinite products [3, Chapter 5, §2.3]. The location and order of the zeros is clear by inspection. | No |
Proposition 3.4. Let \( {n}_{1},\ldots ,{n}_{r} \in \mathbb{Z} \) and \( {z}_{1},\ldots ,{z}_{r} \in \mathbb{C} \) satisfy\n\n\[ \sum {n}_{i} = 0\;\text{ and }\;\sum {n}_{i}{z}_{i} \in \Lambda . \]\n\nThen there exists an elliptic function \( f\left( z\right) \in \mathbb{C}\left( \Lambda \right) \) satisfying\n\n\[ \op... | Proof. Let \( \lambda = \sum {n}_{i}{z}_{i} \in \Lambda \) . Replacing\n\n\[ {n}_{1}\left( {z}_{1}\right) + \cdots + {n}_{r}\left( {z}_{r}\right) \;\text{ by }\;{n}_{1}\left( {z}_{1}\right) + \cdots + {n}_{r}\left( {z}_{r}\right) + \left( 0\right) - \left( \lambda \right) ,\]\n\nwe may assume that \( \sum {n}_{i}{z}_{i... | Yes |
Theorem 3.5. (a) The Laurent series for \( \wp \left( z\right) \) around \( z = 0 \) is given by\n\n\[ \wp \left( z\right) = \frac{1}{{z}^{2}} + \mathop{\sum }\limits_{{k = 1}}^{\infty }\left( {{2k} + 1}\right) {G}_{{2k} + 2}{z}^{2k}. \] | Proof. (a) For all \( z \) with \( \left| z\right| < \left| \omega \right| \) we have\n\n\[ \frac{1}{{\left( z - \omega \right) }^{2}} - \frac{1}{{\omega }^{2}} = \frac{1}{{\omega }^{2}}\left( {\frac{1}{{\left( 1 - z/\omega \right) }^{2}} - 1}\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {n + 1}\right) \fra... | Yes |
Proposition 3.6. Let \( {g}_{2} = {g}_{2}\left( \Lambda \right) \) and \( {g}_{3} = {g}_{3}\left( \Lambda \right) \) be the quantities associated to a lattice \( \Lambda \subset \mathbb{C} \) . (a) The polynomial \[ f\left( x\right) = 4{x}^{3} - {g}_{2}x - {g}_{3} \] has distinct roots, so its discriminant \[ \Delta \l... | Proof. (a) Let \( \left\{ {{\omega }_{1},{\omega }_{2}}\right\} \) be a basis for \( \Lambda \) and let \( {\omega }_{3} = {\omega }_{1} + {\omega }_{2} \) . Then, since \( {\wp }^{\prime }\left( z\right) \) is an odd elliptic function, we see that \[ {\wp }^{\prime }\left( \frac{{\omega }_{i}}{2}\right) = - {\wp }^{\p... | Yes |
Theorem 4.1. (a) With notation as above, the association\n\n\[ \n\\left\\{ {\\alpha \\in \\mathbb{C} : \\alpha {\\Lambda }_{1} \\subset {\\Lambda }_{2}}\\right\\} \\rightarrow \\left\\{ \\begin{array}{l} \\text{ holomorphic maps } \\\\\n\\phi : \\mathbb{C}/{\\Lambda }_{1} \\rightarrow \\mathbb{C}/{\\Lambda }_{2} \\\\\n... | Proof. (a) If \( {\\phi }_{\\alpha } = {\\phi }_{\\beta } \), then\n\n\[ \n{\\alpha z} \\equiv {\\beta z}\\;\\left( {\\;\\operatorname{mod}\\;{\\Lambda }_{2}}\\right) \\;\\text{ for all }z \\in \\mathbb{C}.\n\]\n\nHence the map \( z \\mapsto \\left( {\\alpha - \\beta }\\right) z \) sends \( \\mathbb{C} \) to \( {\\Lamb... | Yes |
Theorem 5.1. (Uniformization Theorem) Let \( A, B \in \mathbb{C} \) be complex numbers satisfying \( {A}^{3} - {27}{B}^{2} \neq 0 \) . Then there exists a unique lattice \( \Lambda \subset \mathbb{C} \) satisfying\n\n\[{g}_{2}\left( \Lambda \right) = A\;\text{ and }\;{g}_{3}\left( \Lambda \right) = B.\] | Proof. The proof may be found in many textbooks; see for example [5, Theorem 2.9], [210, I.3.13], [249, §4.2], [266, I.4.3], or [232, VII Proposition 5]. | No |
Corollary 5.1.1. Let \( E/\mathbb{C} \) be an elliptic curve. There exist a lattice \( \Lambda \subset \mathbb{C} \), unique up to homothety, and a complex analytic isomorphism\n\n\[ \phi : \mathbb{C}/\Lambda \rightarrow E\left( \mathbb{C}\right) ,\;\phi \left( z\right) = \left\lbrack {\wp \left( {z,\Lambda }\right) ,{... | Proof. The existence is immediate from (VI.3.6b) and (VI.5.1), and the uniqueness is (VI.4.1.1). | No |
Let \( E/\mathbb{C} \) be an elliptic curve with Weierstrass coordinate functions \( x \) and \( y \). (a) Let \( \alpha \) and \( \beta \) be closed paths on \( E\left( \mathbb{C}\right) \) giving a basis for \( {H}_{1}\left( {E,\mathbb{Z}}\right) \). Then the periods \[ {\omega }_{1} = {\int }_{\alpha }\frac{dx}{y}\;... | From (VI.5.1.1), there exists some lattice \( {\Lambda }_{1} \) such that the map \[ {\phi }_{1} : \mathbb{C}/{\Lambda }_{1} \rightarrow E\left( \mathbb{C}\right) ,\;{\phi }_{1}\left( z\right) = \left\lbrack {\wp \left( {z,{\Lambda }_{1}}\right) ,{\wp }^{\prime }\left( {z,{\Lambda }_{1}}\right) ,1}\right\rbrack , \] is... | Yes |
Theorem 5.3. The following categories are equivalent:\n\n(a) Objects: Elliptic curves over \( \mathbb{C} \).\n\nMaps: Isogenies.\n\n(b) Objects: Elliptic curves over \( \mathbb{C} \).\n\nMaps: Complex analytic maps taking \( O \) to \( O \).\n\n(c) Objects: Lattices \( \Lambda \subset \mathbb{C} \), up to homothety.\n\... | Proof. The one-to-one correspondence between elliptic curves over \( \mathbb{C} \) and lattices modulo homothety follows from (VI.3.6b), (VI.5.1.1), and (VI.5.2). The matchup of the maps in (a), (b), and (c) is precisely the content of (VI.4.1). | Yes |
Proposition 5.4. Let \( E/\mathbb{C} \) be an elliptic curve and let \( m \geq 1 \) be an integer.\n\n(a) There is an isomorphism of abstract groups\n\n\[ E\left\lbrack m\right\rbrack \cong \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z} \] | Proof. (a) From (VI.5.1.1), we know that \( E\left( \mathbb{C}\right) \) is isomorphic to \( \mathbb{C}/\Lambda \) for some lattice \( \Lambda \subset \mathbb{C} \) . Hence\n\n\[ E\left\lbrack m\right\rbrack \cong \left( \frac{\mathbb{C}}{\Lambda }\right) \left\lbrack m\right\rbrack \cong \frac{\frac{1}{m}\Lambda }{\La... | Yes |
