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Let \( V \) be the algebraic set in \( {\mathbb{P}}^{2} \) given by the single equation\n\n\[ \n{X}^{2} + {Y}^{2} = {Z}^{2} \n\]\n\nThen for any field \( K \) with \( \operatorname{char}\left( K\right) \neq 2 \), the set \( V\left( K\right) \) is isomorphic to \( {\mathbb{P}}^{1}\left( K\right) \). | for example by the map\n\n\[ \n{\mathbb{P}}^{1}\left( K\right) \rightarrow V\left( K\right) ,\;\left\lbrack {s, t}\right\rbrack \mapsto \left\lbrack {{s}^{2} - {t}^{2},{2st},{s}^{2} + {t}^{2}}\right\rbrack .\n\] | Yes |
The algebraic set\n\n\[ V : {X}^{2} + {Y}^{2} = 3{Z}^{2} \]\n\nis defined over \( \mathbb{Q} \). However, \( V\left( \mathbb{Q}\right) = \varnothing \). | To see this, suppose that \( \left\lbrack {x, y, z}\right\rbrack \in V\left( \mathbb{Q}\right) \) with \( x, y, z \in \mathbb{Z} \) and \( \gcd \left( {x, y, z}\right) = 1 \). Then\n\n\[ {x}^{2} + {y}^{2} \equiv 0\;\left( {\;\operatorname{mod}\;3}\right) \]\n\nso the fact that -1 is not a square modulo 3 implies that\n... | Yes |
Proposition 2.6. (a) Let \( V \) be an affine variety. Then \( \bar{V} \) is a projective variety, and\n\n\[ V = \bar{V} \cap {\mathbb{A}}^{n} \] | Proof. See [111, I.2.3] for (a) and (b). | No |
Let \( V \) be the projective variety given by the equation\n\n\[ V : {Y}^{2} = {X}^{3} + {17} \] | This really means that \( V \) is the variety in \( {\mathbb{P}}^{2} \) given by the homogeneous equation\n\n\[ {\bar{Y}}^{2}\bar{Z} = {\bar{X}}^{3} + {17}{\bar{Z}}^{3} \]\n\nthe identification being\n\n\[ X = \bar{X}/\bar{Z},\;Y = \bar{Y}/\bar{Z} \]\n\nThis variety has one point at infinity, namely \( \left\lbrack {0,... | No |
Assume that \( \operatorname{char}\left( K\right) \neq 2 \) and let \( V \) be the variety from (I.2.3), \[ V : {X}^{2} + {Y}^{2} = {Z}^{2}. \] Consider the rational map \[ \phi : V \rightarrow {\mathbb{P}}^{1},\;\phi = \left\lbrack {X + Z, Y}\right\rbrack . \] Clearly \( \phi \) is regular at every point of \( V \) ex... | However, using \[ \left( {X + Z}\right) \left( {X - Z}\right) \equiv - {Y}^{2}\;\left( {{\;\operatorname{mod}\;I}\left( V\right) }\right) \] we have \[ \phi = \left\lbrack {X + Z, Y}\right\rbrack = \left\lbrack {{X}^{2} - {Z}^{2}, Y\left( {X - Z}\right) }\right\rbrack = \left\lbrack {-{Y}^{2}, Y\left( {X - Z}\right) }\... | Yes |
Let \( V \) be the variety\n\n\[ V : {Y}^{2}Z = {X}^{3} + {X}^{2}Z \]\n\nand consider the rational maps\n\n\[ \psi : {\mathbb{P}}^{1} \rightarrow V,\;\psi = \left\lbrack {\left( {{S}^{2} - {T}^{2}}\right) T,\left( {{S}^{2} - {T}^{2}}\right) S,{T}^{3}}\right\rbrack ,\]\n\n\[ \phi : V \rightarrow {\mathbb{P}}^{1},\;\phi ... | Not coincidentally, the point \( \left\lbrack {0,0,1}\right\rbrack \) is a singular point of \( V \) ; see (II.2.1). We emphasize that although the compositions \( \phi \circ \psi \) and \( \psi \circ \phi \) are the identity map wherever they are defined, the maps \( \phi \) and \( \psi \) are not isomorphisms, becaus... | Yes |
Consider the varieties\n\n\\[ \n{V}_{1} : {X}^{2} + {Y}^{2} = {Z}^{2}\\;\\text{ and }\\;{V}_{2} : {X}^{2} + {Y}^{2} = 3{Z}^{2}. \n\\]\n\nThey are not isomorphic over \\( \\mathbb{Q} \\), since \\( {V}_{2}\\left( \\mathbb{Q}\\right) = \\varnothing \\) from (I.2.5), while \\( {V}_{1}\\left( \\mathbb{Q}\\right) \\) contai... | However, the varieties \\( {V}_{1} \\) and \\( {V}_{2} \\) are isomorphic over \\( \\mathbb{Q}\\left( \\sqrt{3}\\right) \\), an isomorphism being given by\n\n\\[ \n\\phi : {V}_{2} \\rightarrow {V}_{1},\\;\\phi = \\left\\lbrack {X, Y,\\sqrt{3}Z}\\right\\rbrack . \n\\] | Yes |
Proposition 1.1. Let \( C \) be a curve and \( P \in C \) a smooth point. Then \( \bar{K}{\left\lbrack C\right\rbrack }_{P} \) is a discrete valuation ring. | Proof. From (I.1.7), the vector space \( {M}_{P}/{M}_{P}^{2} \) is a one-dimensional vector space over the field \( \bar{K} = \bar{K}{\left\lbrack C\right\rbrack }_{P}/{M}_{P} \) . Now use [8, Proposition 9.2] or Exercise 2.1. | No |
Proposition 1.2. Let \( C \) be a smooth curve and \( f \in \bar{K}\left( C\right) \) with \( f \neq 0 \) . Then there are only finitely many points of \( C \) at which \( f \) has a pole or zero. Further, if \( f \) has no poles, then \( f \in \bar{K} \) . | Proof. See [111, I.6.5], [111, II.6.1], or [243, III §1] for the finiteness of the number of poles. To deal with the zeros, look instead at \( 1/f \) . The last statement is \( \lbrack {111} \) , I.3.4a] or [243, I §5, Corollary 1]. | No |
Consider the two curves\n\n\[ \n{C}_{1} : {Y}^{2} = {X}^{3} + X\;\text{ and }\;{C}_{2} : {Y}^{2} = {X}^{3} + {X}^{2}. \n\]\n\n(Remember our convention (I.2.7) concerning affine equations for projective varieties. Each of \( {C}_{1} \) and \( {C}_{2} \) has a single point at infinity.) Let \( P = \left( {0,0}\right) \).... | The maximal ideal \( {M}_{P} \) of \( \bar{K}{\left\lbrack {C}_{1}\right\rbrack }_{P} \) has the property that \( {M}_{P}/{M}_{P}^{2} \) is generated by \( Y \) (I.1.8), so for example,\n\n\[ \n{\operatorname{ord}}_{P}\left( Y\right) = 1,\;{\operatorname{ord}}_{P}\left( X\right) = 2,\;{\operatorname{ord}}_{P}\left( {2{... | Yes |
Proposition 2.1. Let \( C \) be a curve, let \( V \subset {\mathbb{P}}^{N} \) be a variety, let \( P \in C \) be a smooth point, and let \( \phi : C \rightarrow V \) be a rational map. Then \( \phi \) is regular at \( P \) . In particular, if \( C \) is smooth, then \( \phi \) is a morphism. | Proof. Write \( \phi = \left\lbrack {{f}_{0},\ldots ,{f}_{N}}\right\rbrack \) with functions \( {f}_{i} \in \bar{K}\left( C\right) \), and choose a uni-formizer \( t \in \bar{K}\left( C\right) \) for \( C \) at \( P \) . Let\n\n\[ n = \mathop{\min }\limits_{{0 \leq i \leq N}}{\operatorname{ord}}_{P}\left( {f}_{i}\right... | Yes |
Let \( C/K \) be a smooth curve and let \( f \in K\left( C\right) \) be a function. Then \( f \) defines a rational map, which we also denote by \( f \), \[ f : C \rightarrow {\mathbb{P}}^{1},\;P \mapsto \left\lbrack {f\left( P\right) ,1}\right\rbrack . \] | From (II.2.1), this map is actually a morphism. It is given explicitly by \[ f\left( P\right) = \left\{ \begin{array}{ll} \left\lbrack {f\left( P\right) ,1}\right\rbrack & \text{ if }f\text{ is regular at }P, \\ \left\lbrack {1,0}\right\rbrack & \text{ if }f\text{ has a pole at }P. \end{array}\right. \] | Yes |
Theorem 2.3. Let \( \phi : {C}_{1} \rightarrow {C}_{2} \) be a morphism of curves. Then \( \phi \) is either constant or surjective. | Proof. See [111, II.6.8] or [243, I §5, Theorem 4]. | No |
