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Lemma 6.11.10. (1) For all \( x > 0 \) we have \( \left( {x + 1}\right) \log \left( {x + 1}\right) > x\log \left( x\right) \) . | Proof. Since \( \log \left( x\right) \) is an increasing function of \( x \) and \( \log \left( {x + 1}\right) > 0 \) we have \( \left( {x + 1}\right) \log \left( {x + 1}\right) > x\log \left( {x + 1}\right) > x\log \left( x\right) \), so (1) is clear. | Yes |
Proposition 6.11.12. Let \( x \) and \( y \) be nonzero integers and \( p \) and \( q \) be odd primes such that \( {x}^{p} - {y}^{q} = 1 \) . Then if \( p < q \) we have \( p \mid y \) . | Proof. Assume on the contrary that \( p \nmid y \) . It follows from Corollary 6.11.4 that \( x - 1 \) and \( {r}_{p}\left( x\right) \) are coprime, and since their product is a \( q \) th power, they both are. We can thus write \( x - 1 = {a}^{q} \) for some integer \( a \), and \( a \neq 0 \) (otherwise \( y = 0 \) )... | Yes |
Corollary 6.11.13. With the same assumptions as above (and in particular \( p < q) \) we have \( \left| y\right| \geq {p}^{q - 1} + p \) . | Proof. Since by the above proposition we have \( p \mid y \), as in Corollary 6.11.6 we deduce that there exist integers \( a \) and \( v \) with \( a \neq 0 \) and \( v > 0 \) such that \( x - 1 = {p}^{q - 1}{a}^{p},\left( {{x}^{p} - 1}\right) /\left( {x - 1}\right) = p{v}^{q} \) and \( y = {pav} \) . Set \( P\left( X... | Yes |
Proposition 6.11.15. Let \( p, q \) be odd primes and \( x, y \) be nonzero integers such that \( {x}^{p} - {y}^{q} = 1 \) . Then \( \left| x\right| \geq {q}^{p - 1} + q \) and \( \left| y\right| \geq {p}^{q - 1} + p \) . | Proof. Since we can change \( \left( {p, q, x, y}\right) \) into \( \left( {q, p, - y, - x}\right) \), it is enough to prove the statement for \( y \) . If \( p < q \), this is Hyrrö’s result. Otherwise we have \( p \geq q \), hence \( p > q \) since \( p \neq q \) . By Cassels’s Corollary 6.11.6 we have \( y + 1 = \) ... | Yes |
A number \( n \) is a congruent number if and only if there exists a rational point on the curve \( {y}^{2} = x\left( {{x}^{2} - {n}^{2}}\right) \) with \( y \neq 0 \) . More precisely, if \( \left( {a, b, c}\right) \) is a Pythagorean triangle of area \( n \), then the four points \( \left( {a\left( {a \pm c}\right) /... | The proof consists in simple verifications: if for example \( x = a(a + \) \( c)/2, y = {a}^{2}\left( {a + c}\right) /2 \), and \( n = {ab}/2 \) is the area of the triangle, then\n\n\[ x\left( {{x}^{2} - {n}^{2}}\right) = a\frac{a + c}{2}\frac{{a}^{2}{\left( a + c\right) }^{2} - {a}^{2}{b}^{2}}{4} = \frac{{a}^{3}\left(... | Yes |
Proposition 6.12.2. The number \( n = 1 \) is not congruent. | Proof. Assume by contradiction that 1 is a congruent number, so that there exists \( \left( {x, y}\right) \in {\mathbb{Q}}^{2} \) with \( y \neq 0 \) such that \( {y}^{2} = x\left( {{x}^{2} - 1}\right) \) . Writing \( x = p/q \) and \( y = u/v \) with \( \gcd \left( {p, q}\right) = \gcd \left( {u, v}\right) = 1 \), we ... | Yes |
Proposition 7.1.1. A general Weierstrass equation over any field \( K \) defines an absolutely irreducible curve (i.e., irreducible over an algebraic closure of \( K) \) . | Proof. Without loss of generality we may assume that \( K \) is algebraically closed. Let \( {Ax} + {By} + {Cz} = 0 \) be the general equation of a line in the projective plane over \( K \) with \( \left( {A, B, C}\right) \neq \left( {0,0,0}\right) \), and set \( P\left( {x, z}\right) = {x}^{3} + \) \( {a}_{2}{x}^{2}z ... | Yes |
Proposition 7.1.2. If a general Weierstrass equation has singularities it has exactly one, and it is on the line \( {2y} + {a}_{1}x + {a}_{3}z = 0 \) . In particular, if the characteristic of \( K \) is different from 2 and \( {a}_{1} = {a}_{3} = 0 \), it is on the \( x \) -axis. | Proof. A point \( P = \left( {{x}_{0},{y}_{0},{z}_{0}}\right) \in E \) is singular if and only if the three partial derivatives of the equation vanish at \( \left( {{x}_{0},{y}_{0},{z}_{0}}\right) \) (note that because of Euler’s relation \( \sum {x}_{i}\frac{\partial f}{\partial {x}_{i}} = {kf} \) for a homogeneous fu... | Yes |
Lemma 7.1.3. Let \( p \geq 3 \), let \( K = {\mathbb{F}}_{p} \), let \( E \) be a degenerate curve over \( {\mathbb{F}}_{p} \) as above, and let \( {c}_{6} = {c}_{6}\left( E\right) \) be the invariant defined in Section 7.1.2. Then \( E \) has a cusp (respectively a double point with tangents defined over \( {\mathbb{F... | Proof. Since \( E \) is degenerate it has a singular point, and changing coordinates we may assume that it is at the origin, so that we can choose the equation of our curve to be \( {y}^{2} = {x}^{2}\left( {x + a}\right) \) for some \( a \in {\mathbb{F}}_{p} \) . One computes that \( {c}_{6}\left( E\right) = - {64}{a}^... | Yes |
Theorem 7.1.4. Let \( {y}^{2} + {a}_{1}{xy} + {a}_{3}y = {x}^{3} + {a}_{2}{x}^{2} + {a}_{4}x + {a}_{6} \) be the generalized Weierstrass equation of an elliptic curve \( E \) (hence nonsingular). We define an addition law on \( E \) by asking that the point at infinity \( \mathcal{O} \) be the identity element, and tha... | Proof. The proof of (1) is very classical and not too difficult, although associativity is painful if one tries to prove it directly from the formulas given in (2), but it is immediate in terms of divisors; see Exercise 6. The formulas of (2) follow from an immediate computation; see Exercise 5. | No |
Theorem 7.1.6. Let \( E \) and \( {E}^{\prime } \) be two elliptic curves with identity elements \( \mathcal{O} \) and \( {\mathcal{O}}^{\prime } \) respectively, and let \( \phi \) be a morphism of algebraic curves from \( E \) to \( {E}^{\prime } \) (i.e., \( \phi \) is defined by rational functions). Then \( \phi \)... | The condition is of course necessary, but what is remarkable is that the simple fact that \( \phi \) is a morphism of algebraic curves (together with the very weak condition \( \phi \left( \mathcal{O}\right) = {\mathcal{O}}^{\prime } \) ) implies that \( \phi \) preserves the group law. Note that when \( \phi \left( \m... | Yes |
Theorem 7.1.8. Let \( \phi \) be a nonconstant isogeny from \( E \) to \( {E}^{\prime } \) defined over an algebraically closed field \( K \) . Then\n\n(1) The map \( \phi \) is surjective.\n\n(2) \( \phi \) is a finite map; in other words, the fiber over any point of \( {E}^{\prime } \) is constant and finite. | From these properties it is easy to see that \( \phi \) induces an injective map from the function field of \( {E}^{\prime } \) to that of \( E \) over some algebraic closure of the base field. The degree of the corresponding field extension is finite and called the degree of \( \phi \) . If this field extension is sep... | No |