Theorem 5.5. Let \( E/\mathbb{C} \) be an elliptic curve, and let \( {\omega }_{1} \) and \( {\omega }_{2} \) be generators for the lattice \( \Lambda \) associated to \( E \) by (VI.5.1.1). Then one of the following is true:\n\n(i) \( \operatorname{End}\left( E\right) = \mathbb{Z} \).\n\n(ii) The field \( \mathbb{Q}\l... | Proof. Let \( \tau = {\omega }_{2}/{\omega }_{1} \) . Multiplying \( \Lambda \) by \( 1/{\omega }_{1} \) shows that \( \Lambda \) is homothetic to \( \mathbb{Z} + \mathbb{Z}\tau \), so we may replace \( \Lambda \) by \( \mathbb{Z} + \mathbb{Z}\tau \) . Let\n\n\[ \mathcal{R} = \{ \alpha \in \mathbb{C} : {\alpha \Lambda ... | Yes |
Proposition 5.6. Let \( E/\mathbb{C} \) be an elliptic curve, and fix a lattice \( \Lambda \) and an isomorphism \( E\left( \mathbb{C}\right) \cong \mathbb{C}/\Lambda \) .\n\n(a) There is a natural isomorphism\n\n\[ \n{H}_{1}\left( {E\left( \mathbb{C}\right) ,\mathbb{Z}}\right) \overset{ \sim }{ \rightarrow }\Lambda ,\... | Proof. (a) We proved this during the course of proving (VI.5.2a).\n\n(b) From (a) we have\n\n\[ \n{H}_{1}\left( {E\left( \mathbb{C}\right) ,\mathbb{Z}/m\mathbb{Z}}\right) \cong {H}_{1}\left( {E\left( \mathbb{C}\right) ,\mathbb{Z}}\right) \otimes \mathbb{Z}/m\mathbb{Z} \cong \Lambda \otimes \mathbb{Z}/m\mathbb{Z} \cong ... | Yes |
Theorem 6.1. Let \( K \) be a field of characteristic 0 and let \( E/K \) be an elliptic curve. (a) Let \( m \geq 1 \) be an integer. Then\n\n\[ E\left\lbrack m\right\rbrack \cong \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z} \]\n\n(b) The endomorphism ring of \( E \) is either \( \mathbb{Z} \) or an order in a ... | Proof. (a) This is immediate from (VI.5.4) and the Lefschetz principle.\n\n(b) Here we can apply the Lefschetz principle to (VI.5.5), once we note that \( \operatorname{End}\left( E\right) \) is countably (in fact, finitely) generated from (III.7.5). Alternatively, even without (III.7.5), we can argue as follows. If \(... | Yes |
Proposition 1.3. (a) Every elliptic curve \( E/K \) has a minimal Weierstrass equation. | Proof. (a) One can easily find some Weierstrass equation with all \( {a}_{i} \in R \), and among such equations, there exists (at least) one that minimizes \( v\left( \Delta \right) \), since \( v \) is discrete. | No |
There is an exact sequence of abelian groups\n\n\[ 0 \rightarrow {E}_{1}\left( K\right) \rightarrow {E}_{0}\left( K\right) \rightarrow {\widetilde{E}}_{\mathrm{{ns}}}\left( k\right) \rightarrow 0, \]\n\nwhere the right-hand map is reduction modulo \( \pi \) . | We begin by showing that the reduction map is surjective. To do this, we use Hensel’s lemma and the completeness of \( K \) . Thus let\n\n\[ f\left( {x, y}\right) = {y}^{2} + {a}_{1}{xy} + {a}_{3}y - {x}^{3} - {a}_{2}{x}^{2} - {a}_{4}x - {a}_{6} = 0 \]\n\nbe a minimal Weierstrass equation for \( E \), let \( \widetilde... | Yes |
Proposition 2.2. Let \( E/K \) be given by a minimal Weierstrass equation, let \( \widehat{E}/R \) be the formal group associated to \( E \) as in (IV.2.2.3), and let \( w\left( z\right) \in R\llbracket z\rrbracket \) be the power series from (IV.1.1). Then the map \[ \widehat{E}\left( \mathcal{M}\right) \rightarrow {E... | Proof. From (IV.1.1b), the point \( \left( {z/w\left( z\right) , - 1/w\left( z\right) }\right) \), when considered as a pair of power series, satisfies the Weierstrass equation for \( E \) . Since \[ w\left( z\right) = {z}^{3}\left( {1 + \cdots }\right) \in R\llbracket z\rrbracket \] we see that \( w\left( z\right) \) ... | Yes |
Proposition 3.1. Let \( E/K \) be an elliptic curve and let \( m \geq 1 \) be an integer that is relatively prime to \( \operatorname{char}\left( k\right) \) .\n\n(a) The subgroup \( {E}_{1}\left( K\right) \) has no nontrivial points of order \( m \) .\n\n(b) Assume further that the reduced curve \( \widetilde{E}/k \) ... | Proof. From (VII.2.1) we have an exact sequence\n\n\[ 0 \rightarrow {E}_{1}\left( K\right) \rightarrow {E}_{0}\left( K\right) \rightarrow {\widetilde{E}}_{\mathrm{{ns}}}\left( k\right) \rightarrow 0. \]\n\nWe know from (VII.2.2) that \( {E}_{1}\left( K\right) \cong \widehat{E}\left( \mathcal{M}\right) \), where \( \wid... | Yes |
Let \( E/\mathbb{Q} \) be the elliptic curve \( E : {y}^{2} + y = {x}^{3} - x + 1 \). | The discriminant of \( E \) is \( \Delta = - {611} = - {13} \cdot {47} \), so \( \widetilde{E} \) is nonsingular modulo 2 . It is easy to check that \( \widetilde{E}\left( {\mathbb{F}}_{2}\right) = \{ O\} \) and \( E\left( \mathbb{Q}\right) \left\lbrack 2\right\rbrack = \{ O\} \) ; hence (VII.3.1) implies that \( E\lef... | No |
Let \( E/\mathbb{Q} \) be the elliptic curve \( E : {y}^{2} = {x}^{3} + 3 \). It has discriminant \( \Delta = - {2}^{4} \cdot {3}^{5} \), so \( \widetilde{E} \) is nonsingular modulo \( p \) for every prime \( p \geq 5 \). One easily checks that \( \# \widetilde{E}\left( {\mathbb{F}}_{5}\right) = 6\;\text{ and }\;\# \w... | In particular, the point \( \left( {1,2}\right) \in E\left( \mathbb{Q}\right) \) has infinite order, so \( E\left( \mathbb{Q}\right) \) is an infinite set, two facts that are by no means obvious. For a complete analysis of \( E{\left( \mathbb{Q}\right) }_{\text{tors }} \) for curves of the form \( {y}^{2} = {x}^{3} + D... | No |
Let \( E/\mathbb{Q} \) be the elliptic curve\n\n\[ E : {y}^{2} = {x}^{3} + x \]\n\nhaving discriminant \( \Delta = - {64} \) . The point \( \left( {0,0}\right) \in E\left( \mathbb{Q}\right) \) is a point of order 2 . We compute\n\n\[ \# \widetilde{E}\left( {\mathbb{F}}_{3}\right) = 4,\;\# \widetilde{E}\left( {\mathbb{F... | It is not hard to check (Exercise 5.12) that \( \# E\left( {\mathbb{F}}_{p}\right) \) is divisible by 4 for every prime \( p \geq 3 \) . However, we gain additional information by looking at the group structure modulo different primes. Thus\n\n\[ \widetilde{E}\left( {\mathbb{F}}_{3}\right) = \{ O,\left( {0,0}\right) ,\... | No |