Let \( \phi : {C}_{1} \rightarrow {C}_{2} \) be a nonconstant map defined over \( K \) . Then \( K\left( {C}_{1}\right) \) is a finite extension of \( {\phi }^{ * }\left( {K\left( {C}_{2}\right) }\right) \) . | Proof. (a) [111, II.6.8]. | No |
Corollary 2.4.1. Let \( {C}_{1} \) and \( {C}_{2} \) be smooth curves, and let \( \phi : {C}_{1} \rightarrow {C}_{2} \) be a map of degree one. Then \( \phi \) is an isomorphism. | Proof. By definition, \( \deg \phi = 1 \) means that \( {\phi }^{ * }\bar{K}\left( {C}_{2}\right) = \bar{K}\left( {C}_{1}\right) \), so \( {\phi }^{ * } \) is an isomorphism of function fields. Hence from (II.2.4b), corresponding to the inverse map \( {\left( {\phi }^{ * }\right) }^{-1} : \bar{K}\left( {C}_{1}\right) \... | Yes |
Example 2.5.1. Hyperelliptic Curves. We assume that \( \operatorname{char}\left( K\right) \neq 2 \) . We choose a polynomial \( f\left( x\right) \in K\left\lbrack x\right\rbrack \) of degree \( d \) and consider the affine curve \( {C}_{0}/K \) given by the equation\n\n\[ \n{C}_{0} : {y}^{2} = f\left( x\right) = {a}_{0... | If we treat \( {C}_{0} \) as a curve in \( {\mathbb{P}}^{2} \) by homogenizing its affine equation, then one easily checks that the point(s) at infinity are singular whenever \( d \geq 4 \) . On the other hand,(II.2.4c) assures us that there exists some smooth projective curve \( C/K \) whose function field equals \( K... | Yes |
Proposition 2.6. Let \( \phi : {C}_{1} \rightarrow {C}_{2} \) be a nonconstant map of smooth curves.\n\n(a) For every \( Q \in {C}_{2} \) ,\n\n\[ \mathop{\sum }\limits_{{P \in {\phi }^{-1}\left( Q\right) }}{e}_{\phi }\left( P\right) = \deg \left( \phi \right) \] | Proof. (a) Use [111, II.6.9] with \( Y = {C}_{2} \) and \( D = \left( Q\right) \), or see [142, Proposition 2], [233, I Proposition 10], or [243, III §2, Theorem 1]. | No |
Corollary 2.7. A map \( \phi : {C}_{1} \rightarrow {C}_{2} \) is unramified if and only if\n\n\[ \# {\phi }^{-1}\left( Q\right) = \deg \left( \phi \right) \;\text{ for all }Q \in {C}_{2}. \] | Proof. From (II.2.6a), we see that \( \# {\phi }^{-1}\left( Q\right) = \deg \left( \phi \right) \) if and only if\n\n\[ \mathop{\sum }\limits_{{P \in {\phi }^{-1}\left( Q\right) }}{e}_{\phi }\left( P\right) = \# {\phi }^{-1}\left( Q\right) \]\n\nSince \( {e}_{\phi }\left( P\right) \geq 1 \), this occurs if and only if ... | Yes |
Consider the map \[ \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1},\;\phi \left( \left\lbrack {X, Y}\right\rbrack \right) = \left\lbrack {{X}^{3}{\left( X - Y\right) }^{2},{Y}^{5}}\right\rbrack . \] Then \( \phi \) is ramified at the points \( \left\lbrack {0,1}\right\rbrack \) and \( \left\lbrack {1,1}\right\rbr... | Further, \[ {e}_{\phi }\left( \left\lbrack {0,1}\right\rbrack \right) = 3\;\text{ and }\;{e}_{\phi }\left( \left\lbrack {1,1}\right\rbrack \right) = 2, \] so \[ \mathop{\sum }\limits_{{P \in {\phi }^{-1}\left( \left\lbrack {0,1}\right\rbrack \right) }}{e}_{\phi }\left( P\right) = {e}_{\phi }\left( \left\lbrack {0,1}\ri... | Yes |
Proposition 2.11. Let \( K \) be a field of characteristic \( p > 0 \), let \( q = {p}^{r} \), let \( C/K \) be a curve, and let \( \phi : C \rightarrow {C}^{\left( q\right) } \) be the \( {q}^{\text{th }} \) -power Frobenius morphism.\n\n(a) \( {\phi }^{ * }K\left( {C}^{\left( q\right) }\right) = K{\left( C\right) }^{... | Proof. (a) Using the description (I.2.9) of \( K\left( C\right) \) as consisting of quotients \( f/g \) of homogeneous polynomials of the same degree, we see that \( {\phi }^{ * }K\left( {C}^{\left( q\right) }\right) \) is the subfield of \( K\left( C\right) \) given by quotients\n\n\[{\phi }^{ * }\left( \frac{f}{g}\ri... | Yes |
Every map \( \psi : {C}_{1} \rightarrow {C}_{2} \) of (smooth) curves over a field of characteristic \( p > 0 \) factors as\n\n\[ \n{C}_{1}\xrightarrow[]{\phi }{C}_{1}^{\left( q\right) }\xrightarrow[]{\lambda }{C}_{2} \n\]\n\nwhere \( q = {\deg }_{i}\left( \psi \right) \), the map \( \phi \) is the \( {q}^{\text{th }} ... | Proof. Let \( \mathbb{K} \) be the separable closure of \( {\psi }^{ * }K\left( {C}_{2}\right) \) in \( K\left( {C}_{1}\right) \) . Then \( K\left( {C}_{1}\right) /\mathbb{K} \) is purely inseparable of degree \( q \), so \( K{\left( {C}_{1}\right) }^{q} \subset \mathbb{K} \) . From (II.2.11a, c) we have,\n\n\[ \nK{\le... | Yes |
Proposition 3.1. Let \( C \) be a smooth curve and let \( f \in \bar{K}{\left( C\right) }^{ * } \) . (a) \( \operatorname{div}\left( f\right) = 0 \) if and only if \( f \in {\bar{K}}^{ * } \) . | Proof. (a) If \( \operatorname{div}\left( f\right) = 0 \), then \( f \) has no poles, so the associated map \( f : C \rightarrow {\mathbb{P}}^{1} \) as defined in (II.2.2) is not surjective. Then (II.2.3) tells us that the map is constant, so \( f \in {\bar{K}}^{ * } \) . The converse is clear. | Yes |
On \( {\mathbb{P}}^{1} \), every divisor of degree 0 is principal. | To see this, suppose that \( D = \sum {n}_{P}\left( P\right) \) has degree 0 . Writing \( P = \left\lbrack {{\alpha }_{P},{\beta }_{P}}\right\rbrack \in {\mathbb{P}}^{1} \), we see that \( D \) is the divisor of the function\n\n\[ \mathop{\prod }\limits_{{P \in {\mathbb{P}}^{1}}}{\left( {\beta }_{P}X - {\alpha }_{P}Y\r... | No |
Example 3.5. Let \( C \) be a smooth curve, let \( f \in \bar{K}\left( C\right) \) be a nonconstant function, and let \( f : C \rightarrow {\mathbb{P}}^{1} \) be the corresponding map (II.2.2). Then directly from the definitions, | \[ \operatorname{div}\left( f\right) = {f}^{ * }\left( {\left( 0\right) - \left( \infty \right) }\right) . \] | Yes |
Proposition 3.6. Let \( \phi : {C}_{1} \rightarrow {C}_{2} \) be a nonconstant map of smooth curves.\n\n(a) \( \deg \left( {{\phi }^{ * }D}\right) = \left( {\deg \phi }\right) \left( {\deg D}\right) \) for all \( D \in \operatorname{Div}\left( {C}_{2}\right) \) . | Proof. (a) Follows directly from (II.2.6a). | No |
Proposition 4.2. Let \( C \) be a curve.\n\n(a) \( {\Omega }_{C} \) is a 1-dimensional \( \bar{K}\left( C\right) \) -vector space.\n\n(b) Let \( x \in \bar{K}\left( C\right) \) . Then \( {dx} \) is a \( \bar{K}\left( C\right) \) -basis for \( {\Omega }_{C} \) if and only if \( \bar{K}\left( C\right) /\bar{K}\left( x\ri... | Proof. (a) See [164, 27.A, B], [210, II.3.4], or [243, III §4, Theorem 3].\n\n(b) See [164, 27A, B] or [243, III §4, Theorem 4].\n\n(c) Using (a) and (b), choose \( y \in \bar{K}\left( {C}_{2}\right) \) such that \( {\Omega }_{{C}_{2}} = \bar{K}\left( {C}_{2}\right) {dy} \) and such that \( \bar{K}\left( {C}_{2}\right)... | No |
Proposition 4.3. Let \( C \) be a curve, let \( P \in C \), and let \( t \in \bar{K}\left( C\right) \) be a unformizer at \( P \). (a) For every \( \omega \in {\Omega }_{C} \) there exists a unique function \( g \in \bar{K}\left( C\right) \), depending on \( \omega \) and \( t \), satisfying \[ \omega = {gdt}. \] We de... | Proof. (a) This follows from (II.1.4) and (4.2ab). | No |