Proposition 7.1.9. If \( G \) is a finite subgroup of \( E \) there exists a natural elliptic curve \( {E}^{\prime } \) and an isogeny \( \phi \) from \( E \) to \( {E}^{\prime } \) whose kernel (over some algebraic closure) is equal to \( G \) . The elliptic curve \( {E}^{\prime } \) is well defined up to isomorphism ... | Note that the equation of \( {E}^{\prime } \) can be given explicitly by formulas due to Vélu [Vel]. | No |
Proposition 7.2.1. Let \( {y}^{2} = {a}^{2}{x}^{4} + b{x}^{3} + c{x}^{2} + {dx} + e \) be the equation of a hyperelliptic quartic with rational point \( \left( {0,1,0}\right) \) and rational tangents at infinity. We set\n\n\[ \left( {{a}_{1},{a}_{2},{a}_{3},{a}_{4},{a}_{6}}\right) = \left( {\frac{b}{a}, - \frac{{b}^{2}... | Proof. This is a simple verification, which is left to the reader (Exercise 8). | No |
Corollary 7.2.2. Let \( {y}^{2} = {x}^{4} + c{x}^{2} + e \) be the equation of an even hyperelliptic quartic with leading coefficient 1. Set\n\n\[ \nX = 2{x}^{2} - {2y} + c,\;Y = {2x}\left( {2{x}^{2} - y + c}\right) ,\;x = \frac{Y}{2X},\;y = \frac{{Y}^{2}}{4{X}^{2}} - \frac{X - c}{2}.\n\]\n\nThen the maps \( \left( {x,... | Proof. This is a special case of the above proposition, after changing \( X \) into \( X + c \) . | No |
Proposition 7.2.3. The curves \( {x}^{3} + {y}^{3} = c \) and \( {Y}^{2} = {X}^{3} - {432}{c}^{2} \) are birationally equivalent over \( \mathbb{Q} \) via the transformations | \[ X = \frac{12c}{x + y}, Y = \frac{{36c}\left( {y - x}\right) }{x + y}, x = - \frac{Y - {36c}}{6X}, y = \frac{Y + {36c}}{6X}. \] | Yes |
Proposition 7.2.4. Let \( a, b, c \) be nonzero rational numbers, and assume that there exists a rational point \( \left( {{x}_{0},{y}_{0},{z}_{0}}\right) \neq \left( {0,0,0}\right) \) on the cubic \( a{x}^{3} + \) \( b{y}^{3} + c{z}^{3} = 0 \) . Then this cubic is birationally equivalent to the elliptic curve whose af... | Proof. See Exercise 11. | No |
Proposition 7.3.1. Let \( R \) be a principal ideal domain with field of fractions \( K \), let \( E \) be an elliptic curve given by a generalized Weierstrass equation as \( {Y}^{2} + {a}_{1}{XY} + {a}_{3}Y = {X}^{3} + {a}_{2}{X}^{2} + {a}_{4}X + {a}_{6} \) with \( {a}_{i} \in R \), and let \( \left( {X, Y}\right) \in... | Proof. Write \( X = M/A \) and \( Y = N/B \) with \( \gcd \left( {M, A}\right) = \gcd \left( {N, B}\right) = 1 \) . Substituting in the equation and clearing denominators gives \( {A}^{3}{N}^{2} + {a}_{1}{A}^{2}{BMN} + {a}_{3}{A}^{3}{BN} = {B}^{2}{M}^{3} + {a}_{2}{B}^{2}A{M}^{2} + {a}_{4}{B}^{2}{A}^{2}M + {a}_{6}{B}^{2... | Yes |
Theorem 7.3.3. Let \( f\left( X\right) = {X}^{3} + a{X}^{2} + {bX} + c \in k\left\lbrack X\right\rbrack \) be such that \( \gcd \left( {f,{f}^{\prime }}\right) = 1 \), and let \( {E}_{g} \) be the elliptic curve defined over \( K = k\left( T\right) \) by an equation \( g\left( T\right) {Y}^{2} = f\left( X\right) \), wh... | Proof. By Exercise 21, we must show that \( h\left( {P + Q}\right) + h\left( {P - Q}\right) = 2(h\left( P\right) + \) \( h\left( Q\right) ) \) for all \( P \) and \( Q \) in \( {E}_{g}\left( K\right) \) . If \( P \) or \( Q \) is the point at infinity this is clear, so we assume that this is not the case. We thus write... | No |
Lemma 7.3.4. There exists \( u \in {k}^{ * } \) such that\n\n\[ \n{N}_{ + }{N}_{ - } = u{A}_{N, N},\;{D}_{ + }{D}_{ - } = u{A}_{D, D},\;\text{ and }\;\frac{{N}_{ + }{D}_{ - } + {N}_{ - }{D}_{ + }}{2} = u{A}_{N, D}.\n\] | Proof. Define \( {S}_{1} = \gcd \left( {{N}_{ + }{N}_{ - },{D}_{ + }{D}_{ - }}\right) \) and \( {S}_{2} = \gcd \left( {{A}_{N, N},{A}_{D, D}}\right) \) . We thus have\n\n\[ \n{S}_{2}{N}_{ + }{N}_{ - } = {S}_{1}{A}_{N, N},{S}_{2}{D}_{ + }{D}_{ - } = {S}_{1}{A}_{D, D},{S}_{2}\left( {{N}_{ + }{D}_{ - } + {N}_{ - }{D}_{ + ... | Yes |
Lemma 7.3.5. Recall that we have set \( f\left( x\right) = {x}^{3} + a{x}^{2} + {bx} + c \) .\n\n(1) If \( {z}_{P} \) and \( {z}_{Q} \) are not both equal to 0 we have:\n\n(a) If \( {z}_{P} \geq 0 \) and \( {z}_{Q} \geq 0 \) then \( {z}_{ + } \geq 0 \) and \( {z}_{ - } \geq 0 \) .\n\n(b) If \( {z}_{P} \geq 0 \) and \( ... | Proof. From the formulas that we have proved for \( {X}_{ \pm } \) and \( {X}_{ + }{X}_{ - } \) it\n\nfollows that\n\[ \n{X}_{ \pm } = \frac{{A}_{ \mp }}{C} = \frac{B}{{A}_{ \pm }}\n\]\n\nwhere\n\n\[ \n{A}_{ \pm } = \left( {{X}_{P}{X}_{Q} + b}\right) \left( {{X}_{P} + {X}_{Q}}\right) + {2a}{X}_{P}{X}_{Q} + {2c} \pm {2\... | Yes |
Lemma 7.3.6. There exists \( u \in {k}^{ * } \) such that \( {N}_{ + } = u{A}_{N} \) and \( {D}_{ + } = u{A}_{D} \) . | Proof. This immediately follows from the fact that \( {A}_{N} \) and \( {A}_{D} \) are co-prime and is left to the reader (Exercise 24). | No |
Lemma 7.3.7. Keep all the above notation.\n\n(1) If \( {z}_{P} > 0 \) then \( {z}_{ + } > 0 \).\n\n(2) If \( {z}_{P} < 0 \) and \( c \neq 0 \) then \( {z}_{ + } \leq 0 \).\n\n(3) If \( {z}_{P} < 0 \) and \( c = 0 \) then \( {z}_{ + } > 0 \).\n\n(4) If \( {z}_{P} = 0 \) and \( f\left( {\ell }_{P}\right) \neq 0 \) then \... | Proof. Identical to that of Lemma 7.3.5 and left to the reader (Exercise 24). | No |
Lemma 7.3.8. Assume that \( g\left( T\right) \) is not equal to a constant times the square of a rational function, and let \( P \in {E}_{g}\left( K\right) \) . The following are equivalent:\n\n(1) \( h\left( P\right) = 0 \) .\n\n(2) \( P \in {E}_{g}\left( k\right) \) .\n\n(3) Either \( P = \mathcal{O} \) or there exis... | Proof. Since the result is trivial when \( P = \mathcal{O} \), assume that this is not the case and write as usual \( P = \left( {{X}_{P},{Y}_{P}}\right) \) . Clearly \( h\left( P\right) = 0 \) if and only if \( {X}_{P} \in k \) , so we can write \( {X}_{P} = u \) . We thus have \( g\left( T\right) {Y}_{P}^{2} = f\left... | Yes |
Lemma 7.3.9. Let \( q \) be an odd prime power, let \( \rho \) be the unique multiplicative character of order 2 on \( {\mathbb{F}}_{q} \), and let \( {y}^{2} = f\left( x\right) \) be the equation of an elliptic curve over \( {\mathbb{F}}_{q} \), where \( f\left( X\right) \) is a polynomial of degree 3 . Then \( \left|... | Proof. For a given \( X \in {\mathbb{F}}_{q} \) it is clear that the number of \( y \in {\mathbb{F}}_{q} \) such that \( {y}^{2} = X \) is equal to \( 1 + \rho \left( X\right) \) . Counting the point at infinity separately it follows that\n\n\[ \n\left| {E\left( {\mathbb{F}}_{q}\right) }\right| = 1 + \mathop{\sum }\lim... | Yes |