Theorem 3.4. Assume that \( \operatorname{char}\left( K\right) = 0 \) and that \( p = \operatorname{char}\left( k\right) > 0 \) . Let \( E/K \) 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\nwith all \( {a}_{i} \in R ... | Proof. If \( x\left( P\right) \in R \), there is nothing to prove, so we assume that \( v\left( {x\left( P\right) }\right) < 0 \) . If the equation for \( E \) is not minimal and if \( \left( {{x}^{\prime },{y}^{\prime }}\right) \) are coordinates for a minimal equation, then we see from (VII.1.3d) that\n\n\[ v\left( {... | Yes |
Proposition 4.1. Let \( E/K \) be an elliptic curve such that the reduced curve \( \widetilde{E}/k \) is nonsingular.\n\n(a) Let \( m \geq 1 \) be an integer that is relatively prime to \( \operatorname{char}\left( k\right) \), i.e., satisfying \( v\left( m\right) = 0 \) . Then \( E\left\lbrack m\right\rbrack \) is unr... | Proof. (a) Let \( {K}^{\prime }/K \) be a finite extension satisfying \( E\left\lbrack m\right\rbrack \subset E\left( {K}^{\prime }\right) \), and let\n\n\[ \n{R}^{\prime } = \text{the ring of integers of}{K}^{\prime }\text{,} \n\]\n\n\[ \n{\mathcal{M}}^{\prime } = \text{the maximal ideal of}{R}^{\prime }\text{,} \n\]\... | Yes |
Proposition 5.1. Let \( E/K \) be an elliptic curve given by a minimal 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\nLet \( \Delta \) be the discriminant of this equation, and let \( {c}_{4} \) be the usual expression involving \( {a}_{1},\ldots ,... | Proof. The reduction type of \( E \) follows from (III.1.4) applied to the reduced Weierstrass equation over the field \( k \) . Then the group \( {\widetilde{E}}_{\mathrm{{ns}}}\left( \bar{k}\right) \) is given by (III.2.5). | Yes |
Let \( p \geq 5 \) be a prime. Then the elliptic curve\n\n\[ \n{E}_{1} : {y}^{2} = {x}^{3} + p{x}^{2} + 1 \n\]\n\nhas good reduction over \( {\mathbb{Q}}_{p} \), while\n\n\[ \n{E}_{2} : {y}^{2} = {x}^{3} + {x}^{2} + p \n\]\n\nhas (split) multiplicative reduction over \( {\mathbb{Q}}_{p} \), and\n\n\[ \n{E}_{3} : {y}^{2... | If we go to the extension field \( \mathbb{Q}\left( \sqrt[6]{p}\right) \), then \( {E}_{3} \) attains good reduction, since the substitution\n\n\[ \nx \mapsto \sqrt[3]{p}{x}^{\prime },\;y \mapsto \sqrt{p}{y}^{\prime }, \n\]\n\nyields a minimal Weierstrass equation having good reduction. On the other hand, the curve \( ... | Yes |
If \( K \) is a finite extension of \( {\mathbb{Q}}_{p} \) and if \( E/K \) has complex multiplication, then one can show that \( E \) has potential good reduction. | See Exercise 7.10. | No |
Corollary 6.2. The subgroup \( {E}_{0}\left( K\right) \) has finite index in \( E\left( K\right) \) . | Proof. The finiteness of \( E\left( K\right) /{E}_{0}\left( K\right) \) follows from the existence of the Néron model, which is a group scheme over \( \operatorname{Spec}\left( R\right) \) whose generic fiber is \( E/K \) ; see [266, IV §§5, 6]. The specific description of \( E\left( K\right) /{E}_{0}\left( K\right) \)... | No |
Proposition 6.3. Let \( K \) be a finite extension of \( {\mathbb{Q}}_{p} \), so in particular \( \operatorname{char}\left( K\right) = 0 \) and \( k \) is a finite field. Then \( E\left( K\right) \) contains a subgroup of finite index that is isomorphic to \( {R}^{ + } \), the additive group of \( R \) . | Proof. From (VII.6.2) we know that \( E\left( K\right) /{E}_{0}\left( K\right) \) is finite, and (VII.2.1) tells us that \( {E}_{0}\left( K\right) /{E}_{1}\left( K\right) \) is isomorphic to the finite group \( {\widetilde{E}}_{\mathrm{{ns}}}\left( k\right) \) . (This is where we use the fact that \( k \) is finite.) I... | Yes |
Corollary 7.2. Let \( {E}_{1}/K \) and \( {E}_{2}/K \) be elliptic curves that are isogenous over \( K \) . Then \( {E}_{1} \) has good reduction over \( K \) if and only if \( {E}_{2} \) has good reduction over \( K \) . | Proof. Let \( \phi : {E}_{1} \rightarrow {E}_{2} \) be a nonzero isogeny defined over \( K \), and let \( m \geq 2 \) be an integer that is relatively prime to both \( \operatorname{char}\left( k\right) \) and \( \deg \phi \) . Then the induced map\n\n\[ \phi : {E}_{1}\left\lbrack m\right\rbrack \rightarrow {E}_{2}\lef... | Yes |
Corollary 7.3. Let \( E/K \) be an elliptic curve. Then \( E \) has potential good reduction if and only if the inertia group \( {I}_{v} \) acts on the Tate module \( {T}_{\ell }\left( E\right) \) through a finite quotient for some (all) prime(s) \( \ell \neq \operatorname{char}\left( k\right) \) . | Proof. Suppose that \( E \) has potential good reduction, and let \( {K}^{\prime }/K \) be a finite extension such that \( E \) has good reduction over \( {K}^{\prime } \) . Extending \( {K}^{\prime } \), we may assume that \( {K}^{\prime }/K \) is a Galois extension. Let \( {v}^{\prime } \) be the valuation on \( {K}^... | Yes |
Lemma 1.1.1. Let \( L/K \) be a finite Galois extension. If \( E\left( L\right) /{mE}\left( L\right) \) is finite, then \( E\left( K\right) /{mE}\left( K\right) \) is also finite. | Proof. The inclusion \( E\left( K\right) \hookrightarrow E\left( L\right) \) induces a natural map\n\n\[ E\left( K\right) /{mE}\left( K\right) \rightarrow E\left( L\right) /{mE}\left( L\right) \]\n\nLet \( \Phi \) be the kernel of this map, so\n\n\[ \Phi = \frac{E\left( K\right) \cap {mE}\left( L\right) }{{mE}\left( K\... | Yes |