We are going to show that there are no holomorphic differentials on \( {\mathbb{P}}^{1} \) . | First, if \( t \) is a coordinate function on \( {\mathbb{P}}^{1} \), then\n\n\[ \operatorname{div}\left( {dt}\right) = - 2\left( \infty \right) \]\n\nTo see this, note that for all \( \alpha \in \bar{K} \), the function \( t - \alpha \) is a uniformizer at \( \alpha \), so\n\n\[ {\operatorname{ord}}_{\alpha }\left( {d... | Yes |
Example 4.6. Let \( C \) be the curve\n\n\[ C : {y}^{2} = \left( {x - {e}_{1}}\right) \left( {x - {e}_{2}}\right) \left( {x - {e}_{3}}\right) ,\]\n\nwhere we continue with the notation from (II.3.3). Then\n\n\[ \operatorname{div}\left( {dx}\right) = \left( {P}_{1}\right) + \left( {P}_{2}\right) + \left( {P}_{3}\right) ... | We thus see that\n\n\[ \operatorname{div}\left( {{dx}/y}\right) = 0. \]\n\nHence the differential \( {dx}/y \) is both holomorphic and nonvanishing. | No |
Proposition 5.2. Let \( D \in \operatorname{Div}\left( C\right) \). (a) If \( \deg D < 0 \), then \[ \mathcal{L}\left( D\right) = \{ 0\} \;\text{ and }\;\ell \left( D\right) = 0. \] | Proof. (a) Let \( f \in \mathcal{L}\left( D\right) \) with \( f \neq 0 \). Then (II.3.1b) tells us that \[ 0 = \deg \operatorname{div}\left( f\right) \geq \deg \left( {-D}\right) = - \deg D \] so \( \deg D \geq 0 \). | No |
Let \( {K}_{C} \in \operatorname{Div}\left( C\right) \) be a canonical divisor on \( C \), say\n\n\[ \n{K}_{C} = \operatorname{div}\left( \omega \right)\n\]\n\nThen each function \( f \in \mathcal{L}\left( {K}_{C}\right) \) has the property that\n\n\[ \n\operatorname{div}\left( f\right) \geq - \operatorname{div}\left( ... | Since every differential on \( C \) has the form \( {f\omega } \) for some \( f \), we have established an isomorphism of \( \bar{K} \) -vector spaces,\n\n\[ \n\mathcal{L}\left( {K}_{C}\right) \cong \left\{ {\omega \in {\Omega }_{C} : \omega }\right. \text{is holomorphic}\} \text{.} \n\] | Yes |
Theorem 5.4. (Riemann-Roch) Let \( C \) be a smooth curve and let \( {K}_{C} \) be a canonical divisor on \( C \) . There is an integer \( g \geq 0 \), called the genus of \( C \), such that for every divisor \( D \in \operatorname{Div}\left( C\right) \) , | Proof. For a fancy proof using Serre duality, see [111, IV §1]. A more elementary proof, due to Weil, is given in [136, Chapter 1]. | No |
Corollary 5.5. (a) \( \ell \left( {K}_{C}\right) = g \) . | Proof. (a) Use (II.5.4) with \( D = 0 \) . Note that \( \mathcal{L}\left( 0\right) = \bar{K} \) from (II.1.2), so \( \ell \left( 0\right) = 1 \) . | Yes |
Example 5.6. Let \( C = {\mathbb{P}}^{1} \). Then (II.4.5) says that there are no holomorphic differentials on \( C \), so using the identification from (II.5.3), we see that \( \ell \left( {K}_{C}\right) = 0 \). Then (II.5.5a) says that \( {\mathbb{P}}^{1} \) has genus 0, and the Riemann-Roch theorem reads | \[ \ell \left( D\right) - \ell \left( {-2\left( \infty \right) - D}\right) = \deg D + 1. \] In particular, if \( \deg D \geq - 1 \), then \[ \ell \left( D\right) = \deg D + 1 \] (See Exercise 2.3b.) | No |
Let \( C \) be the curve \[ C : {y}^{2} = \left( {x - {e}_{1}}\right) \left( {x - {e}_{2}}\right) \left( {x - {e}_{3}}\right) ,\] where we continue with the notation of (II.3.3) and (II.4.6). We have seen in (II.4.6) that \[ \operatorname{div}\left( {{dx}/y}\right) = 0, \] so the canonical class on \( C \) is trivial, ... | We consider several special cases. (i) Let \( P \in C \). Then \( \ell \left( \left( P\right) \right) = 1 \). But \( \mathcal{L}\left( \left( P\right) \right) \) certainly contains the constant functions, which have no poles, so this shows that there are no functions on \( C \) having a single simple pole. (ii) Recall ... | Yes |
Proposition 5.8. Let \( C/K \) be a smooth curve and let \( D \in {\operatorname{Div}}_{K}\left( C\right) \) . Then \( \mathcal{L}\left( D\right) \) has a basis consisting of functions in \( K\left( C\right) \) . | Proof. Since \( D \) is defined over \( K \), we have\n\n\[ \n{f}^{\sigma } \in \mathcal{L}\left( {D}^{\sigma }\right) = \mathcal{L}\left( D\right) \;\text{ for all }f \in \mathcal{L}\left( D\right) \text{ and all }\sigma \in {G}_{\bar{K}/K}.\n\]\n\nThus \( {G}_{\bar{K}/K} \) acts on \( \mathcal{L}\left( D\right) \), a... | No |
Lemma 5.8.1. Let \( V \) be a \( \bar{K} \) -vector space, and assume that \( {G}_{\bar{K}/K} \) acts continuously on \( V \) in a manner compatible with its action on \( \bar{K} \) . Let\n\n\[ \n{V}_{K} = {V}^{{G}_{\bar{K}/K}} = \left\{ {\mathbf{v} \in V : {\mathbf{v}}^{\sigma } = \mathbf{v}\text{ for all }\sigma \in ... | Proof. It is clear that \( {V}_{K} \) is a \( K \) -vector space, so it suffices to show that every \( \mathbf{v} \in V \) is a \( \bar{K} \) -linear combination of vectors in \( {V}_{K} \) . Let \( \mathbf{v} \in V \) and let \( L/K \) be a finite Galois extension such that \( \mathbf{v} \) is fixed by \( {G}_{\bar{K}... | Yes |
Theorem 5.9. (Hurwitz) Let \( \phi : {C}_{1} \rightarrow {C}_{2} \) be a nonconstant separable map of smooth curves of genera \( {g}_{1} \) and \( {g}_{2} \), respectively. Then\n\n\[ 2{g}_{1} - 2 \geq \left( {\deg \phi }\right) \left( {2{g}_{2} - 2}\right) + \mathop{\sum }\limits_{{P \in {C}_{1}}}\left( {{e}_{\phi }\l... | Proof. Let \( \omega \in {\Omega }_{{C}_{2}} \) be a nonzero differential, let \( P \in {C}_{1} \), and let \( Q = \phi \left( P\right) \) . Since \( \phi \) is separable,(II.4.2c) tells us that \( {\phi }^{ * }\omega \neq 0 \) . We need to relate the values of \( {\operatorname{ord}}_{P}\left( {{\phi }^{ * }\omega }\r... | Yes |
Let \( P = \left( {{x}_{0},{y}_{0}}\right) \) be a point satisfying a Weierstrass equation \[ f\left( {x, y}\right) = {y}^{2} + {a}_{1}{xy} + {a}_{3}y - {x}^{3} - {a}_{2}{x}^{2} - {a}_{4}x - {a}_{6} = 0, \] and assume that \( P \) is a singular point on the curve \( f\left( {x, y}\right) = 0 \). Then from (I.1.5) we ha... | It follows that there are \( \alpha ,\beta \in \bar{K} \) such that the Taylor series expansion of \( f\left( {x, y}\right) \) at \( P \) has the form \[ f\left( {x, y}\right) - f\left( {{x}_{0},{y}_{0}}\right) \] \[ = \left( {\left( {y - {y}_{0}}\right) - \alpha \left( {x - {x}_{0}}\right) }\right) \left( {\left( {y -... | Yes |