Theorem 7.3.10 (Hasse). For all elliptic curves \( E \) over a finite field \( {\mathbb{F}}_{q} \) we have the inequality \[ q + 1 - 2{q}^{1/2} \leq \left| {E\left( {\mathbb{F}}_{q}\right) }\right| \leq q + 1 + 2{q}^{1/2}. \] | Proof. We first assume that the characteristic of \( {\mathbb{F}}_{q} \) is not equal to 2, and we will see below how to modify the proof in characteristic 2 . In this case we know that \( E \) has a plane model whose affine equation is \( {y}^{2} = f\left( x\right) \) with \( f\left( x\right) = {x}^{3} + a{x}^{2} + {b... | Yes |
Proposition 7.3.11. We have \( {h}_{n} = {n}^{2} + {a}_{q}n + q \), where \( {a}_{q} = q + 1 - \left| {E\left( {\mathbb{F}}_{q}\right) }\right| \) . | Proof. By Theorem 7.3.3 we have\n\n\[ \n{h}_{n} = {n}^{2}h\left( P\right) + {2nB}\left( {P, Q}\right) + h\left( Q\right) = {n}^{2} + {a}_{q}n + q \n\] \n\nfor some constant \( {a}_{q} = {2B}\left( {P, Q}\right) \) to be determined, where \( B\left( {P, Q}\right) \) is the bilinear form associated with \( h \). To compu... | Yes |
Proposition 7.3.16. Let \( E \) be an elliptic curve defined over \( {\mathbb{F}}_{p} \) by a Weierstrass equation of the form \( {y}^{2} = f\left( x\right) \) for some cubic polynomial \( f \), where \( p \) is an odd prime, let \( D \in {\mathbb{F}}_{p}^{ * } \), let \( {E}_{D} \) be the quadratic twist of \( E \) by... | Proof. By Lemma 7.3.9 we have\n\n\[ \n{a}_{p}\left( E\right) = - \mathop{\sum }\limits_{{x \in {\mathbb{F}}_{p}}}\left( \frac{f\left( x\right) }{p}\right) .\n\]\n\nOn the other hand, the equation of \( {E}_{D} \) is \( D{y}^{2} = f\left( x\right) \), which is isomorphic to \( {Y}^{2} = {Df}\left( x\right) \) since \( D... | Yes |
There exists a unique formal power series \( s\left( T\right) = \mathop{\sum }\limits_{{k \geq 0}}{u}_{k}{T}^{k} \in R\left\lbrack \left\lbrack T\right\rbrack \right\rbrack \) with \( {u}_{0} \neq 0 \) such that if we set \( \left( {x\left( T\right), y\left( T\right) }\right) = \left( {s\left( T\right) /{T}^{2}, - s\le... | We use a Hensel-type argument with the \( T \) -adic valuation instead of a \( \mathfrak{p} \) -adic one. Set \( f\left( {x, y}\right) = {y}^{2} + {a}_{1}{xy} + {a}_{3}y - \left( {{x}^{3} + {a}_{2}{x}^{2} + {a}_{4}x + {a}_{6}}\right) \). Identifying coefficients of \( {T}^{-6} \) in \( f\left( {x\left( T\right), y\left... | Yes |
Proposition 7.3.18. We have \( {x}^{\prime }\left( T\right) /\left( {{2y}\left( T\right) + {a}_{1}x\left( T\right) + {a}_{3}}\right) \in R\left\lbrack \left\lbrack T\right\rbrack \right\rbrack \) . | Proof. Since \( {x}^{\prime }\left( T\right) = - 2{T}^{-3} + O\left( {T}^{-2}\right) \) we can write\n\n\[ g\left( T\right) = {x}^{\prime }\left( T\right) /\left( {{2y}\left( T\right) + {a}_{1}x\left( T\right) + {a}_{3}}\right) = \mathop{\sum }\limits_{{k \geq 0}}{g}_{k}{T}^{k}. \]\n\nBecause of the factor 2 in the den... | Yes |
Proposition 7.3.19. (1) There exists a unique formal power series in two variables \( F\left( {{T}_{1},{T}_{2}}\right) \in R\left\lbrack \left\lbrack {{T}_{1},{T}_{2}}\right\rbrack \right\rbrack \) such that \( F\left( {{T}_{1},{T}_{2}}\right) = {T}_{1} + {T}_{2} + O\left( {T}^{2}\right) \) and satisfying\n\n\[ \left( ... | Proof. For \( i = 1 \) and 2 let \( {P}_{i} = \left( {x\left( {T}_{i}\right), y\left( {T}_{i}\right) }\right) \), and let \( {mx} + {ny} = 1 \) be the equation of the line through \( {P}_{1} \) and \( {P}_{2} \) (since \( {T}_{1} \) and \( {T}_{2} \) are formal variables and \( x\left( T\right) = {T}^{-2} + O\left( {T}... | Yes |
Proposition 7.3.20. Let \( F\left( {{T}_{1},{T}_{2}}\right) \in R\left\lbrack \left\lbrack {{T}_{1},{T}_{2}}\right\rbrack \right\rbrack \) be a formal group.\n\n(1) There exists a unique formal power series \( L \in K\left\lbrack \left\lbrack T\right\rbrack \right\rbrack \) such that \( L\left( T\right) = \) \( T + O\l... | Proof. Denote by \( {F}_{1}^{\prime } \) and \( {F}_{2}^{\prime } \) the (formal) partial derivative of \( F \) with respect to the first and second variables respectively. Assuming the existence of \( L \), we differentiate with respect to \( {T}_{1} \) its defining equation. We obtain \( {F}_{1}^{\prime }\left( {{T}_... | No |
Proposition 7.3.22. (1) For any \( m \in \mathbb{Z} \) we have\n\n\[{\left( \left\lbrack m\right\rbrack T\right) }^{\prime } = m{L}^{\prime }\left( T\right) /{L}^{\prime }\left( {\left\lbrack m\right\rbrack T}\right) .\] | Proof. If we differentiate the formula \( E\left( {L\left( T\right) }\right) = T \) with respect to \( T \) we obtain \( {E}^{\prime }\left( T\right) = 1/{L}^{\prime }\left( {E\left( T\right) }\right) \), so replacing \( T \) by \( {mL}\left( T\right) = L\left( {\left\lbrack m\right\rbrack T}\right) \) gives \( {E}^{\p... | Yes |
Proposition 7.3.23. Let \( E \) be an elliptic curve defined over \( {K}_{\mathfrak{p}} \), and let \( r \in \bar{E}\left( {\mathbb{F}}_{\mathfrak{p}}\right) \) be a nonsingular point. Then there exists \( R \in E\left( {K}_{\mathfrak{p}}\right) \) such that \( r = \bar{R} \) . | Proof. Let \( E \) be given by a plane model \( f\left( {x, y, z}\right) = 0 \) chosen as above, and write \( r = \bar{P} \), where \( P = \left( {{x}_{0},{y}_{0},{z}_{0}}\right) \) is any lift of \( r \) to \( {\mathbb{Z}}_{\mathfrak{p}} \) . Since \( r \) is a nonsingular point of \( \bar{E} \) the partial derivative... | Yes |
Proposition 7.3.24. The group \( {G}_{0} \) is a subgroup of \( G \) of finite index, and the natural reduction map modulo \( \mathfrak{p} \) induces a group homomorphism from \( {G}_{0} \) to \( {\bar{G}}_{0} \subset \bar{E}\left( {\mathbb{F}}_{\mathfrak{p}}\right) \) . In particular, if \( \bar{E} \) is nonsingular, ... | Proof. Left to the reader (Exercise 7). | No |
Lemma 7.3.26. A point \( P = \left( {x, y, z}\right) \neq \mathcal{O} \) in canonical coordinates is in the kernel of the reduction if and only if there exists an integer \( N \geq 1 \) such that \( v\left( x\right) = N, v\left( y\right) = 0 \), and \( v\left( z\right) = {3N} \), or equivalently, \( \left| x\right| = {... | Proof. Since \( \bar{O} = \left( {0,1,0}\right) \), in canonical coordinates \( P \) is in the kernel of reduction modulo \( \mathfrak{p} \) if and only if \( \left| x\right| < 1,\left| z\right| < 1 \), and \( \left| y\right| = 1 \) . It follows from the non-Archimedean inequality and \( \left| {a}_{i}\right| \leq 1 \)... | Yes |