Proposition 1.2. (a) The Kummer pairing is well-defined. | Proof of (VIII.1.2). Most of this proposition follows immediately from basic properties of group cohomology; see (VIII §2). For the convenience of the reader, we give a direct proof here.\n\n(a) We must show that \( \kappa \left( {P,\sigma }\right) \) is in \( E\left\lbrack m\right\rbrack \) and that its value does not... | Yes |
Proposition 1.5. Let\n\n\\[ \nL = K\\left( {{\\left\\lbrack m\\right\\rbrack }^{-1}E\\left( K\\right) }\\right) \n\\]\n\nbe the field defined in (VIII.1.2d).\n\n(a) The extension \\( L/K \\) is abelian and has exponent \\( m \\), i.e., the Galois group \\( {G}_{L/K} \\) is abelian and every element of \\( {G}_{L/K} \\)... | PROOF. (a) This follows immediately from (VIII.1.2), which implies that there is an injection\n\n\\[ \n{G}_{L/K} \\rightarrow \\operatorname{Hom}\\left( {E\\left( K\\right), E\\left\\lbrack m\\right\\rbrack }\\right) ,\\;\\sigma \\mapsto \\kappa \\left( {\\cdot ,\\sigma }\\right) .\n\\]\n\n(b) Let \\( v \\in {M}_{K} \\... | Yes |
Proposition 2.1. Let\n\n\[ \nS = \left\{ {v \in {M}_{K}^{0} : E\text{ has bad reduction at }v}\right\} \cup \left\{ {v \in {M}_{K}^{0} : v\left( m\right) \neq 0}\right\} \cup {M}_{K}^{\infty }. \]\n\nThen the image of \( E\left( K\right) \) in \( {H}^{1}\left( {{G}_{\bar{K}/K}, E\left\lbrack m\right\rbrack }\right) \) ... | Proof. Let \( P \in E\left( K\right) \) and, as above, let\n\n\[ \n{c}_{\sigma } = {Q}^{\sigma } - Q \]\n\nbe the cocycle representing \( \delta \left( P\right) \) for some point \( Q \) satisfying \( \left\lbrack m\right\rbrack Q = P \) . Then (VIII.1.5b) says that the field \( K\left( Q\right) \) is unramified at \( ... | Yes |
Theorem 3.1. (Descent Theorem) Let \( A \) be an abelian group. Suppose that there exists \( a \) (height) function\n\n\[ h : A \rightarrow \mathbb{R} \]\n\nwith the following three properties:\n\n(i) Let \( Q \in A \) . There is a constant \( {C}_{1} \), depending on \( A \) and \( Q \), such that\n\n\[ h\left( {P + Q... | Proof. Choose elements \( {Q}_{1},\ldots ,{Q}_{r} \in A \) to represent the finitely many cosets in \( A/{mA} \), and let \( P \in A \) be an arbitrary element. The idea is to show that the difference between \( P \) and an appropriate linear combination of \( {Q}_{1},\ldots ,{Q}_{r} \) is a multiple of a point whose h... | Yes |
Let \( E/\mathbb{Q} \) be an elliptic curve given by a Weierstrass equation\n\n\[ E : {y}^{2} = {x}^{3} + {Ax} + B\;\text{ with }A, B \in \mathbb{Z}. \]\n\n(a) Let \( {P}_{0} \in E\left( \mathbb{Q}\right) \). There is a constant \( {C}_{1} \) that depends on \( {P}_{0}, A \), and \( B \) such that\n\n\[ {h}_{x}\left( {... | Proof. We may assume that \( {C}_{1} > \max \left\{ {{h}_{x}\left( {P}_{0}\right) ,{h}_{x}\left( {\left\lbrack 2\right\rbrack {P}_{0}}\right) }\right\} \), which ensures that (a) is true if \( {P}_{0} = O \) or if \( P \in \left\{ {O, \pm {P}_{0}}\right\} \). In all other cases we write\n\n\[ P = \left( {x, y}\right) =... | Yes |
Let \( P \in {\mathbb{P}}^{N}\left( \mathbb{Q}\right) \) be a point with rational coordinates. Since \( \mathbb{Z} \) is a principal ideal domain, we can find homogeneous coordinates \[ P = \left\lbrack {{x}_{0},\ldots ,{x}_{N}}\right\rbrack \] satisfying \[ {x}_{0},\ldots ,{x}_{N} \in \mathbb{Z}\;\text{ and }\;\gcd \l... | With this definition, it is clear that for any constant \( C \), the set \[ \left\{ {P \in {\mathbb{P}}^{N}\left( \mathbb{Q}\right) : H\left( P\right) \leq C}\right\} \] is a finite set. Indeed, it has at most \( {\left( 2C + 1\right) }^{N} \) elements. This is the sort of finiteness property that is needed for the des... | Yes |
Proposition 5.4. Let \( P \in {\mathbb{P}}^{N}\left( K\right) \) .\n\n(a) The height \( {H}_{K}\left( P\right) \) does not depend on the choice of homogeneous coordinates for \( P \) .\n\n(b) The height satisfies\n\n\[ {H}_{K}\left( P\right) \geq 1 \]\n\n(c) Let \( L/K \) be a finite extension. Then\n\n\[ {H}_{L}\left(... | Proof. (a) Any other choice of homogeneous coordinates for \( P \) has the form \( \left\lbrack {\lambda {x}_{0},\ldots ,\lambda {x}_{N}}\right\rbrack \) for some \( \lambda \in {K}^{ * } \) . Using the product formula (VIII.5.3), we have\n\n\[ \mathop{\prod }\limits_{{v \in {M}_{K}}}\max {\left\{ {\left| \lambda {x}_{... | Yes |
Corollary 5.8. Let \( A \in {\mathrm{{GL}}}_{N + 1}\left( \overline{\mathbb{Q}}\right) \), so multiplication by the matrix \( A \) induces an automorphism \( A : {\mathbb{P}}^{N} \rightarrow {\mathbb{P}}^{N} \) . There are positive constants \( {C}_{1} \) and \( {C}_{2} \), depending on the entries of the matrix \( A \... | Proof. This is (VIII.5.6) for morphisms of degree one. | No |
Theorem 5.10. Let \( P \in {\mathbb{P}}^{N}\left( \overline{\mathbb{Q}}\right) \) and let \( \sigma \in {G}_{\overline{\mathbb{Q}}/\mathbb{Q}} \) . Then\n\n\[ H\left( {P}^{\sigma }\right) = H\left( P\right) \] | Proof. Let \( K/\mathbb{Q} \) be a field such that \( P \in {\mathbb{P}}^{N}\left( K\right) \) . The field \( K \) may not be Galois over \( \mathbb{Q} \), but in any case \( \sigma \) gives an isomorphism \( \sigma : K\overset{ \sim }{ \rightarrow }{K}^{\sigma } \), and \( \sigma \) likewise identifies the sets of abs... | Yes |
Proposition 6.1. Let \( E/K \) be an elliptic curve, and let \( f \in K\left( E\right) \) be a nonconstant function. Then for any constant \( C \), the set\n\n\[ \n\left\{ {P \in E\left( K\right) : {h}_{f}\left( P\right) \leq C}\right\} \n\]\n\nis a finite set of points. | Proof. The function \( f \in K\left( E\right) \) is defined over \( K \), so it maps points \( P \in E\left( K\right) \) to points \( f\left( P\right) \in {\mathbb{P}}^{1}\left( K\right) \) . Hence \( f \) gives a finite-to-one map from the set in question to the set\n\n\[ \n\left\{ {Q \in {\mathbb{P}}^{1}\left( K\righ... | Yes |