Proposition 1.4. (a) The curve given by a Weierstrass equation satisfies:\n\n(i) It is nonsingular if and only if \( \Delta \neq 0 \) .\n\n(ii) It has a node if and only if \( \Delta = 0 \) and \( {c}_{4} \neq 0 \) .\n\n(iii) It has a cusp if and only if \( \Delta = {c}_{4} = 0 \) .\n\nIn cases (ii) and (iii), there is... | Proof. Let \( E \) be given by the Weierstrass equation\n\n\[ E : f\left( {x, y}\right) = {y}^{2} + {a}_{1}{xy} + {a}_{3}y - {x}^{3} - {a}_{2}{x}^{2} - {a}_{4}x - {a}_{6} = 0. \]\n\nWe start by showing that the point at infinity is never singular. Thus we look at the curve in \( {\mathbb{P}}^{2} \) with homogeneous equ... | Yes |
Proposition 1.5. Let \( E \) be an elliptic curve. Then the invariant differential \( \omega \) associated to a Weierstrass equation for \( E \) is holomorphic and nonvanishing, i.e., \( \operatorname{div}\left( \omega \right) = 0 \) . | Proof. Let \( P = \left( {{x}_{0},{y}_{0}}\right) \in E \) and\n\n\[ E : F\left( {x, y}\right) = {y}^{2} + {a}_{1}{xy} + {a}_{3}y - {x}^{3} - {a}_{2}{x}^{2} - {a}_{4}x - {a}_{6} = 0, \]\n\nso\n\n\[ \omega = \frac{d\left( {x - {x}_{0}}\right) }{{F}_{y}\left( {x, y}\right) } = - \frac{d\left( {y - {y}_{0}}\right) }{{F}_{... | Yes |
Proposition 1.6. If a curve \( E \) given by a Weierstrass equation is singular, then there exists a rational map \( \phi : E \rightarrow {\mathbb{P}}^{1} \) of degree one, i.e., the curve \( E \) is birational to \( {\mathbb{P}}^{1} \) . (Note that since \( E \) is singular, we cannot use (II.2.4.1) to conclude that \... | Proof. Making a linear change of variables, we may assume that the singular point is \( \left( {x, y}\right) = \left( {0,0}\right) \) . Checking partial derivatives, we see that the Weierstrass equation has the form\n\n\[ E : {y}^{2} + {a}_{1}{xy} = {x}^{3} + {a}_{2}{x}^{2}. \]\n\nThen the rational map\n\n\[ E \rightar... | Yes |
Every elliptic curve is isomorphic (over \( \bar{K} \) ) to an elliptic curve in Legendre form\n\n\[ \n{E}_{\lambda } : {y}^{2} = x\left( {x - 1}\right) \left( {x - \lambda }\right) \n\]\n\nfor some \( \lambda \in \bar{K} \) with \( \lambda \neq 0,1 \) . | Since \( \operatorname{char}\left( K\right) \neq 2 \), we know that \( E \) has a Weierstrass equation of the form\n\n\[ \n{y}^{2} = 4{x}^{3} + {b}_{2}{x}^{2} + 2{b}_{4}x + {b}_{6} \n\]\n\nReplacing \( \left( {x, y}\right) \) by \( \left( {x,{2y}}\right) \) and factoring the cubic yields an equation of the form\n\n\[ \... | Yes |
Proposition 2.2. The composition law (III.2.1) has the following properties: (a) If a line \( L \) intersects \( E \) at the (not necessarily distinct) points \( P, Q, R \), then\n\n\[ \n\\left( {P \\oplus Q}\\right) \\oplus R = O.\n\]\n\n(b) \( P \\oplus O = P \) for all \( P \\in E \) .\n\n(c) \( P \\oplus Q = Q \\op... | Proof. All of this is easy except for the associativity (e).\n\n(a) This is obvious from (III.2.1), or look at Figure 3.3 and note that the tangent line to \( E \) at \( O \) intersects \( E \) with multiplicity 3 at \( O \) .\n\n(b) Taking \( Q = O \) in (III.2.1), we see that the lines \( L \) and \( {L}^{\\prime } \... | No |
Corollary 2.3.1. With notation as in (III.2.3), a function \( f \in \bar{K}\left( E\right) = \bar{K}\left( {x, y}\right) \) is said to be even if \( f\left( P\right) = f\left( {-P}\right) \) for all \( P \in E \) . Then \[ f\text{is even if and only if}\;f \in \bar{K}\left( x\right) \text{.} \] | Proof. From (III.2.3), if \( P = \left( {{x}_{0},{y}_{0}}\right) \), then \( - P = \left( {{x}_{0}, - {y}_{0} - {a}_{1}{x}_{0} - {a}_{3}}\right) \) . It follows immediately that every element of \( \bar{K}\left( x\right) \) is even. Suppose now that \( f \in \bar{K}\left( {x, y}\right) \) is even. Using the Weierstrass... | Yes |
Proposition 2.5. Let \( E \) be a curve given by a Weierstrass equation with \( \Delta = 0 \), so \( E \) has a singular point \( S \) . Then the composition law (III.2.1) makes \( {E}_{\mathrm{{ns}}} \) into an abelian group. | Proof. We first observe that \( {E}_{\mathrm{{ns}}} \) is closed under the composition law (III.2.1), since if a line \( L \) intersects \( {E}_{\mathrm{{ns}}} \) at two (not necessarily distinct) points, then \( L \) cannot contain the point \( S \) . This is true because \( S \) is a singular point of \( E \), so \( ... | No |
Proposition 3.1. Let \( E \) be an elliptic curve defined over \( K \) .\n\n(a) There exist functions \( x, y \in K\left( E\right) \) such that the map\n\n\[ \phi : E \rightarrow {\mathbb{P}}^{2},\;\phi = \left\lbrack {x, y,1}\right\rbrack \]\n\ngives an isomorphism of \( E/K \) onto a curve given by a Weierstrass equa... | Proof. (a) We look at the vector spaces \( \mathcal{L}\left( {n\left( O\right) }\right) \) for \( n = 1,2,\ldots \) By the Riemann-Roch theorem, more specifically from (II.5.5c) with \( g = 1 \), we have\n\n\[ \ell \left( {n\left( O\right) }\right) = \dim \mathcal{L}\left( {n\left( O\right) }\right) = n\;\text{ for all... | Yes |
Corollary 3.1.1. Let \( E/K \) be an elliptic curve with Weierstrass coordinate functions \( x \) and \( y \) . Then\n\n\[ K\left( E\right) = K\left( {x, y}\right) \;\text{ and }\;\left\lbrack {K\left( E\right) : K\left( x\right) }\right\rbrack = 2.\] | Proof. These two facts were proven during the course of proving (III.3.1a). | No |
Lemma 3.3. Let \( C \) be a curve of genus one and let \( P, Q \in C \) . Then\n\n\[ \left( P\right) \sim \left( Q\right) \;\text{ if and only if }\;P = Q. \] | Proof. Suppose that \( \left( P\right) \sim \left( Q\right) \) and choose \( f \in \bar{K}{\left( C\right) }^{ * } \) such that\n\n\[ \operatorname{div}\left( f\right) = \left( P\right) - \left( Q\right) \]\n\nThen \( f \in \mathcal{L}\left( \left( Q\right) \right) \) . The Riemann-Roch theorem (II.5.5c) tells us that\... | Yes |
Proposition 3.4. Let \( \left( {E, O}\right) \) be an elliptic curve.\n\n(a) For every degree-0 divisor \( D \in {\operatorname{Div}}^{0}\left( E\right) \) there exists a unique point \( P \in E \) satisfying\n\n\[ D \sim \left( P\right) - \left( O\right) \] | Proof. (a) Since \( E \) has genus one, the Riemann-Roch theorem (II.5.5c) says that\n\n\[ \dim \mathcal{L}\left( {D + \left( O\right) }\right) = 1 \]\n\nLet \( f \in \bar{K}\left( E\right) \) be a nonzero element of \( \mathcal{L}\left( {D + \left( O\right) }\right) \), so \( f \) is a basis for this one-dimensional v... | Yes |
Corollary 3.5. Let \( E \) be an elliptic curve and let \( D = \sum {n}_{P}\left( P\right) \in \operatorname{Div}\left( E\right) \) . Then \( D \) is a principal divisor if and only if\n\n\[ \mathop{\sum }\limits_{{P \in E}}{n}_{P} = 0\;\text{ and }\;\mathop{\sum }\limits_{{P \in E}}\left\lbrack {n}_{P}\right\rbrack P ... | Proof. From (II.3.1b), every principal divisor has degree 0 . Next let \( D \in {\operatorname{Div}}^{0}\left( E\right) \) . We use (III.3.4a, e) to deduce that\n\n\[ D \sim 0\; \Leftrightarrow \;\sigma \left( D\right) = O\; \Leftrightarrow \;\mathop{\sum }\limits_{{P \in E}}\left\lbrack {n}_{P}\right\rbrack \sigma \le... | Yes |