Lemma 7.3.28. (1) The sets \( {G}_{N} \) are groups such that \( G \supset {G}_{0} \supset {G}_{1} \supset \) \( \cdots {G}_{N} \supset \cdots \) and we have \( \mathop{\bigcap }\limits_{N}{G}_{N} = \{ \mathcal{O}\} \) . (2) The quotient \( {G}_{0}/{G}_{1} \) is isomorphic to the group \( \overline{{G}_{0}} \) of nonsi... | Proof. For \( N \geq 1 \) make the change of variables \( x = {\pi }^{N}X, y = Y \), and \( z = {\pi }^{3N}Z \) in the Weierstrass equation. We obtain a new curve \( {E}_{N} \) with equation \[ {Y}^{2}Z + {a}_{1}{\pi }^{N}{XYZ} + {a}_{3}{\pi }^{3N}Y{Z}^{2} = {X}^{3} + {a}_{2}{\pi }^{2N}{X}^{2}Z + {a}_{4}{\pi }^{4N}X{Z}... | Yes |
Corollary 7.3.29. If \( P \in {G}_{1} \) is a torsion point, its order is a power of \( p \) . | Proof. Assume that \( {kP} = \mathcal{O} \) with \( k \neq 0 \), write \( k = {p}^{u}m \) with \( p \nmid m \), and consider \( Q = {p}^{u}P \) . If \( Q = \mathcal{O} \) then the order of \( P \) divides \( {p}^{u} \) so is a power of \( p \) . Otherwise, by Lemma 7.3.28 there exists a unique \( N \geq 1 \) such that ... | Yes |
Proposition 7.3.30. For all \( t \in \mathfrak{p}{\mathbb{Z}}_{\mathfrak{p}} \) set \( P\left( t\right) = \left( {{ts}\left( t\right) , - s\left( t\right) ,{t}^{3}}\right) \in {\mathbb{P}}^{2}\left( {K}_{\mathfrak{p}}\right) \) , where \( s\left( T\right) \in {\mathbb{Z}}_{\mathfrak{p}}\left\lbrack \left\lbrack T\right... | Proof. Note first that since \( \left| t\right| < 1 \) and \( s\left( T\right) \in {\mathbb{Z}}_{\mathfrak{p}}\left\lbrack \left\lbrack T\right\rbrack \right\rbrack \) the series \( s\left( t\right) \) converges, and more precisely \( s\left( t\right) - 1 \in \mathfrak{p}{\mathbb{Z}}_{\mathfrak{p}} \), which also shows... | Yes |
Theorem 7.3.31. Let \( N \geq 1 \) . If \( P \in {G}_{N} \) is a torsion point of order \( {p}^{k} \) with \( k \geq 1 \) then\n\n\[ \n{p}^{k} \leq \frac{{pe}\left( {\mathfrak{p}/p}\right) }{\left( {p - 1}\right) N},\;\text{ or equivalently,}\;N \leq \frac{e\left( {\mathfrak{p}/p}\right) }{{p}^{k} - {p}^{k - 1}}.\n\] | Proof. We may clearly assume that \( P \) has exact order \( k \) . We prove the result by induction on \( k \geq 1 \) . By the above proposition we let \( P = P\left( t\right) \) with \( t \neq 0 \), and assume first that \( {pP}\left( t\right) = \mathcal{O} \), or equivalently by the proposition, that \( \left\lbrack... | Yes |
Lemma 7.3.32. Let \( G \) be an abelian group and \( {G}_{t} \) its torsion subgroup. If \( H \) is a torsion-free subgroup of \( G \) the natural map from \( {G}_{t} \) to \( G/H \) is injective. | Proof. Indeed, if \( g \in {G}_{t} \) maps to the class of 0 in \( G/H \), in other words to \( H \), we have \( g \in {G}_{t} \cap H = \{ 0\} \) since \( H \) is torsion-free, so \( g = 0 \) . | Yes |
Corollary 7.3.33. Assume that \( K = \mathbb{Q} \), so that \( \mathfrak{p} = p\mathbb{Z} \) .\n\n(1) If \( p > 2 \) or \( {a}_{1} \) is even the group \( {G}_{1} \) is torsion-free.\n\n(2) If \( p = 2 \) and \( {a}_{1} \) is odd the group \( {G}_{2} \) is torsion-free, and the group \( {G}_{1} \) is either torsion-fre... | Proof. (1) and (2). Since \( e\left( {\mathfrak{p}/p}\right) = 1 \) for all \( p \) we have \( e\left( {\mathfrak{p}/p}\right) /\left( {{p}^{k} - {p}^{k - 1}}\right) < \) 1 for all \( {p}^{k} \geq 3 \), and \( e\left( {\mathfrak{p}/p}\right) /\left( {{p}^{k} - {p}^{k - 1}}\right) = 1 \) for \( {p}^{k} = 2 \) . Thus \( ... | Yes |
Proposition 7.3.35. Let again \( K = \mathbb{Q} \) and \( \mathfrak{p} = p\mathbb{Z} \), and assume that \( p \nmid \) \( \operatorname{disc}\left( E\right) \) . The torsion subgroup \( {G}_{t} \) of \( E\left( {\mathbb{Q}}_{p}\right) \) is either isomorphic to a subgroup of \( \bar{E}\left( {\mathbb{F}}_{p}\right) \) ... | Proof. First note that by definition of the discriminant \( p \nmid \operatorname{disc}\left( E\right) \) means that the reduced curve \( \bar{E} \) is nonsingular (this is true even for \( p = 2 \) ), in other words that \( G = E\left( {\mathbb{Q}}_{p}\right) = {G}_{0} \) . By Proposition 7.3.28, since \( \bar{E} \) i... | Yes |
Theorem 8.1.1 (Mordell). Let \( E \) be an elliptic curve defined over \( \mathbb{Q} \) . Then \( E\left( \mathbb{Q}\right) \) is a finitely generated abelian group. In other words, the torsion subgroup \( {E}_{t}\left( \mathbb{Q}\right) \) of all \( T \in E\left( \mathbb{Q}\right) \) such that there exists a nonzero i... | The proof of this theorem is not too difficult, and we will give it below (see Theorem 8.2.7 and Corollary 8.3.8). | No |
Theorem 8.1.3 (Faltings). Let \( E \) and \( {E}^{\prime } \) be two elliptic curves defined over \( \mathbb{Q} \) . If \( E \) and \( {E}^{\prime } \) are isogenous over \( \mathbb{Q} \) then \( L\left( {E, s}\right) = L\left( {{E}^{\prime }, s}\right) \), and conversely, if \( L\left( {E, s}\right) = L\left( {{E}^{\p... | The first part of this theorem is not difficult, but the converse (known previously as the isogeny conjecture) is a deep theorem of Faltings, proved in the same paper as the proof of Mordell's conjecture. Note that this theorem is important mainly for theoretical reasons. | No |
Corollary 8.1.9. Let \( E \) be an elliptic curve defined over \( \mathbb{Q} \), let \( r \) be its (algebraic) rank, denote by \( {r}_{an} \) its analytic rank, and let \( \varepsilon \left( E\right) \) the sign of the functional equation. Then\n\n(1) If \( r \geq 2 \), then \( {r}_{an} \geq 2 \), in other words \( L\... | Proof. Immediate and left to the reader (Exercise 1). | No |
Theorem 8.1.10 (Nagell, Lutz). Let \( E \) be given by a Weierstrass equation \( {y}^{2} = {x}^{3} + a{x}^{2} + {bx} + c = f\left( x\right) \) with \( a, b \), and \( c \) in \( \mathbb{Z} \) . If \( T = \left( {x, y}\right) \in \) \( {E}_{t}\left( \mathbb{Q}\right) \smallsetminus \{ \mathcal{O}\} \), then either \( T ... | Proof. The statement concerning points of order 2 is clear, so assume that \( T \) is not of order 2 . Since the natural map from \( E\left( \mathbb{Q}\right) \) to \( E\left( {\mathbb{Q}}_{p}\right) \) is injective, it follows from Corollary 7.3.33 (3) that \( \left( {x, y}\right) \in {\mathbb{Z}}_{p}^{2} \) for all \... | Yes |
Corollary 8.1.11. If \( P = \left( {x, y}\right) \) is a rational point on an elliptic curve given as above (i.e., with integral coefficients), then \( P \) is a nontorsion point if and only if there exists \( k \) such that \( k \cdot P \) has nonintegral coordinates. | Proof. If \( k \cdot P \) has nonintegral coordinates, then it cannot be a torsion point by the above theorem, so \( P \) is also nontorsion. Conversely, if \( k \cdot P \) has integral coordinates for all \( k \), these points cannot be distinct; otherwise, we would have an infinity of integral points, which is imposs... | Yes |