Lemma 6.3. Let \( f, g \in K\left( E\right) \) be even functions. Then\n\n\[ \left( {\deg g}\right) {h}_{f} = \left( {\deg f}\right) {h}_{g} + O\left( 1\right) . \] | Proof. Let \( x, y \in K\left( E\right) \) be Weierstrass coordinates for \( E/K \) . We know from (III.2.3.1) that the subfield of \( K\left( E\right) \) consisting of even functions is exactly \( K\left( x\right) \) , so we can find a rational function \( r\left( X\right) \in K\left( X\right) \) such that there is a ... | Yes |
Let \( E/K \) be an elliptic curve, and let \( f \in K\left( E\right) \) be an even function.\n\n(a) Let \( Q \in E\left( \bar{K}\right) \) . Then\n\n\[ \n{h}_{f}\left( {P + Q}\right) \leq 2{h}_{f}\left( P\right) + O\left( 1\right) \;\text{ for all }P \in E\left( \bar{K}\right) ,\n\]\n\nwhere the \( O\left( 1\right) \)... | Proof. (a) This follows immediately from (VIII.6.2), since \( {h}_{f}\left( {P - Q}\right) \geq 0 \) .\n\n(b) Since \( f \) is even, it suffices to consider \( m \geq 0 \) . Further, the result is trivial for \( m = \) 0 and \( m = 1 \) . We use induction to complete the proof. Suppose that the desired result is known ... | Yes |
Theorem 6.7. (Mordell-Weil theorem) Let \( K \) be a number field, and let \( E/K \) be an elliptic curve. Then the group \( E\left( K\right) \) is finitely generated. | Proof. Choose any even nonconstant function \( f \in K\left( E\right) \), for example, \( f \) could be the \( x \) -coordinate on a Weierstrass equation. The Mordell-Weil theorem follows immediately from the weak Mordell-Weil theorem (VIII.1.1) with \( m = 2 \) and the descent theorem (VIII.3.1) as soon as we show tha... | Yes |
Corollary 7.2. ([152], [190]) Let \( E/\mathbb{Q} \) be an elliptic curve with Weierstrass equation\n\n\[ \n{y}^{2} = {x}^{3} + {Ax} + B,\;A, B \in \mathbb{Z}.\n\]\n\nSuppose that \( P \in E\left( \mathbb{Q}\right) \) is a nonzero torsion point.\n\n(a) \( x\left( P\right), y\left( P\right) \in \mathbb{Z} \).\n\n(b) Eit... | Proof. (a) Let \( P \) have exact order \( m \) . If \( m = 2 \), then \( y\left( P\right) = 0 \), so \( x\left( P\right) \in \mathbb{Z} \), since it is the root of a monic polynomial with integer coefficients. If \( m > 2 \), the desired result follows immediately from (VIII.7.1), since the quantity \( {r}_{v} \) in (... | Yes |
The Weierstrass equation\n\n\[ E : {y}^{2} = {x}^{3} - {43x} + {166} \]\n\nhas\n\n\[ 4{A}^{3} + {27}{B}^{2} = {425984} = {2}^{15} \cdot {13}. \]\n\nHence any torsion point in \( E\left( \mathbb{Q}\right) \) has its \( y \)-coordinate in the set\n\n\[ \{ 0, \pm 1, \pm 2, \pm 4, \pm 8, \pm {16}, \pm {32}, \pm {64}, \pm {... | A little bit of work with a calculator reveals the points\n\n\[ \{ \left( {3, \pm 8}\right) ,\left( {-5, \pm {16}}\right) ,\left( {{11}, \pm {32}}\right) \} .\n\nOn the other hand, since \( E \) has good reduction modulo 3, we know that \( {E}_{\text{tors }}\left( \mathbb{Q}\right) \) injects into \( \widetilde{E}\left... | Yes |
Lemma 8.1. With notation as above, the ideal class in \( K \) of the ideal \( {\mathfrak{a}}_{\Delta } \) is independent of \( \Delta \) . | Proof. Suppose that we take a different Weierstrass equation for \( E \) over \( K \), say with discriminant \( {\Delta }^{\prime } \) . Then \( \Delta = {u}^{12}{\Delta }^{\prime } \) for some \( u \in {K}^{ * } \), so directly from the definitions we see that\n\n\[ \left( {\Delta }^{\prime }\right) {\mathfrak{a}}_{{\... | Yes |
Example 8.4. The Weierstrass equation\n\n\[ E : {y}^{2} = {x}^{3} + {16} \]\n\nhas discriminant \( \Delta = - {2}^{12}{3}^{3} \) and it is not minimal at 2 . The substitution\n\n\[ x = 4{x}^{\prime }, y = 8{y}^{\prime } + 4, \]\n\ngives the global minimal equation\n\n\[ {\left( {y}^{\prime }\right) }^{2} + {y}^{\prime ... | The substitution\n\n\[ x = 4{x}^{\prime }, y = 8{y}^{\prime } + 4, \]\n\ngives the global minimal equation\n\n\[ {\left( {y}^{\prime }\right) }^{2} + {y}^{\prime } = {\left( {x}^{\prime }\right) }^{3} \] | Yes |
Let \( K = \mathbb{Q}\left( \sqrt{-{10}}\right) \), so \( K \) has class number 2, the class group being generated by the prime ideal \( \mathfrak{p} = \left( {5,\sqrt{-{10}}}\right) \). Let \( E/K \) be the elliptic curve given by the equation \( E : {y}^{2} = {x}^{3} + {125} \). This equation has discriminant \( \Del... | For \( \mathfrak{p} \), the change of coordinates \( x = {\left( \sqrt{-{10}}\right) }^{2}{x}^{\prime },\;y = {\left( \sqrt{-{10}}\right) }^{3}{y}^{\prime } \) gives an equation \( {\left( {y}^{\prime }\right) }^{2} = {\left( {x}^{\prime }\right) }^{3} - \frac{1}{8} \) that has good reduction at \( \mathfrak{p} \). Hen... | Yes |
Proposition 8.7. Let \( S \subset {M}_{K} \) be a finite set of absolute values containing \( {M}_{K}^{\infty } \) and all finite places dividing 2 and 3. Assume further that the ring of \( S \) -integers \( {R}_{S} \) is a principal ideal domain. Then every elliptic curve \( E/K \) has a Weierstrass equation of the fo... | Proof. Choose any Weierstrass equation for \( E/K \) of the form\n\n\[ E : {y}^{2} = {x}^{3} + {Ax} + B, \]\n\nand let \( \Delta = - {16}\left( {4{A}^{3} + {27}{B}^{2}}\right) \) . For each \( v \in {M}_{K} \) with \( v \notin S \), choose \( {u}_{v} \in {K}^{ * } \) such that the substitution\n\n\[ x = {u}_{v}^{2}{x}^... | Yes |