For each \( m \in \mathbb{Z} \) we define the multiplication-by- \( m \) isogeny\n\n\[ \left\lbrack m\right\rbrack : E \rightarrow E \]\n\nin the natural way. Thus if \( m > 0 \), then\n\n\[ \left\lbrack m\right\rbrack \left( P\right) = \underset{m\text{ terms }}{\underbrace{P + P + \cdots + P}}. \]\n\nFor \( m < 0 \),... | Using (III.3.6), an easy induction shows that \( \left\lbrack m\right\rbrack \) is a morphism, hence an isogeny, since it clearly sends \( O \) to \( O \) . | No |
Proposition 4.2. (a) Let \( E/K \) be an elliptic curve and let \( m \in \mathbb{Z} \) with \( m \neq 0 \) . Then the multiplication-by-m map\n\n\[ \left\lbrack m\right\rbrack : E \rightarrow E \]\n\nis nonconstant. | Proof. (a) We start by showing that \( \left\lbrack 2\right\rbrack \neq \left\lbrack 0\right\rbrack \) . The duplication formula (III.2.3d) says that if a point \( P = \left( {x, y}\right) \in E \) has order 2, then it must satisfy\n\n\[ 4{x}^{3} + {b}_{2}{x}^{2} + 2{b}_{4}x + {b}_{6} = 0. \]\n\nIf \( \operatorname{cha... | No |
Assume that \( \operatorname{char}\left( K\right) \neq 2 \) and let \( i \in \bar{K} \) be a primitive fourth root of unity, i.e., \( {i}^{2} = - 1 \) . Then, as noted in (III.3.2), the elliptic curve \( E/K \) given by the equation\n\n\[ E : {y}^{2} = {x}^{3} - x \]\n\nhas endomorphism ring \( \operatorname{End}\left(... | Thus \( E \) has complex multiplication. Clearly \( \left\lbrack i\right\rbrack \) is defined over \( K \) if and only if \( i \in K \) . Hence even if \( E \) is defined over \( K \), it may happen that \( {\operatorname{End}}_{K}\left( E\right) \) is strictly smaller than \( \operatorname{End}\left( E\right) \) .\n\n... | Yes |
Again assume that \( \operatorname{char}\left( K\right) \neq 2 \) and let \( a, b \in K \) satisfy \( b \neq 0 \) and \( r = {a}^{2} - {4b} \neq 0 \) . Consider the two elliptic curves\n\n\[ \n{E}_{1} : {y}^{2} = {x}^{3} + a{x}^{2} + {bx} \n\]\n\n\[ \n{E}_{2} : {Y}^{2} = {X}^{3} - {2a}{X}^{2} + {rX}. \n\]\n\nThere are ... | A direct computation shows that \( \widehat{\phi } \circ \phi = \left\lbrack 2\right\rbrack \) on \( {E}_{1} \) and \( \phi \circ \widehat{\phi } = \left\lbrack 2\right\rbrack \) on \( {E}_{2} \) . The maps \( \phi \) and \( \widehat{\phi } \) are examples of dual isogenies, which we discuss further in (III §6). | Yes |
Let \( K \) be a field of characteristic \( p > 0 \), let \( q = {p}^{r} \), and let \( E/K \) be an elliptic curve given by a Weierstrass equation. We recall from (II §2) that the curve \( {E}^{\left( q\right) }/K \) is defined by raising the coefficients of the equation for \( E \) to the \( {q}^{\text{th }} \) power... | Since \( {E}^{\left( q\right) } \) is the zero locus of a Weierstrass equation, it will be an elliptic curve provided that its equation is nonsingular. Writing everything out in terms of Weierstrass coefficients and using the fact that the \( {q}^{\text{th }} \) -power map \( K \rightarrow K \) is a homomorphism, it is... | Yes |
Let \( E/K \) be an elliptic curve and let \( Q \in E \). Then we can define a translation-by-Q map \[ {\tau }_{Q} : E \rightarrow E,\;P \mapsto P + Q. \] | The map \( {\tau }_{Q} \) is clearly an isomorphism, since \( {\tau }_{-Q} \) provides an inverse. Of course, it is not an isogeny unless \( Q = O \). | Yes |
Theorem 4.8. Let\n\n\\[ \n\\phi : {E}_{1} \\rightarrow {E}_{2} \n\\]\n\nbe an isogeny. Then\n\n\\[ \n\\phi \\left( {P + Q}\\right) = \\phi \\left( P\\right) + \\phi \\left( Q\\right) \\;\\text{ for all }P, Q \\in {E}_{1}. \n\\] | Proof. If \\( \\phi \\left( P\\right) = O \\) for all \\( P \\in E \\), there is nothing to prove. Otherwise, \\( \\phi \\) is a finite map, so by (II.3.7), it induces a homomorphism\n\n\\[ \n{\\phi }_{ * } : {\\operatorname{Pic}}^{0}\\left( {E}_{1}\\right) \\rightarrow {\\operatorname{Pic}}^{0}\\left( {E}_{2}\\right) ... | Yes |
Corollary 4.9. Let \( \phi : {E}_{1} \rightarrow {E}_{2} \) be a nonzero isogeny. Then\n\n\[ \ker \phi = {\phi }^{-1}\left( O\right) \]\n\n is a finite group. | Proof. It is a subgroup of \( {E}_{1} \) from (III.4.8), and it is finite (of order at most \( \deg \phi \) ) from (II.2.6a). | Yes |
Theorem 4.10. Let \( \phi : {E}_{1} \rightarrow {E}_{2} \) be a nonzero isogeny.\n\n(a) For every \( Q \in {E}_{2} \) ,\n\n\[ \n\# {\phi }^{-1}\left( Q\right) = {\deg }_{s}\phi \n\]\n\nFurther, for every \( P \in {E}_{1} \) ,\n\n\[ \n{e}_{\phi }\left( P\right) = {\deg }_{i}\phi \n\] | Proof. (a) From (II.2.6b) we know that\n\n\[ \n\# {\phi }^{-1}\left( Q\right) = {\deg }_{s}\phi \;\text{ for all but finitely many }Q \in {E}_{2}.\n\]\n\nBut for any \( Q,{Q}^{\prime } \in {E}_{2} \), if we choose some \( R \in {E}_{1} \) with \( \phi \left( R\right) = {Q}^{\prime } - Q \), then the fact that \( \phi \... | Yes |
Corollary 4.11. Let\n\n\\[ \n\\phi : {E}_{1} \\rightarrow {E}_{2}\\;\\text{ and }\\;\\psi : {E}_{1} \\rightarrow {E}_{3} \n\\]\n\nbe nonconstant isogenies, and assume that \\( \\phi \\) is separable. If\n\n\\[ \n\\ker \\phi \\subset \\ker \\psi \n\\]\n\nthen there is a unique isogeny\n\n\\[ \n\\lambda : {E}_{2} \\right... | Proof. Since \\( \\phi \\) is separable,(III.4.10c) says that \\( \\bar{K}\\left( {E}_{1}\\right) \\) is a Galois extension of \\( {\\phi }^{ * }\\bar{K}\\left( {E}_{2}\\right) \\) . Then the inclusion \\( \\ker \\phi \\subset \\ker \\psi \\) and the identification (III.4.10b) imply that every element of \\( \\operator... | Yes |
Proposition 4.12. Let \( E \) be an elliptic curve and let \( \Phi \) be a finite subgroup of \( E \). There are a unique elliptic curve \( {E}^{\prime } \) and a separable isogeny\n\n\[ \phi : E \rightarrow {E}^{\prime }\;\text{ satisfying }\;\ker \phi = \Phi . \] | Proof of (III.4.12). As in (III.4.10b), each point \( T \in \Phi \) gives rise to an automorphism \( {\tau }_{T}^{ * } \) of \( \bar{K}\left( E\right) \). Let \( \bar{K}{\left( E\right) }^{\Phi } \) be the subfield of \( \bar{K}\left( E\right) \) fixed by every element of \( \Phi \). Then Galois theory tells us that \(... | Yes |
Proposition 5.1. Let \( E \) and \( \omega \) be as above, let \( Q \in E \), and let \( {\tau }_{Q} : E \rightarrow E \) be the translation-by-Q map (III.4.7). Then \[ {\tau }_{Q}^{ * }\omega = \omega \] | Proof. One can prove this proposition by a straightforward, but messy and unenlightening, calculation as follows. Write \( x\left( {P + Q}\right) \) and \( y\left( {P + Q}\right) \) in terms of \( x\left( P\right), x\left( Q\right), y\left( P\right) \), and \( y\left( Q\right) \) using the addition formula (III.2.3c). ... | No |