Proposition 8.1.12. Assume that \( E \) is given by an equation of the form \( {y}^{2} = {x}^{3} + a{x}^{2} + {bx} \) with a and \( b \) integral, in other words that up to translation of the \( x \) -coordinate the curve has a rational 2-torsion point. Then if \( \left( {x, y}\right) \in \) \( {E}_{t}\left( \mathbb{Q}... | Proof. Assume that \( T = \left( {x, y}\right) \in {E}_{t}\left( \mathbb{Q}\right) \) with \( y \neq 0 \) . Then \( {2T} \in {E}_{t}\left( \mathbb{Q}\right) \) and \( {2T} \neq \mathcal{O} \) . The \( x \) -coordinate of \( {2T} \) is equal to \( {\left( b - {x}^{2}\right) }^{2}/\left( {{4x}\left( {{x}^{2} + {ax} + b}\... | Yes |
Proposition 8.1.13. Let \( E \) be given by \( {y}^{2} = {x}^{3} + a{x}^{2} + {bx} + c = f\left( x\right) \), and let \( \ell \) be a prime number such that \( \ell \nmid \operatorname{disc}\left( E\right) = - {16D} \), where \( D = \operatorname{disc}\left( f\right) \) is as in Theorem 8.1.10. Then \( {E}_{t}\left( \m... | Proof. Since \( \ell \neq 2 \) and \( \ell \nmid \operatorname{disc}\left( f\right) \) the reduction \( \bar{E} \) of the curve \( E \) modulo \( \ell \) is again an elliptic curve. By the Nagell-Lutz theorem all the points of \( {E}_{t}\left( \mathbb{Q}\right) \) different from the point at infinity \( \mathcal{O} \) ... | Yes |
Let \( d \) be a nonzero integer.\n\n(1) Let \( E \) be given by \( {y}^{2} = {x}^{3} - {dx} \). Then \( {E}_{t}\left( \mathbb{Q}\right) \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \) if and only if \( d \) has the form \( d = {m}^{2},{E}_{t}\left( \mathbb{Q}\right) \simeq \mathbb{Z}/4\mathbb{Z} \) if a... | The proof of both statements relies on the essential fact that the two types of curves under consideration have complex multiplication by \( \mathbb{Z}\left\lbrack i\right\rbrack \) and \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) respectively, where \( {i}^{2} + 1 = 0 \) and \( {\rho }^{2} + \rho + 1 = 0 \), but th... | Yes |
Corollary 8.1.15. Let \( a, b, c \) be nonzero rational numbers, and assume that there exists a (projective) rational point \( \left( {{x}_{0} : {y}_{0} : {z}_{0}}\right) \) on the cubic \( a{x}^{3} + b{y}^{3} + \) \( c{z}^{3} = 0 \) . (1) If there are only a finite number of such points then either \( b/a, c/a \), or ... | Proof. By Proposition 7.2.4 we know that the cubic is birationally equivalent to the elliptic curve \( E \) whose affine Weierstrass equation is \( {Y}^{2} = \) \( {X}^{3} - {432}{\left( abc\right) }^{2} \) . It follows that if there are \( k \) projective points on our cubic, the curve \( E \) has rank 0 and \( \left|... | Yes |
Theorem 8.1.16 (Mazur). The group \( {E}_{t}\left( \mathbb{Q}\right) \) is isomorphic either to \( \mathbb{Z}/N\mathbb{Z} \) with \( 1 \leq N \leq {10} \) or \( N = {12} \), or to \( \left( {\mathbb{Z}/N\mathbb{Z}}\right) \times \left( {\mathbb{Z}/2\mathbb{Z}}\right) \) with \( N = 2,4,6 \), or 8 . | For instance, if we find a point of order 7 in \( {E}_{t}\left( \mathbb{Q}\right) \), it is not necessary to go any further \( \left( {{E}_{t}\left( \mathbb{Q}\right) \text{has order 7}}\right) \) . If we find a point of order 5, then since it is trivial to check whether there exists a point of order 2 (the points with... | No |
Theorem 8.1.17. The above limit exists, and defines a nonnegative function \( \widehat{h}\left( P\right) \) on \( E\left( \mathbb{Q}\right) \) with the following properties:\n\n(1) (Quadratic form.) The function \( \widehat{h}\left( P\right) \) is a quadratic form on \( E\left( \mathbb{Q}\right) \) ; in other words, if... | We refer to [Sil-Tat] for proofs of the above properties, which are not difficult. | No |
Theorem 8.1.18. Let \( E \) be an elliptic curve defined over \( \mathbb{Q} \) by a generalized Weierstrass equation. With the usual notation, set\n\n\[ \mu \left( E\right) = \frac{\log \left( \left| {\operatorname{disc}\left( E\right) }\right| \right) + {\log }^{ + }\left( {j\left( E\right) }\right) }{6} + {\log }^{ +... | As a direct application, we see that in the computation of the torsion subgroup, for instance using Theorem 8.1.10, then if \( P = \left( {x, y}\right) \in {E}_{t}\left( \mathbb{Q}\right) \) we have \( \widehat{h}\left( P\right) = 0 \), hence \( h\left( P\right) = h\left( x\right) \leq h\left( {j\left( E\right) }\right... | Yes |
Proposition 8.2.2. Denote by \( I = \phi \left( {E\left( \mathbb{Q}\right) }\right) \) the image of the rational points of \( E \) in \( \widehat{E}\left( \mathbb{Q}\right) \) . Then\n\n(1) \( \widehat{\mathcal{O}} \in I \), and \( \widehat{T} \in I \) if and only if \( \operatorname{disc}\left( E\right) \) is a square... | Proof. Since \( \phi \left( \mathcal{O}\right) = \widehat{\mathcal{O}} \) the first statement is trivial. Since \( x = 0 \) implies \( y = 0 \), hence \( \left( {x, y}\right) = T \) so \( \phi \left( \left( {x, y}\right) \right) = \widehat{\mathcal{O}} \), for the other statements we may assume \( x \neq 0 \) . Then \(... | Yes |
Proposition 8.2.4. (1) The 2-descent map \( \alpha \) is a group homomorphism. | Proof. (1) Clearly if \( P = \left( {x, y}\right) \neq T \) then \( \alpha \left( {-P}\right) = \alpha \left( \left( {x, - y}\right) \right) = x \) ; hence \( \alpha \left( P\right) \alpha \left( {-P}\right) = {x}^{2} \in {\mathbb{Q}}^{*2} \), and \( \alpha \left( T\right) \alpha \left( {-T}\right) = \alpha {\left( T\r... | Yes |
Lemma 8.2.5. Let \( A \) and \( B \) be abelian groups written additively, and let \( \phi \) from \( A \) to \( B \) and \( \widehat{\phi } \) from \( B \) to \( A \) be two group homomorphisms. Assume that the indexes \( \left\lbrack {B : \phi \left( A\right) }\right\rbrack \) and \( \left\lbrack {A : \widehat{\phi }... | Proof. We have\n\n\[ \left\lbrack {A : \widehat{\phi } \circ \phi \left( A\right) }\right\rbrack = \left\lbrack {A : \widehat{\phi }\left( B\right) }\right\rbrack \left\lbrack {\widehat{\phi }\left( B\right) : \widehat{\phi }\left( {\phi \left( A\right) }\right) }\right\rbrack . \]\n\nOn the other hand, it is clear tha... | Yes |
Corollary 8.2.6. The group \( E\left( \mathbb{Q}\right) /{2E}\left( \mathbb{Q}\right) \) is finite. More precisely, its cardinality divides \( {2}^{s + t + 2} \), where \( t \) is the number of distinct prime divisors of \( b \) , and \( s \) is the number of distinct prime divisors of \( {a}^{2} - {4b} \) . | Proof. By Proposition 8.2.4, we have \( \left\lbrack {E\left( \mathbb{Q}\right) : \widehat{\phi }\left( {\widehat{E}\left( \mathbb{Q}\right) }\right) }\right\rbrack \mid {2}^{t + 1} \) . Applying the proposition to \( \widehat{E} \) and \( \phi \), we have \( \left\lbrack {\widehat{E}\left( \mathbb{Q}\right) : \phi \le... | Yes |