Proposition 9.1. (Tate) Let \( E/K \) be an elliptic curve, let \( f \in K\left( E\right) \) be a nonconstant even function, and let \( P \in E\left( \bar{K}\right) \) . Then the limit\n\n\[\n\frac{1}{\deg \left( f\right) }\mathop{\lim }\limits_{{N \rightarrow \infty }}{4}^{-N}{h}_{f}\left( {\left\lbrack {2}^{N}\right\... | Proof. We prove that the limit exists by showing that the sequence is Cauchy. Applying (VIII.6.4b) with \( m = 2 \), there is a constant \( C \) such that for all \( Q \in E\left( \bar{K}\right) \) ,\n\n\[\n\left| {{h}_{f}\left( {\left\lbrack 2\right\rbrack Q}\right) - 4{h}_{f}\left( Q\right) }\right| \leq C.\n\]\n\nFo... | Yes |
Theorem 9.3. (Néron, Tate) Let \( E/K \) be an elliptic curve, and let \( \widehat{h} \) be the canonical height on \( E \) . (a) For all \( P, Q \in E\left( \bar{K}\right) \) we have \[ \widehat{h}\left( {P + Q}\right) + \widehat{h}\left( {P - Q}\right) = 2\widehat{h}\left( P\right) + 2\widehat{h}\left( Q\right) \;\te... | Proof. We start with (e) and then return to (a)-(d). (e) In the course of proving (VIII.9.1) we found a constant \( C \), depending on \( f \), such that for all integers \( N \geq M \geq 0 \) and all points \( P \in E\left( \bar{K}\right) \) , \[ \left| {{4}^{-N}{h}_{f}\left( {\left\lbrack {2}^{N}\right\rbrack P}\righ... | Yes |
Lemma 9.5. Let \( V \) be a finite-dimensional real vector space and let \( L \subset V \) be a lattice, i.e., \( L \) is a discrete subgroup of \( V \) containing a basis for \( V \). Let \( q : V \rightarrow \mathbb{R} \) be a quadratic form, and suppose that \( q \) has the following properties:\n\n(i) For \( P \in ... | Proof. Choose a basis for \( V \) such that for a vector \( \mathbf{x} = \left( {{x}_{1},\ldots ,{x}_{r}}\right) \in V \), the quadratic form \( q \) has the form\n\n\[ \nq\left( \mathbf{x}\right) = \mathop{\sum }\limits_{{i = 1}}^{s}{x}_{i}^{2} - \mathop{\sum }\limits_{{i = 1}}^{t}{x}_{s + i}^{2} \n\]\n\nwhere \( s + ... | Yes |
Proposition 9.6. The canonical height extends to a positive definite quadratic form on the real vector space \( E\left( K\right) \otimes \mathbb{R} \) . | Proof. We consider the lattice \( E\left( K\right) /{E}_{\text{tors }}\left( K\right) \) inside the vector space \( E\left( K\right) \otimes \mathbb{R} \) and apply (VIII.9.5) to get the desired result. Condition (i) of (VIII.9.5) is exactly (VIII.9.3cd). Condition (ii) of (VIII.9.5) follows from (VIII.9.3e), which say... | Yes |
Proposition 11.2. Szpiro's conjecture (easily) implies Fermat's last theorem for all sufficiently large exponents, i.e., if \( n \) is sufficiently large, then the Fermat equation \( {a}^{n} + {b}^{n} = {c}^{n} \) has no solutions with \( a, b, c \in \mathbb{Z} \) and \( {abc} \neq 0 \) . | Proof. Suppose that \( {a}^{n} + {b}^{n} = {c}^{n} \) with \( a, b, c \in \mathbb{Z} \) and \( {abc} \neq 0 \) . We consider the elliptic curve (sometimes called a Frey curve)\n\n\[ E : {y}^{2} = x\left( {x + {a}^{n}}\right) \left( {x - {b}^{n}}\right) . \]\n\nThis Weierstrass equation for \( E \) has discriminant\n\n\... | Yes |
Lemma 11.3. Let \( A, B, C \in \mathbb{Z} \) be nonzero integers satisfying\n\n\[ A + B = C\;\text{ and }\;\gcd \left( {A, B, C}\right) = 1, \]\n\nand let \( E/\mathbb{Q} \) be the elliptic curve\n\n\[ E : {y}^{2} = x\left( {x + A}\right) \left( {x - B}\right) . \]\n\n(a) The minimal discriminant \( {\Delta }_{E} \) of... | Proof. (a) The given Weierstrass equation for \( E \) has discriminant\n\n\[ \Delta = {16}{A}^{2}{B}^{2}{\left( A + B\right) }^{2} = {16}{A}^{2}{B}^{2}{C}^{2} \]\n\nand associated quantities\n\n\[ {c}_{4} = {16}\left( {{A}^{2} + {AB} + {B}^{2}}\right) \;\text{ and }\;{c}_{6} = - {32}\left( {2{A}^{3} + 3{A}^{2}B + {3A}{... | Yes |
Proposition 11.5. (a) If Szpiro's conjecture (VIII.11.1) is true, then the ABC conjecture (VIII.11.4) is true with exponent \( \frac{3}{2} \) . | Proof. (a) Let \( A, B, C \) be as in the statement of the \( {ABC} \) conjecture. Relabeling if necessary, we may assume that \( C > B > A > 0 \), so in particular\n\n\[ \n{2B} > A + B = C.\text{.}\n\]\n\nWe consider the elliptic curve\n\n\[ \nE : {y}^{2} = x\left( {x + A}\right) \left( {x - B}\right) .\n\]\n\nFrom (V... | Yes |
Proposition 1.2. (Dirichlet) Let \( \alpha \in \mathbb{R} \) with \( \alpha \notin \mathbb{Q} \) . Then there are infinitely many rational numbers \( p/q \in \mathbb{Q} \) such that\n\n\[ \left| {\frac{p}{q} - \alpha }\right| \leq \frac{1}{{q}^{2}} \] | Proof. Let \( Q \) be a (large) integer and look at the set of real numbers\n\n\[ \{ {q\alpha } - \left\lbrack {q\alpha }\right\rbrack : q = 0,1,\ldots, Q\} \]\n\nwhere \( \left\lbrack {\cdot \cdot }\right\rbrack \) denotes greatest integer. Since \( \alpha \) is irrational, this set contains \( Q + 1 \) distinct numbe... | Yes |
Proposition 1.3. (Liouville [151]) Let \( \alpha \in \overline{\mathbb{Q}} \) have degree \( d \geq 2 \) over \( \mathbb{Q} \), i.e., \( \left\lbrack {\mathbb{Q}\left( \alpha \right) : \mathbb{Q}}\right\rbrack = d \) . There is a constant \( C > 0 \), depending on \( \alpha \), such that for all rational numbers \( p/q... | Proof. We may assume that \( \alpha \in \mathbb{R} \), since otherwise \( C = \operatorname{Im}\left( \alpha \right) \) works. Let\n\n\[ f\left( T\right) = {a}_{0}{T}^{d} + {a}_{1}{T}^{d - 1} + \cdots + {a}_{d} \in \mathbb{Z}\left\lbrack T\right\rbrack \]\n\nbe a minimal polynomial for \( \alpha \), and let\n\n\[ {C}_{... | Yes |