Theorem 5.2. Let \( E \) and \( {E}^{\prime } \) be elliptic curves, let \( \omega \) be an invariant differential on \( E \), and let\n\n\[ \phi ,\psi : {E}^{\prime } \rightarrow E \]\n\nbe isogenies. Then\n\n\[ {\left( \phi + \psi \right) }^{ * }\omega = {\phi }^{ * }\omega + {\psi }^{ * }\omega \] | Proof. If \( \phi = \left\lbrack 0\right\rbrack \) or \( \psi = \left\lbrack 0\right\rbrack \), the result is clear. Next, if \( \phi + \psi = \left\lbrack 0\right\rbrack \), then using the fact that\n\n\[ {\psi }^{ * } = {\left( -\phi \right) }^{ * } = {\phi }^{ * } \circ {\left\lbrack -1\right\rbrack }^{ * },\]\n\nit... | Yes |
Let \( \omega \) be an invariant differential on an elliptic curve \( E \) . Let \( m \in \mathbb{Z} \) . Then\n\n\[{\left\lbrack m\right\rbrack }^{ * }\omega = {m\omega }.\] | Proof. The assertion is true for \( m = 0 \), since \( \left\lbrack 0\right\rbrack \) is the constant map, and it is true for \( m = 1 \), since [1] is the identity map. We use (III.5.2) with \( \phi = \left\lbrack m\right\rbrack \) and \( \psi = \left\lbrack 1\right\rbrack \) to obtain\n\n\[{\left\lbrack m + 1\right\r... | Yes |
Corollary 5.4. Let \( E/K \) be an elliptic curve and let \( m \in \mathbb{Z} \) . Assume that \( m \neq 0 \) in \( K \) . Then the multiplication-by-m map on \( E \) is a finite separable endomorphism. | Proof. Let \( \omega \) be an invariant differential on \( E \) . Then (III.5.3) and our assumption on \( m \) implies that\n\n\[{\left\lbrack m\right\rbrack }^{ * }\omega = {m\omega } \neq 0,\]\n\nso certainly \( \left\lbrack m\right\rbrack \neq \left\lbrack 0\right\rbrack \) . Hence \( \left\lbrack m\right\rbrack \) ... | Yes |
Let \( E \) be an elliptic curve defined over a finite field \( {\mathbb{F}}_{q} \) of characteristic \( p \), let \( \phi : E \rightarrow E \) be the \( {q}^{\text{th }} \) -power Frobenius morphism (III.4.6), and let \( m, n \in \mathbb{Z} \) . Then the map\n\n\[ m + {n\phi } : E \rightarrow E \]\n\nis separable if a... | Proof. Let \( \omega \) be an invariant differential on \( E \) . From (II.4.2c) we know that a map \( \psi : E \rightarrow E \) is inseparable if and only if \( {\psi }^{ * }\omega = 0 \) . We apply this criterion to the map \( \psi = m + {n\phi } \) . Using (III.5.2) and (III.5.3), we compute\n\n\[ {\left( m + n\phi ... | Yes |
Corollary 5.6. Let \( E/K \) be an elliptic curve and let \( \omega \) be a nonzero invariant differential on \( E \) . We define a map from \( \operatorname{End}\left( E\right) \) to \( \bar{K} \) in the following way:\n\n\[ \operatorname{End}\left( E\right) \rightarrow \bar{K},\;\phi \mapsto {a}_{\phi }\;\text{ such ... | Proof. As in the proof of (III.5.1), the fact that \( {\Omega }_{E} \) is a one-dimensional \( \bar{K}\left( E\right) \) - vector space (II.4.2) implies that \( {\phi }^{ * }\omega = {a}_{\phi }\omega \) for some function \( {a}_{\phi } \in \bar{K}\left( E\right) \) . We claim that \( {a}_{\phi } \in \bar{K} \) . This ... | Yes |
Corollary 6.3. Let \( {E}_{1} \) and \( {E}_{2} \) be elliptic curves. The degree map\n\n\[ \deg : \operatorname{Hom}\left( {{E}_{1},{E}_{2}}\right) \rightarrow \mathbb{Z} \]\n\nis a positive definite quadratic form. | Proof. Everything is clear except for the fact that the pairing\n\n\[ \langle \phi ,\psi \rangle = \deg \left( {\phi + \psi }\right) - \deg \left( \phi \right) - \deg \left( \psi \right) \]\n\nis bilinear. To verify this, we use the injection\n\n\[ \left\lbrack \;\right\rbrack : \mathbb{Z} \rightarrow \operatorname{End... | Yes |
Corollary 6.4. Let \( E \) be an elliptic curve and let \( m \in \mathbb{Z} \) with \( m \neq 0 \). (a) \( \deg \left\lbrack m\right\rbrack = {m}^{2} \). | Proof. (a) This was proven in (III.6.2d). We record it again here in order to point out that there are other ways of proving that \( \left\lbrack m\right\rbrack \) has degree \( {m}^{2} \) ; see for example exercises 3.7,3.8, and 3.11. | No |
Proposition 7.1. The Tate module has the following structure:\n\n(a) \( \;{T}_{\ell }\left( E\right) \cong {\mathbb{Z}}_{\ell } \times {\mathbb{Z}}_{\ell } \) as a \( {\mathbb{Z}}_{\ell } \) -module, if \( \ell \neq \operatorname{char}\left( K\right) \) .\n\n(b) \( \;{T}_{p}\left( E\right) \cong \{ 0\} \) or \( {\mathb... | Proof. This follows immediately from (III.6.4b, c). | No |
Let \( {E}_{1} \) and \( {E}_{2} \) be elliptic curves. Then\n\n\[ \operatorname{Hom}\left( {{E}_{1},{E}_{2}}\right) \]\n\n is a free \( \mathbb{Z} \) -module of rank at most 4 . | Proof. We know from (III.4.2b) that \( \operatorname{Hom}\left( {{E}_{1},{E}_{2}}\right) \) is torsion-free. This implies that\n\n\[ {\operatorname{rank}}_{\mathbb{Z}}\operatorname{Hom}\left( {{E}_{1},{E}_{2}}\right) = {\operatorname{rank}}_{{\mathbb{Z}}_{\ell }}\operatorname{Hom}\left( {{E}_{1},{E}_{2}}\right) \otimes... | Yes |
Theorem 7.9. (Serre) Let \( K \) be a number field and let \( E/K \) be an elliptic curve without complex multiplication.\n\n(a) \( {\rho }_{\ell }\left( {G}_{\bar{K}/K}\right) \) is of finite index in \( \operatorname{Aut}\left( {{T}_{\ell }\left( E\right) }\right) \) for all primes \( \ell \) .\n\n(b) \( {\rho }_{\el... | Proof. See [237] and [231]. | No |
Proposition 8.1. The Weil \( {e}_{m} \) -pairing has the following properties: (a) It is bilinear: | PROOF. (a) Linearity in the first factor is easy:\n\n\[ {e}_{m}\left( {{S}_{1} + {S}_{2}, T}\right) = \frac{g\left( {X + {S}_{1} + {S}_{2}}\right) }{g\left( X\right) } = \frac{g\left( {X + {S}_{1} + {S}_{2}}\right) }{g\left( {X + {S}_{1}}\right) }\frac{g\left( {X + {S}_{1}}\right) }{g\left( X\right) } \]\n\n\[ = {e}_{m... | Yes |
Corollary 8.1.1. There exist points \( S, T \in E\left\lbrack m\right\rbrack \) such that \( {e}_{m}\left( {S, T}\right) \) is a primitive \( {m}^{\text{th }} \) root of unity. In particular, if \( E\left\lbrack m\right\rbrack \subset E\left( K\right) \), then \( {\mathbf{\mu }}_{m} \subset {K}^{ * } \) . | Proof. The image of \( {e}_{m}\left( {S, T}\right) \) as \( S \) and \( T \) range over \( E\left\lbrack m\right\rbrack \) is a subgroup of \( {\mathbf{\mu }}_{m} \) , say equal to \( {\mathbf{\mu }}_{d} \) . It follows that\n\n\[ 1 = {e}_{m}{\left( S, T\right) }^{d} = {e}_{m}\left( {\left\lbrack d\right\rbrack S, T}\r... | Yes |