Theorem 8.2.7 (Mordell). Let \( E \) be an elliptic curve defined over \( \mathbb{Q} \), and assume known that for some \( m \geq 2 \) we know that \( E\left( \mathbb{Q}\right) /{mE}\left( \mathbb{Q}\right) \) is finite (by the above corollary this is true for \( m = 2 \) when \( E \) has a rational 2-torsion point). T... | Proof. Assume by contradiction that the subgroup \( H \) of \( E\left( \mathbb{Q}\right) \) generated by \( S \) is not equal to \( E\left( \mathbb{Q}\right) \), and let \( {Q}_{1} \in E\left( \mathbb{Q}\right) \smallsetminus H \) . The set of points in \( E\left( \mathbb{Q}\right) \smallsetminus H \) of height less th... | Yes |
A divisor \( {b}_{1} \) of \( b \) is such that the quartic \( {Y}^{2} = {b}_{1}{X}^{4} + a{Z}^{2}{X}^{2} + \left( {b/{b}_{1}}\right) {Z}^{4} \) is solvable with pairwise coprime \( X, Y \), and \( Z \) with \( {XZ} \neq 0 \) if and only if the quartic \( {Y}^{2} = s\left( {b}_{1}\right) {X}^{4} + a{Z}^{2}{X}^{2} + \le... | Assume that \( {b}_{1} \) is such that the quartic \( {Y}^{2} = {b}_{1}{X}^{4} + a{Z}^{2}{X}^{2} + \left( {b/{b}_{1}}\right) {Z}^{4} \) is solvable with pairwise coprime \( X, Y \), and \( Z \) with \( {XZ} \neq 0 \) and write \( {b}_{1} = s\left( {b}_{1}\right) {f}^{2} \). Then\n\n\[{\left( Yf\right) }^{2} = s\left( {... | Yes |
Corollary 8.2.12. Let \( {b}_{1} \) be a divisor of \( b \) such that both \( {b}_{1} \) and \( b/{b}_{1} \) are squarefree (which is in particular the case if \( b \) is squarefree). If \( \left( {X, Y, Z}\right) \) satisfies \( {Y}^{2} = {b}_{1}{X}^{4} + a{Z}^{2}{X}^{2} + \left( {b/{b}_{1}}\right) {Z}^{4} \) with \( ... | Proof. Note that \( {b}_{1} = s\left( {b}_{1}\right) \) . Thus from the proof of the above proposition we see that if we set \( f = \gcd \left( {X, Y}\right) \) then \( {f}^{2} \mid b/{b}_{1} \), so that \( f = 1 \) since we assume \( b/{b}_{1} \) squarefree. As in the above proof we also deduce that \( \gcd \left( {Y,... | No |
Corollary 8.2.13. The group \( \alpha \left( {E\left( \mathbb{Q}\right) }\right) \) is equal to the set of classes modulo squares of 1, of \( s\left( b\right) \), and of \( {b}_{1} \) and \( b/{b}_{1} \) for all positive and negative divisors \( {b}_{1} \) of \( b \) such that \( {b}_{1} \) is squarefree, \( \left| {b}... | Proof. Denote by \( G \) the set of classes modulo squares of the elements described in this corollary. Clearly the classes of 1 and the squarefree part of \( b \) belong to \( \alpha \left( {E\left( \mathbb{Q}\right) }\right) \) . If \( {b}_{1} \) is squarefree we have \( s\left( {b}_{1}\right) = {b}_{1} \), so the pr... | Yes |
Proposition 8.2.14. (1) The curve \( {y}^{2} = {x}^{3} - 1 \) has rank 0 and torsion group of order 2. | Proof. We treat both curves simultaneously. The point \( \mathcal{O} \) and either \( \left( {1,0}\right) \) for the first curve or \( \left( {-1,0}\right) \) for the second clearly are the only points of order dividing 2. By the Nagell-Lutz Theorem 8.1.10 any other torsion point is such that \( y \) is integral and \(... | Yes |
Proposition 8.2.15. Let \( p \) be a positive or negative prime number, and let \( {E}_{p} \) be the elliptic curve with equation \( {y}^{2} = {x}^{3} - {px} \). The torsion group of \( {E}_{p} \) has order 2, and if \( {r}_{p} \) is the rank of \( {E}_{p}\left( \mathbb{Q}\right) \) we have the following results:\n\n(1... | Proof. The points \( \mathcal{O} \) and \( \left( {0,0}\right) \) are clearly the only points of order dividing 2. By the Nagell-Lutz Theorem 8.1.10 and its refinement Proposition 8.1.12, any other torsion point has integral \( x \) and \( y \) with \( x\left| {p\text{and}{y}^{2}}\right| 4{p}^{3} \), in other words \( ... | Yes |
Proposition 8.2.16. For \( p = - {73} \) we have \( {r}_{p} = 2 \), generators of \( E\left( \mathbb{Q}\right) \) modulo torsion being \( \left( {9/{16},{411}/{64}}\right) \) and \( \left( {4/9,{154}/{27}}\right) \) . | Proof. Since \( p < 0 \) we already know that \( \left| {\alpha \left( {E\left( \mathbb{Q}\right) }\right) }\right| = 2 \), so we consider only \( \widehat{E} \) whose equation is \( {y}^{2} = {x}^{3} - {292x} \) . The squarefree divisors \( {b}_{1} \) of \( b = - {292} = - {2}^{2} \cdot {73} \) less than \( {\left| b\... | Yes |
Proposition 8.3.1. (1) The map \( \\alpha \) is a group homomorphism from \( E\\left( \\mathbb{Q}\\right) \) to \( {K}^{ * }/{K}^{*2} \) . | Proof. (1). We treat the generic case, leaving the (easy) special cases to the reader. Clearly if \( P = \\left( {x, y}\\right) \) then\n\n\\[ \n\\alpha \\left( {-P}\\right) \\alpha \\left( P\\right) = \\alpha \\left( \\left( {x, - y}\\right) \\right) \\alpha \\left( \\left( {x, y}\\right) \\right) = {\\left( x - \\the... | No |
Corollary 8.3.2. The map \( \alpha \) induces an injective group homomorphism from \( E\left( \mathbb{Q}\right) /{2E}\left( \mathbb{Q}\right) \) to \( {K}^{ * }/{K}^{*2} \) (which by abuse of notation we will still denote by \( \alpha \) ). In addition, if the image of \( \alpha \) is finite then \( E\left( \mathbb{Q}\... | Proof. The first statement is clear. For the second we note that by assumption \( E \) has no rational 2-torsion; hence if \( d = {\dim }_{{\mathbb{F}}_{2}}\left( {\operatorname{Im}\left( \alpha \right) }\right) \) we have \( \left| {E\left( \mathbb{Q}\right) /{2E}\left( \mathbb{Q}\right) }\right| = \left| {\operatorna... | Yes |
Proposition 8.3.4. Let \( K \) be a number field of signature \( \left( {{r}_{1},{r}_{2}}\right) \), let \( T \) be a finite set of finite places of \( K \), and denote by \( t \) its cardinality.\n\n(1) The group \( {U}_{T}\left( K\right) \) is a finitely generated abelian group of rank \( {r}_{1} + \) \( {r}_{2} + t ... | Proof. (1). Although the proof is well known and easy we repeat it here. We have a natural exact sequence\n\n\[ 1 \rightarrow U\left( K\right) \rightarrow {U}_{T}\left( K\right) \rightarrow \mathcal{A} \rightarrow {Cl}\left( K\right) \rightarrow C{l}_{T}\left( K\right) \rightarrow 1, \]\n\nwhere the map starting from \... | Yes |