Theorem 1.4. (Roth’s Theorem) For every \( \epsilon > 0 \), every number field \( K \) of degree \( d \) has approximation exponent\n\n\[ \tau \left( d\right) = 2 + \epsilon \] | Proof. See (IX §8) for a brief sketch of the proof. A nice exposition for \( K = \mathbb{Q} \) and the usual archimedean absolute value is given in [221, Chapter V]. For the general case, see [114, Part D] or [139, Chapter 7]. | No |
How do theorems on Diophantine approximation lead to results about Diophantine equations? Consider the simple example of trying to solve the equation\n\n\\[ \n{x}^{3} - 2{y}^{3} = a \n\\]\n\nin integers \\( x, y \\in \\mathbb{Z} \\), where \\( a \\in \\mathbb{Z} \\) is fixed. Suppose that \\( \\left( {x, y}\\right) \\)... | Let \\( \\zeta \\) be a primitive cube root of unity, and factor the equation as\n\n\\[ \n\\left( {\\frac{x}{y} - \\sqrt[3]{2}}\\right) \\left( {\\frac{x}{y} - \\zeta \\sqrt[3]{2}}\\right) \\left( {\\frac{x}{y} - {\\zeta }^{2}\\sqrt[3]{2}}\\right) = \\frac{a}{{y}^{3}}.\n\\]\n\nThe second and third factors in the produc... | Yes |
Proposition 2.2. Let \( Q \in C\left( {K}_{v}\right) \) and let \( F \in {K}_{v}\left( C\right) \) be a function that vanishes at \( Q \) . Then the limit\n\n\[ \mathop{\lim }\limits_{\substack{{P \in C\left( {K}_{v}\right) } \\ {P \rightarrow Q} }}\frac{\log {\left| F\left( P\right) \right| }_{v}}{\log {d}_{v}\left( {... | Proof. Let \( {t}_{Q} \) be the function vanishing only at \( Q \) that we are using to define \( {d}_{v}\left( {\cdot, Q}\right) \) . Let \( e = {\operatorname{ord}}_{Q}\left( {t}_{Q}\right) \) and \( f = {\operatorname{ord}}_{Q}\left( F\right) \) . Then the function \( \phi = {F}^{e}/{t}_{Q}^{f} \) has neither a zero... | Yes |
Proposition 2.3. Let \( {C}_{1}/K \) and \( {C}_{2}/K \) be curves, and let \( \phi : {C}_{1} \rightarrow {C}_{2} \) be a finite map defined over \( K \) . Let \( Q \in {C}_{1}\left( {K}_{v}\right) \), and let \( {e}_{\phi }\left( Q\right) \) be the ramification index of \( \phi \) at \( Q \) (II §2). Then\n\n\[ \matho... | Proof. Let \( {t}_{Q} \in {K}_{v}\left( {C}_{1}\right) \) be a function that vanishes to order \( {e}_{1} \geq 1 \) at \( Q \) and has no other zeros, and similarly let \( {t}_{\phi \left( Q\right) } \in {K}_{v}\left( {C}_{2}\right) \) be a function that vanishes to order \( {e}_{2} \geq 1 \) at \( \phi \left( Q\right)... | Yes |
Corollary 2.4. (of (IX.1.4)) Fix an absolute value \( v \in {M}_{K} \) . Let \( C/K \) be a curve, let \( f \in K\left( C\right) \) be a nonconstant function, and let \( Q \in C\left( \bar{K}\right) \) . Then\n\n\[ \mathop{\liminf }\limits_{\substack{{P \in C\left( K\right) } \\ {P \rightarrow Q} }}\frac{\log {d}_{v}\l... | Proof. Replacing \( f \) by \( 1/f \) if necessary, we may assume that \( f\left( Q\right) \neq \infty \) . (Note that \( {H}_{K}\left( {\left( {1/f}\right) \left( P\right) }\right) = {H}_{K}\left( {f\left( P\right) }\right) \) .) The function \( f - f\left( Q\right) \) vanishes at \( Q \), say to order \( e \), so (IX... | Yes |
Corollary 3.2.1. Let \( E/K \) be an elliptic curve with Weierstrass coordinate functions \( x \) and \( y \), let \( S \subset {M}_{K} \) be a finite set of places containing \( {M}_{K}^{\infty } \), and let \( {R}_{S} \) be the ring of \( S \) -integers of \( K \) . Then\n\n\[ \left\{ {P \in E\left( K\right) : x\left... | Proof. We apply (IX.3.1) with the function \( f = x \) . Suppose that there is a sequence of distinct points \( {P}_{1},{P}_{2},\ldots \in E\left( K\right) \) with every \( x\left( {P}_{i}\right) \in {R}_{S} \) . The definition of height then tells us that\n\n\[ {h}_{x}\left( {P}_{i}\right) = \frac{1}{\left\lbrack K : ... | Yes |
Corollary 3.2.2. Let \( C/K \) be a curve of genus one, let \( f \in K\left( C\right) \) be a nonconstant function, and let \( S \) and \( {R}_{S} \) be as in (IX.3.2.1). Then \[ \left\{ {P \in C\left( K\right) : f\left( P\right) \in {R}_{S}}\right\} \] is a finite set. Further, (IX.3.2.2) follows formally from (IX.3.2... | Proof. We are clearly proving something stronger if we extend the field \( K \) and enlarge the set \( S \) . We may thus assume that \( C\left( K\right) \) contains a pole \( Q \) of \( f \), and taking \( Q \) to be the identity element, we view \( \left( {C, Q}\right) \) as an elliptic curve defined over \( K \) . L... | Yes |
Consider the Diophantine equation\n\n\[ \n{y}^{2} = {x}^{3} + {Ax} + B \n\]\n\nwhere \( A, B \in \mathbb{Z} \) and \( 4{A}^{3} + {27}{B}^{2} \neq 0 \) . The corollary (IX.3.2.1) says that this equation has only finitely many solutions \( x, y \in \mathbb{Z} \) . What does (IX.3.1) say in this situation, say if we take ... | Label the nonzero rational points \( {P}_{1},{P}_{2},\ldots \in E\left( \mathbb{Q}\right) \) in order of nondecreasing height, and write\n\n\[ \nx\left( {P}_{i}\right) = \frac{{a}_{i}}{{b}_{i}} \in \mathbb{Q} \n\]\n\nas a fraction in lowest terms. Then\n\n\[ \n\log {d}_{v}\left( {{P}_{i}, O}\right) = \frac{1}{2}\log \m... | Yes |
Corollary 4.3.1. Let \( C/K \) be a curve of genus one and let \( f \in K\left( C\right) \) be a nonconstant function. Then there are only finitely many points \( P \in C\left( K\right) \) such that \( f\left( P\right) \in {R}_{S} \) . | Proof. The reduction procedure described in (IX.3.2.2) says that it suffices to consider the case that \( f \) is the \( x \) -coordinate of a Weierstrass equation. The case \( f = x \) is covered by (IX.4.3). | No |
Theorem 5.2. (Baker) Let \( {\alpha }_{1},\ldots ,{\alpha }_{n} \in {K}^{ * } \) and let \( {\beta }_{1},\ldots ,{\beta }_{n} \in K \) . For any constant \( \kappa \), define\n\n\[ \n\tau \left( \kappa \right) = \tau \left( {\kappa ;{\alpha }_{1},\ldots ,{\alpha }_{n},{\beta }_{1},\ldots ,{\beta }_{n}}\right) = h\left(... | Proof. See [11] or [135, VIII, Theorem 1.1]. | No |