Proposition 8.2. Let \( \phi : {E}_{1} \rightarrow {E}_{2} \) be an isogeny of elliptic curves. Then for all \( m \) - torsion points \( S \in {E}_{1}\left\lbrack m\right\rbrack \) and \( T \in {E}_{2}\left\lbrack m\right\rbrack \) , \n\n\[ \n{e}_{m}\left( {S,\widehat{\phi }\left( T\right) }\right) = {e}_{m}\left( {\ph... | Proof. Let \n\n\[ \n\operatorname{div}\left( f\right) = m\left( T\right) - m\left( O\right) \;\text{ and }\;f \circ \left\lbrack m\right\rbrack = {g}^{m} \n\] \n\nbe as usual. Then \n\n\[ \n{e}_{m}\left( {{\phi S}, T}\right) = \frac{g\left( {X + {\phi S}}\right) }{g\left( X\right) }. \n\] \n\nChoose a function \( h \in... | Yes |
Proposition 8.6. Let \( \phi \in \operatorname{End}\left( E\right) \), and let \( {\phi }_{\ell } : {T}_{\ell }\left( E\right) \rightarrow {T}_{\ell }\left( E\right) \) be the map that \( \phi \) induces on the Tate module of \( E \) . Then\n\n\[ \det \left( {\phi }_{\ell }\right) = \deg \left( \phi \right) \;\text{ an... | Proof. Let \( \left\{ {{v}_{1},{v}_{2}}\right\} \) be a \( {\mathbb{Z}}_{\ell } \) -basis for \( {T}_{\ell }\left( E\right) \) and write\n\n\[ {\phi }_{\ell }\left( {v}_{1}\right) = a{v}_{1} + b{v}_{2},\;{\phi }_{\ell }\left( {v}_{2}\right) = c{v}_{1} + d{v}_{2}, \]\n\nso the matrix of \( {\phi }_{\ell } \) relative to... | Yes |
Let \( \mathcal{K} \) be an imaginary quadratic field and let \( \mathcal{O} \) be its ring of integers. Then for each integer \( f \geq 1 \), the ring \( \mathbb{Z} + f\mathcal{O} \) is an order of \( \mathcal{K} \). | In fact, these are all of the orders of \( \mathcal{K} \); see Exercise 3.20. | No |
Corollary 9.4. The endomorphism ring of an elliptic curve \( E/K \) is either \( \mathbb{Z} \), an order in an imaginary quadratic field, or an order in a quaternion algebra. If \( \operatorname{char}\left( K\right) = 0 \), then only the first two are possible. | Proof. We have proven in (III.4.2b), (III.6.2), and (III.6.3) all of the facts needed to apply (III.9.3) to the ring \( \operatorname{End}\left( E\right) \). This proves the first part of the corollary. If \( \operatorname{char}\left( K\right) = 0 \), then (III.5.6c) says that \( \operatorname{End}\left( E\right) \) is... | No |
Theorem 9.5. Let \( \mathcal{K} \) be a quaternion algebra.\n\n(a) We have \( {\operatorname{inv}}_{p}\left( \mathcal{K}\right) = 0 \) for all but finitely many \( p \), and\n\n\[ \mathop{\sum }\limits_{p}{\operatorname{inv}}_{p}\left( \mathcal{K}\right) \in \mathbb{Z} \]\n\n(Note that the sum includes \( p = \infty \)... | Proof. This is a very special case of the fact that the central simple algebras over a field \( K \) are classified by the Brauer group \( \operatorname{Br}\left( K\right) = {H}^{2}\left( {{G}_{\bar{K}/K},{\bar{K}}^{ * }}\right) \left\lbrack {{233},\mathrm{\;X}§5}\right\rbrack \) , and the fundamental exact sequence fr... | Yes |
Theorem 10.1. Let \( E/K \) be an elliptic curve. Then its automorphism group \( \operatorname{Aut}\left( E\right) \) is a finite group of order dividing 24. More precisely, the order of \( \operatorname{Aut}\left( E\right) \) is given by the following table:\n\n<table><thead><tr><th># \( \mathrm{{Aut}}\left( E\right) ... | Proof. We restrict attention to \( \operatorname{char}\left( K\right) \neq 2,3 \) ; see (III.1.3) and (A.1.2c). Then \( E \) is given by an equation\n\n\[ E : {y}^{2} = {x}^{3} + {Ax} + B, \]\n\nand every automorphism of \( E \) has the form\n\n\[ x = {u}^{2}{x}^{\prime },\;y = {u}^{3}{y}^{\prime }, \]\n\nfor some \( u... | Yes |
Corollary 10.2. Let \( E/K \) be a curve over a field of characteristic not equal to 2 or 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 there is a natural isomo... | Proof. While proving (III.10.1), we showed that the map\n\n\[ \left\lbrack \;\right\rbrack : {\mathbf{\mu }}_{n} \rightarrow E,\;\left\lbrack \zeta \right\rbrack \left( {x, y}\right) = \left( {{\zeta }^{2}x,{\zeta }^{3}y}\right) ,\]\n\nis an isomorphism of abstract groups. It is clear that this map commutes with the ac... | Yes |
Proposition 1.1. (a) The procedure described above gives a power series\n\n\[ w\\left( z\\right) = {z}^{3}\\left( {1 + {A}_{1}z + {A}_{2}{z}^{2} + \\cdots }\\right) \\in \\mathbb{Z}\\left\\lbrack {{a}_{1},\\ldots ,{a}_{6}}\\right\\rbrack \\llbracket z\\rrbracket .\n\]\n\n(b) The series \( w\\left( z\\right) \) is the u... | Proof. Parts (a) and (b) are special cases of Hensel's lemma, which we prove later in this section (IV.1.2). To prove the present proposition, use (IV.1.2) with\n\n\[ R = \\mathbb{Z}\\left\\lbrack {{a}_{1},\\ldots ,{a}_{6}}\\right\\rbrack \\llbracket z\\rrbracket ,\\;I = \\left( z\\right) ,\n\]\n\n\[ F\\left( w\\right)... | Yes |
Example 2.2.1. The formal additive group, denoted by \( {\widehat{\mathbb{G}}}_{a} \), is defined by | \[ F\left( {X, Y}\right) = X + Y \] | Yes |
Example 2.2.2. The formal multiplicative group, denoted by \( {\widehat{\mathbb{G}}}_{m} \), is defined by | \[ F\left( {X, Y}\right) = X + Y + {XY} = \left( {1 + X}\right) \left( {1 + Y}\right) - 1. \] | No |
Proposition 2.3. Let \( \mathcal{F} \) be a formal group over the ring \( R \) and let \( m \in \mathbb{Z} \) . (a) \( \left\lbrack m\right\rbrack \left( T\right) = {mT} + \) (higher-order terms). (b) If \( m \in {R}^{ * } \), then \( \left\lbrack m\right\rbrack : \mathcal{F} \rightarrow \mathcal{F} \) is an isomorphis... | Proof. (a) For \( m \geq 0 \), the stated result is a trivial induction using the recursive definition of \( \left\lbrack m\right\rbrack \) and the fact that \( F\left( {X, Y}\right) = X + Y + \cdots \) . Then, using \[ 0 = F\left( {T, i\left( T\right) }\right) = T + i\left( T\right) + \cdots , \] we see that \( i\left... | Yes |
Lemma 2.4. Let \( a \in {R}^{ * } \) and let \( f\left( T\right) \in R\llbracket T\rrbracket \) be a power series of the form\n\n\[ f\left( T\right) = {aT} + \left( \text{ higher-order terms }\right) .\n\]\n\nThen there is a unique power series \( g\left( T\right) \in R\llbracket T\rrbracket \) satisfying\n\n\[ f\left(... | Proof. We construct a sequence of polynomials \( {g}_{n}\left( T\right) \in R\left\lbrack T\right\rbrack \) satisfying\n\n\[ f\left( {{g}_{n}\left( T\right) }\right) \equiv T\left( {\;\operatorname{mod}\;{T}^{n + 1}}\right) \;\text{ and }\;{g}_{n + 1}\left( T\right) \equiv {g}_{n}\left( T\right) \left( {\;\operatorname... | Yes |
Example 3.1.2. The multiplicative group \( {\widehat{\mathbb{G}}}_{m}\left( \mathcal{M}\right) \) is the group of 1 -units, i.e., the set \( 1 + \mathcal{M} \) with group law multiplication. Notice that we again have an exact sequence, | \[ 0 \rightarrow {\widehat{\mathbb{G}}}_{m}\left( \mathcal{M}\right) \xrightarrow[]{\;z \mapsto 1 + z\;}{\mathbb{R}}^{ * } \rightarrow {k}^{ * } \rightarrow 1. \] | Yes |