Proposition 8.3.5. Let \( P = \left( {x, y}\right) \in E\left( \mathbb{Q}\right) \smallsetminus \{ \mathcal{O}\} \), assume that \( \mathfrak{q} \) is a prime ideal of \( K \) such that \( {v}_{\mathfrak{q}}\left( {x - \theta }\right) \) is odd, and denote by \( q \) the prime number below \( \mathfrak{q} \) . Then \( ... | Proof. Set \( \gamma = x - \theta \in K \) . We can write \( {y}^{2} = {\gamma C} \) with \( C = {\gamma }^{2} + {3\theta \gamma } + \) \( 3{\theta }^{2} + a \) . If \( {v}_{\mathfrak{q}}\left( \gamma \right) \) were negative we would have \( {v}_{\mathfrak{q}}\left( \gamma \right) = {v}_{\mathfrak{q}}\left( {y}^{2}\ri... | Yes |
Lemma 8.3.6. If \( q \mid I\left( \theta \right) \) then \( \mathfrak{q} \) is the only ideal \( {\mathfrak{q}}_{i} \) above \( q \) such that \( x - \theta \in \) \( {\mathfrak{q}}_{i} \), and it has residual degree 1 . | Proof of Lemma 8.3.6. By Proposition 3.3.22, since \( q \nmid I\left( \theta \right) \) the decomposition of \( q{\mathbb{Z}}_{K} \) into prime ideals copies the decomposition of the polynomial \( R\left( X\right) = {X}^{3} + {aX} + b \) modulo \( q \) . Thus, write \( R\left( X\right) \equiv \mathop{\prod }\limits_{i}... | Yes |
Corollary 8.3.7. Denote by \( T \) the set of prime ideals \( \mathfrak{q} \) of \( K \) such that \( \mathfrak{q} \) \( \left( {3{\theta }^{2} + a}\right) \) and \( q \mid I\left( \theta \right) \), where \( q \) is the prime number below \( \mathfrak{q} \) . The image of \( \alpha \) is equal to the group of \( \bar{... | Proof. Let \( P = \left( {x, y}\right) \in E\left( \mathbb{Q}\right) \smallsetminus \{ \mathcal{O}\} \), so that \( \alpha \left( P\right) = x - \theta \) . By the proposition, if \( {v}_{\mathfrak{q}}\left( {x - \theta }\right) \) is odd we have \( \mathfrak{q} \mid \left( {3{\theta }^{2} + a}\right) \) and \( q \mid ... | Yes |
Corollary 8.3.8 (Mordell). The group \( E\left( \mathbb{Q}\right) /{2E}\left( \mathbb{Q}\right) \) is finite and the group \( E\left( \mathbb{Q}\right) \) is finitely generated. More precisely, \( \left| {E\left( \mathbb{Q}\right) /{2E}\left( \mathbb{Q}\right) }\right| \) divides \( {2}^{{r}_{1} + {r}_{2} + t + s} \) ,... | Proof. The finiteness and the bound for \( E\left( \mathbb{Q}\right) /{2E}\left( \mathbb{Q}\right) \) follow from the above proposition and Proposition 8.3.4. The statement for \( E\left( \mathbb{Q}\right) \) follows from Theorem 8.2.7. | Yes |
Proposition 8.4.2. Let \( E \) be an elliptic curve defined over a perfect commutative field \( K \) of characteristic different from 2 and having a \( K \) -rational subgroup of order 3, necessarily of the form \( \mathcal{T} = \{ \mathcal{O}, T, - T\} \) .\n\n(1) The abscissa \( x\left( T\right) \) of \( T \) is in \... | Proof. We have necessarily \( \mathcal{T} = \{ \mathcal{O},\left( {x, y}\right) ,\left( {x, - y}\right) \} \) with \( x = x\left( T\right) \) and \( y = y\left( T\right) \) . Let \( L \) be as in the definition of a \( K \) -rational subgroup. If \( \sigma \in \) \( \operatorname{Gal}\left( {L/K}\right) \) we must have... | Yes |
Proposition 8.4.3. For any \( P = \left( {x, y}\right) \in E \) set\n\n\[ \phi \left( P\right) = \left( {\widehat{x},\widehat{y}}\right) = \left( {\frac{{x}^{3} + {4d}\left( {\left( {{a}^{2}/3}\right) {x}^{2} + {abx} + {b}^{2}}\right) }{{x}^{2}},\frac{y\left( {{x}^{3} - {4db}\left( {{ax} + {2b}}\right) }\right) }{{x}^{... | Proof. As in the 2-descent case, it is enough to check the given formulas. However, this is not satisfactory and does not explain how they have been obtained. I give here a partial justification. For \( P = \left( {x, y}\right) \in E \) we will set with evident notation \( \widehat{x} = x\left( P\right) + x\left( {P + ... | No |
Lemma 8.4.5. Set \( u = \left( {\gamma + \widehat{x}/\gamma }\right) /2 \) and \( v = \left( {\gamma - \widehat{x}/\gamma }\right) /\left( {2\sqrt{\widehat{d}}}\right) \). (1) \( u \) and \( v \) are in \( \mathbb{Q} \). (2) We have \[ {\left( \widehat{x}/\gamma \right) }^{3} = \widehat{y} + \left( {\widehat{a}\widehat... | Proof. (1). This is trivial if \( \sqrt{\widehat{d}} \in \mathbb{Q} \), so assume that this is not the case. Then denoting by \( \sigma \left( \gamma \right) \) the conjugate of \( \gamma \) in the quadratic field \( {K}_{\widehat{d}} \) we have \[ {\left( \gamma \sigma \left( \gamma \right) \right) }^{3} = {\widehat{y... | Yes |
Proposition 8.4.8. (1) The 3-descent map \( \alpha \) is a group homomorphism. | Proof. If \( P = \left( {x, y}\right) \neq T \), then \( \alpha \left( {-P}\right) = \alpha \left( \left( {x, - y}\right) \right) = - y - \left( {{ax} + b}\right) \sqrt{d} \), so \( \alpha \left( P\right) \alpha \left( {-P}\right) = - \left( {{y}^{2} - d{\left( ax + b\right) }^{2}}\right) = {\left( -x\right) }^{3} \in ... | Yes |
Corollary 8.5.2. For \( n \neq 0 \) let \( {E}_{n} \) be the elliptic curve with affine equation \( {y}^{2} = {x}^{3} - {n}^{2}x \) . When \( p \mid {2n} \) or \( p \equiv 3\left( {\;\operatorname{mod}\;4}\right) \) we have \( {a}_{p}\left( {E}_{n}\right) = 0 \), and when \( p \nmid {2n} \) and \( p \equiv 1\left( {\;\... | Proof. Clear from the proposition, and also from Proposition 7.3.16 together with the proposition applied to \( N = 1 \) . | No |
Corollary 8.5.4. When \( p = 3 \) or \( p \equiv 2\left( {\;\operatorname{mod}\;3}\right) \) we have \( {a}_{p}\left( {E}_{1}\right) = 0 \), and when \( p \equiv 1\left( {\;\operatorname{mod}\;3}\right) \) we have\n\n\[ \n{a}_{p}\left( {E}_{1}\right) = \left\{ \begin{array}{ll} b - {2a} & \text{ if }b\text{ is even,} \... | Proof. See Exercise 19. | No |
Proposition 8.5.5. Let \( E \) be an elliptic curve defined over \( \mathbb{Q} \) and having \( {CM} \) by the imaginary quadratic order \( {\mathcal{O}}_{D} \) of discriminant \( D \), and let \( p \) be a prime of good reduction and not dividing \( D \) . If \( \left( \frac{D}{p}\right) = - 1 \) we have \( {a}_{p}\le... | We can write \( \pi = \left( {a + b\sqrt{D}}\right) /2 \) with \( a \equiv {bD}\left( {\;\operatorname{mod}\;2}\right) \), so we have \( {a}_{p}\left( E\right) = \) \( a \), where \( {a}^{2} + \left| D\right| {b}^{2} = {4p} \) . An important algorithmic fact is that, given \( D \) and \( p \), by using a variant of Euc... | No |
Proposition 8.5.6. Assume that \( D \leq - 7 \) is a fundamental discriminant such that \( {\mathcal{O}}_{D} \) has class number 1, in other words \( D = - 7, - 8, - {11}, - {19} \) , \( - {43}, - {67} \), or -163, and let \( j = j\left( \tau \right) \) with \( \tau = \left( {-\delta + \sqrt{D}}\right) /2 \) as above, ... | Proof. Assuming the fundamental theorem of CM, which tells us that \( j \in \mathbb{Z} \), the proof is a simple verification: we compute numerically the seven values of \( j \) to sufficient accuracy (this can be done very simply), round to the nearest integer, and check. | No |