Let \( V \) be a finite-dimensional vector space over \( \mathbb{R} \) . Given any basis \( \mathbf{e} = \) \( \left\{ {{e}_{1},\ldots ,{e}_{n}}\right\} \) for \( V \), let \( \parallel \cdot {\parallel }_{\mathbf{e}} \) be the sup norm with respect to \( \mathbf{e} \), i.e., \[ \parallel x{\parallel }_{\mathbf{e}} = {... | Proof. Let \( A = \left( {a}_{ij}\right) \) be the change of basis matrix from e to \( \mathbf{f} \), so \( {e}_{i} = \mathop{\sum }\limits_{j}{a}_{ij}{f}_{j} \) , and let \( \parallel A\parallel = \max \left\{ \left| {a}_{ij}\right| \right\} \) . Then for any \( x = \mathop{\sum }\limits_{i}{x}_{i}{e}_{i} \in V \) we ... | Yes |
Theorem 6.1. (Shafarevich [242]) Let \( S \subset {M}_{K} \) be a finite set of places containing \( {M}_{K}^{\infty } \) . Then up to isomorphism over \( K \), there are only finitely many elliptic curves \( E/K \) having good reduction at all primes not in \( S \) . | Proof. Clearly we are proving something stronger if we enlarge \( S \), so we may assume that \( S \) contains all primes of \( K \) lying over 2 and 3 . Enlarging \( S \) further, we may also assume that the ring of \( S \) -integers \( {R}_{S} \) has class number one.\n\nUnder these assumptions, we see from (VIII.8.7... | Yes |
Corollary 6.2. Fix an elliptic curve \( E/K \) . Then there are only finitely many elliptic curves \( {E}^{\prime }/K \) that are \( K \) -isogenous to \( E \) . | Proof. If \( E \) and \( {E}^{\prime } \) are isogenous over \( K \), then (VII.7.2) says that \( E \) and \( {E}^{\prime } \) have the same set of primes of bad reduction. Now apply (IX.6.1). | No |
Corollary 6.3. (Serre) Let \( E/K \) be an elliptic curve with no complex multiplication. Then for all but finitely many primes \( \ell \), the group of \( \ell \) -torsion points \( E\left\lbrack \ell \right\rbrack \) has no nontrivial \( {G}_{\bar{K}/K} \) -invariant subgroups. (In other words, the representation of ... | Proof. Suppose that \( {\Phi }_{\ell } \subset E\left\lbrack \ell \right\rbrack \) is a nontrivial \( {G}_{\bar{K}/K} \) -invariant subgroup of \( E\left\lbrack \ell \right\rbrack \) . We know that \( E\left\lbrack \ell \right\rbrack \cong {\left( \mathbb{Z}/\ell \mathbb{Z}\right) }^{2} \), so \( {\Phi }_{\ell } \) is ... | Yes |
For \( K = \mathbb{Q} \), results of Mazur [166] and Kenku [125] give a statement that is far more precise than (IX.6.2). They show that for a given elliptic curve \( E/\mathbb{Q} \) , there are at most eight \( \mathbb{Q} \) -isomorphism classes of elliptic curves \( {E}^{\prime }/\mathbb{Q} \) that are \( \mathbb{Q} ... | \[ 1 \leq \deg \phi \leq {19}\;\text{ or }\;\deg \phi \in \{ {21},{25},{27},{37},{43},{67},{163}\} . \] It is no coincidence that the possibilities for \( \deg \phi \) are values of \( d \) for which \( \mathbb{Q}\left( \sqrt{-d}\right) \) has class number one. The class number one condition means that the elliptic cur... | Yes |
Proposition 7.1. (a) (V.A. Lebesgue) The equation\n\n\\[ \n{y}^{2} = {x}^{3} + 7 \n\\]\n\nhas no solutions in integers \\( x, y \\in \\mathbb{Z} \\) . | Proof. (a) Suppose that \\( x, y \\in \\mathbb{Z} \\) satisfy \\( {y}^{2} = {x}^{3} + 7 \\) . We first observe that \\( x \\) must be odd, since no integer of the form \\( {8k} + 7 \\) is a square. Next we rewrite the equation as\n\n\\[ \n{y}^{2} + 1 = {x}^{3} + 8 = \\left( {x + 2} \\right) \\left( {{x}^{2} - {2x} + 4}... | Yes |
Theorem 1.1. (a) With notation as above, there is a bilinear pairing\n\n\\[ \nb : E\\left( K\\right) /{mE}\\left( K\\right) \\times E\\left\\lbrack m\\right\\rbrack \\rightarrow {K}^{ * }/{\\left( {K}^{ * }\\right) }^{m}\n\\]\n\nsatisfying\n\n\\[ \n{e}_{m}\\left( {{\\delta }_{E}\\left( P\\right), T}\\right) = {\\delta ... | (d) The pairing in (a) may be computed as follows. For each \\( T \\in E\\left\\lbrack m\\right\\rbrack \\), choose functions \\( {f}_{T},{g}_{T} \\in K\\left( E\\right) \\) satisfying the conditions\n\n\\[ \n\\operatorname{div}\\left( {f}_{T}\\right) = m\\left( T\\right) - m\\left( O\\right) \\;\\text{ and }\\;{f}_{T}... | Yes |
Proposition 1.4. (Complete 2-Descent) Let \( E/K \) be an elliptic curve given by a Weierstrass equation\n\n\[ \n{y}^{2} = \\left( {x - {e}_{1}}\\right) \\left( {x - {e}_{2}}\\right) \\left( {x - {e}_{3}}\\right) \\;\\text{ with }{e}_{1},{e}_{2},{e}_{3} \\in K.\n\]\n\nLet \( S \\subset {M}_{K} \) be a finite set of pla... | Proof. As explained above, this is a special case of (X.1.1). | No |
We use (X.1.4) to compute \( E\left( \mathbb{Q}\right) /{2E}\left( \mathbb{Q}\right) \) for the elliptic curve\n\n\[ E : {y}^{2} = {x}^{3} - {12}{x}^{2} + {20x} = x\left( {x - 2}\right) \left( {x - {10}}\right) .\n\] | This equation has discriminant\n\n\[ \Delta = {409600} = {2}^{14}{5}^{2} \]\n\nso it has good reduction except at 2 and 5 . Reducing the equation modulo 3, we easily check that \( \# \widetilde{E}\left( {\mathbb{F}}_{3}\right) = 4 \) . Since \( E\left\lbrack 2\right\rbrack \subset {E}_{\text{tors }}\left( \mathbb{Q}\ri... | Yes |
The map \( \xi \) is a 1-cocycle, i.e., \[ {\xi }_{\sigma \tau } = {\left( {\xi }_{\sigma }\right) }^{\tau }{\xi }_{\tau }\;\text{ for all }\sigma ,\tau \in {G}_{\bar{K}/K}. \] | We compute \[ {\xi }_{\sigma \tau } = {\phi }^{\sigma \tau }{\phi }^{-1} = {\left( {\phi }^{\sigma }{\phi }^{-1}\right) }^{\tau }\left( {{\phi }^{\tau }{\phi }^{-1}}\right) = {\left( {\xi }_{\sigma }\right) }^{\tau }{\xi }_{\tau }. \] | Yes |
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