Let \( \widehat{E} \) be the formal group associated to an elliptic curve \( E/K \) as described in (IV.2.2.3), where \( K \) is the field of fractions of the complete local ring \( R \) . Then, as noted in (IV \( §1 \) ), the power series \( x\left( z\right) \) and \( y\left( z\right) \) give a well-defined map \[ \ma... | More precisely, they imply that \( {P}_{F\left( {z,{z}^{\prime }}\right) } = {P}_{z} + {P}_{{z}^{\prime }} \) for distinct \( z,{z}^{\prime } \in \mathcal{M} \) . For \( z = {z}^{\prime } \), we can let \( {z}^{\prime } \mapsto z \) and use the fact that the map \( z \mapsto {P}_{z} \) and the addition law on \( E\left... | Yes |
Proposition 3.2. Let \( \mathcal{F}/R \) be a formal group defined over a complete local ring. (a) For each \( n \geq 1 \), the map\n\n\[ \frac{\mathcal{F}\left( {\mathcal{M}}^{n}\right) }{\mathcal{F}\left( {\mathcal{M}}^{n + 1}\right) } \rightarrow \frac{{\mathcal{M}}^{n}}{{\mathcal{M}}^{n + 1}} \]\n\ninduced by the i... | Proof. (a) Since the underlying sets are the same, it suffices to show that the map is a homomorphism. But this is clear, since for any \( x, y \in {\mathcal{M}}^{n} \) we have\n\n\[ x{ \oplus }_{\mathcal{F}}y = F\left( {x, y}\right) \]\n\n\[ = x + y + \text{(higher-order terms)} \]\n\n\[ \equiv x + y\;\left( {\;\opera... | Yes |
Example 4.1.2. On the multiplicative group \( {\widehat{\mathbb{G}}}_{m} \), the following is an invariant differential: | \[ \omega = \frac{dT}{1 + T} = \left( {1 - T + {T}^{2} - {T}^{3} + \cdots }\right) {dT}. \] | Yes |
Proposition 4.2. Let \( \mathcal{F}/R \) be a formal group. There exists a unique normalized invariant differential on \( \mathcal{F}/R \) . It is given by the formula\n\n\[ \n\omega = {F}_{X}{\left( 0, T\right) }^{-1}{dT}.\n\]\n\nEvery invariant differential on \( \mathcal{F}/R \) is of the form \( {a\omega } \) for s... | Proof. Suppose that \( P\left( T\right) {dT} \) is an invariant differential on \( \mathcal{F}/R \), so it satisfies\n\n\[ \nP\left( {F\left( {T, S}\right) }\right) {F}_{X}\left( {T, S}\right) = P\left( T\right) \]\n\nPutting \( T = 0 \) and remembering that \( F\left( {0, S}\right) = S \) gives\n\n\[ \nP\left( S\right... | Yes |
Corollary 4.3. Let \( \mathcal{F}/R \) and \( \mathcal{G}/R \) be formal groups with normalized differentials \( {\omega }_{\mathcal{F}} \) and \( {\omega }_{\mathcal{G}} \) . Let \( f : \mathcal{F} \rightarrow \mathcal{G} \) be a homomorphism. Then\n\n\[{\omega }_{\mathcal{G}} \circ f = {f}^{\prime }\left( 0\right) {\... | Proof. Let \( F\left( {X, Y}\right) \) and \( G\left( {X, Y}\right) \) be the formal group laws for \( \mathcal{F} \) and \( \mathcal{G} \) . We claim that \( {\omega }_{\mathcal{G}} \circ f \) is an invariant differential for \( \mathcal{F} \) . To prove this, we compute\n\n\[ \left( {{\omega }_{\mathcal{G}} \circ f}\... | Yes |
Corollary 4.4. Let \( \mathcal{F}/R \) be a formal group and let \( p \in \mathbb{Z} \) be a prime. There are power series \( f\left( T\right), g\left( T\right) \in R\llbracket T\rrbracket \) with \( f\left( 0\right) = g\left( 0\right) = 0 \) such that\n\n\[ \left\lbrack p\right\rbrack \left( T\right) = {pf}\left( T\ri... | Proof. Let \( \omega \left( T\right) \) be the normalized invariant differential on \( \mathcal{F} \) . From (IV.2.3a) we have \( {\left\lbrack p\right\rbrack }^{\prime }\left( 0\right) = p \), so (IV.4.3) implies that\n\n\[ {p\omega }\left( T\right) = \left( {\omega \circ \left\lbrack p\right\rbrack }\right) \left( T\... | Yes |
The formal group law and invariant differential of the formal multiplicative group \( \mathcal{F} = {\widehat{\mathbb{G}}}_{m} \) are | \[ {F}_{\mathcal{F}}\left( {X, Y}\right) = X + Y + {XY}\;\text{ and }\;{\omega }_{\mathcal{F}}\left( T\right) = {\left( 1 + T\right) }^{-1}{dT}. \] | Yes |
Let \( R \) be a torsion-free ring and let \( \mathcal{F}/R \) be a formal group. Then \[ {\log }_{\mathcal{F}} : \mathcal{F} \rightarrow {\widehat{\mathbb{G}}}_{a} \] is an isomorphism of formal groups over \( K = R \otimes \mathbb{Q} \) . | Proof. Let \( \omega \left( T\right) \) be the normalized invariant differential on \( \mathcal{F}/R \), so \[ \omega \left( {F\left( {T, S}\right) }\right) = \omega \left( T\right) \] Integrating with respect to \( T \) gives \[ {\log }_{\mathcal{F}}F\left( {T, S}\right) = {\log }_{\mathcal{F}}\left( T\right) + C\left... | No |
Lemma 5.4. Let \( R \) be a torsion-free ring, let \( K = R \otimes \mathbb{Q} \), and let\n\n\[ f\left( T\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{a}_{n}}{n!}{T}^{n} \in K\llbracket T\rrbracket \]\n\nbe a power series with \( {a}_{n} \in R \) and \( {a}_{1} \in {R}^{ * } \). Then there is a unique powe... | Proof. Differentiating \( f\left( {g\left( T\right) }\right) = T \) gives\n\n\[ {f}^{\prime }\left( {g\left( T\right) }\right) {g}^{\prime }\left( T\right) = 1 \]\n\nand evaluating at \( T = 0 \) shows that\n\n\[ {b}_{1} = {g}^{\prime }\left( 0\right) = \frac{1}{{f}^{\prime }\left( 0\right) } = \frac{1}{{a}_{1}} \in {R... | Yes |
Proposition 5.5. Let \( R \) be a torsion-free ring and let \( \mathcal{F}/R \) be a formal group. Then\n\n\[{\log }_{\mathcal{F}}\left( T\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{a}_{n}}{n}{T}^{n}\;\text{ and }\;{\exp }_{\mathcal{F}}\left( T\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{b}_{... | Proof. The expression for \( {\log }_{\mathcal{F}} \) follows directly from the definition of the formal logarithm, and then (IV.5.4) implies that \( {\exp }_{\mathcal{F}} \) has the specified form. | No |
Theorem 6.1. Let \( R \) be a discrete valuation ring that is complete with respect to its maximal ideal \( \mathcal{M} \), let \( p = \operatorname{char}\left( {R/\mathcal{M}}\right) \), and let \( v \) be the valuation on \( R \) . Let \( \mathcal{F}/R \) be a formal group, and suppose that \( x \in \mathcal{F}\left(... | Proof. The statement is trivial (and uninteresting) if \( \operatorname{char}\left( R\right) \neq 0 \) or if \( p = 0 \), since then \( v\left( p\right) = \infty \), so we may assume that \( \operatorname{char}\left( R\right) = 0 \) and that \( p > 0 \) . From (IV.4.4) we know that there are power series \( f\left( T\r... | Yes |
Lemma 6.2. Let \( v \) be a valuation and let \( p \in \mathbb{Z} \) be a prime such that \( 0 < v\left( p\right) < \infty \) . Then for all integers \( n \geq 1 \) , \[ v\left( {n!}\right) \leq \frac{\left( {n - 1}\right) v\left( p\right) }{p - 1}. \] | Proof. We compute \[ v\left( {n!}\right) = \mathop{\sum }\limits_{{i = 1}}^{\infty }\left\lbrack \frac{n}{{p}^{i}}\right\rbrack v\left( p\right) \leq \mathop{\sum }\limits_{{i = 1}}^{\left\lbrack {\log }_{p}n\right\rbrack }\frac{{nv}\left( p\right) }{{p}^{i}} = \frac{{nv}\left( p\right) }{p - 1}\left( {1 - {p}^{-\left\... | Yes |
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