Theorem 8.5.8. Let \( D \leq - 7 \) be one of the 11 discriminants considered above, let \( E = {E}_{j} \) be the corresponding basic \( {CM} \) curve, and let \( {D}_{0} \) be the discriminant of the quadratic field \( \mathbb{Q}\left( \sqrt{D}\right) \), in other words the unique fundamental discriminant such that \(... | Since \( D < 0 \), the properties of the Legendre-Kronecker symbol tell us that \( \left( \frac{D}{-{a}_{p}\left( E\right) }\right) = - \left( \frac{D}{{a}_{p}\left( E\right) }\right) \), so in all cases this theorem allows us to distinguish between \( {a}_{p}\left( E\right) \) and its opposite, which was the goal of t... | Yes |
Corollary 8.5.11. Assume that \( \varepsilon \left( E\right) = 1 \) . Then\n\n\[ L\left( {E,1}\right) = 2\mathop{\sum }\limits_{{n \geq 1}}\frac{{a}_{n}\left( E\right) }{n}{e}^{-{2\pi n}/\sqrt{N}}. \] | Proof. We simply choose \( {t}_{0} = 1 \) in the proposition, and note that\n\n\[ \Gamma \left( {1, x}\right) = {\int }_{x}^{\infty }{e}^{-t}{dt} = {e}^{-x}. \]\n\nWe see that we obtain an exceedingly simple and fast formula for \( L\left( {E,1}\right) \) . Note however that it is useful only when \( N \) is not too la... | Yes |
Proposition 8.5.13. Set\n\n\[ \n\\omega = \\log \\left( \\frac{2\\pi }{{t}_{0}\\sqrt{N}}\\right) \n\]\n\nWe have the formula\n\n\[ \n\\frac{{L}^{\\left( r\\right) }\\left( {E, s}\\right) }{r!} = \\mathop{\\sum }\\limits_{{n \\geq 1}}\\frac{{a}_{n}\\left( E\\right) }{{n}^{s}}{\\Gamma }_{r}\\left( {s,\\frac{2\\pi n}{{t}_... | Proof. It immediately follows from the first formula of Proposition 8.5.15 below that\n\n\[ \n\\frac{d}{ds}{\\Gamma }_{r}\\left( {s, x}\\right) = {\\Gamma }_{r}\\left( {s, x}\\right) \\log \\left( x\\right) + \\left( {r + 1}\\right) {\\Gamma }_{r + 1}\\left( {s, x}\\right) . \n\]\n\nUsing Proposition 8.5.10, the above ... | Yes |
Corollary 8.5.14. Assume that \( \varepsilon \left( E\right) = {\left( -1\right) }^{r} \) and in addition that \( {L}^{\left( k\right) }\left( {E,1}\right) = \) 0 when \( 0 \leq k \leq r - 1, k \equiv r\left( {\;\operatorname{mod}\;2}\right) \) . Then\n\n\[ \frac{{L}^{\left( r\right) }\left( {E,1}\right) }{r!} = 2\math... | Proof. Since \( \varepsilon \left( E\right) = {\left( -1\right) }^{r} \), the functional equation implies that \( {L}^{\left( k\right) }\left( {E,1}\right) = \) 0 for all \( k ≢ r\left( {\;\operatorname{mod}\;2}\right) \) . Thus the hypotheses of the corollary and the above proposition applied with \( {t}_{0} = 1 \) gi... | Yes |
Proposition 8.5.15. For \( r \geq 0 \) we have\n\n\[ \n{\Gamma }_{r}\left( {s, x}\right) = {x}^{s}{\int }_{1}^{\infty }\frac{\log {\left( t\right) }^{r}}{r!}{e}^{-{xt}}{t}^{s}\frac{dt}{t} = {\int }_{x}^{\infty }\frac{\log {\left( t/x\right) }^{r}}{r!}{e}^{-t}{t}^{s}\frac{dt}{t}.\n\] | Proof. We prove this by induction on \( r \), calling \( {g}_{r}\left( x\right) \) the first integral on the right-hand side. It is clear that it is true for \( r = 0 \) . It is also clear that \( {g}_{r}\left( x\right) \) tends exponentially fast to 0 as \( x \) tends to infinity, so by definition of \( {\Gamma }_{r}\... | Yes |
Proposition 8.5.17. Recall that\n\n\[ \n{G}_{r}\left( x\right) = \mathop{\sum }\limits_{{n \geq 1}}{\left( -1\right) }^{n - 1}\frac{{x}^{n}}{{n}^{r}n!}.\n\]\n\nSet \( {H}_{k}\left( n\right) = \mathop{\sum }\limits_{{1 \leq j \leq n}}1/{j}^{k} \), and define arithmetic functions \( {A}_{k}\left( n\right) \) by a formal ... | Proof. We prove the proposition by induction on \( r \), the case \( r = 0 \) being\nclear since\n\[ \n{G}_{0}\left( x\right) = 1 - {e}^{-x} = {e}^{-x}\mathop{\sum }\limits_{{n \geq 1}}\frac{{x}^{n}}{n!}.\n\]\n\nDefine, by induction on \( k,{A}_{0}\left( n\right) = 1 \) for \( n \geq 0 \), and for \( k \geq 1 \) ,\n\n\... | Yes |
Proposition 8.6.2. If \( \gamma \in {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) (in particular if \( \gamma \in {\Gamma }_{0}\left( N\right) \) ) then \( \Delta \left( {\gamma \left( \tau \right) }\right) = \) \( \Delta \left( \tau \right) \), and if \( \gamma \in {\Gamma }_{0}\left( N\right) \) and \( \tau \) is a ... | Proof. This comes from the fundamental group equality\n\n\[ \n{\Gamma }_{0}\left( N\right) = \Gamma \cap {\left( \begin{matrix} N & 0 \\ 0 & 1 \end{matrix}\right) }^{-1}\Gamma \left( \begin{matrix} N & 0 \\ 0 & 1 \end{matrix}\right) \n\]\n\nwith \( \Gamma = {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \), and is left to... | No |
Proposition 8.6.3. Let \( \tau \in \mathcal{H} \) be a quadratic irrationality and let \( \left( {A, B, C}\right) \) be the quadratic form with discriminant \( D \) associated with \( \tau \) . Then \( \tau \) is a Heegner point of level \( N \) if and only if \( N \mid A \) and one of the following equivalent conditio... | Proof. We have \( \tau = \left( {-B + \sqrt{D}}\right) /\left( {2A}\right) \), hence \( {N\tau } = \left( {-{NB} + N\sqrt{D}}\right) /\left( {2A}\right) \) . For this to have the same discriminant, it must be of the form \( \left( {-{B}^{\prime } + }\right. \) \( \sqrt{D})/\left( {2{A}^{\prime }}\right) \) ; hence by i... | No |
Corollary 8.6.4. If \( \tau \) is a Heegner point of level \( N \) and discriminant \( D \) , then so is \( W\left( \tau \right) = - 1/\left( {N\tau }\right) \) . | Proof. Indeed, if \( \left( {A, B, C}\right) \) is the quadratic form associated with \( \tau \) then clearly \( \left( {{CN}, - B, A/N}\right) \) is the quadratic form associated with \( - 1/\left( {N\tau }\right) \), so the result follows from the proposition. Equivalently, since \( \Delta \left( {-1/\tau }\right) = ... | Yes |
Proposition 8.6.5. Let \( \tau \) be a Heegner point of discriminant \( D \) and level \( N \) . If \( D \) is a fundamental discriminant the condition \( \gcd \left( {N, B, F}\right) = 1 \) of the above proposition is automatically satisfied and for all \( p \mid \gcd \left( {D, N}\right) \) we have \( p\parallel N \)... | Proof. Let \( p \) be a prime dividing \( \gcd \left( {N, B, F}\right) \) . Since \( {B}^{2} - {4NF} = D \) we deduce that \( {p}^{2} \mid D \), which implies that \( p = 2 \) and \( D/4 \equiv 2 \) or 3 modulo 4 since \( D \) is fundamental. But then \( {\left( B/2\right) }^{2} = \left( {D/4}\right) + {NF} \equiv \lef... | Yes |
Proposition 8.6.6. There is a one-to-one correspondence between on the one hand classes modulo \( {\Gamma }_{0}\left( N\right) \) of Heegner points of discriminant \( D \) and level \( N \), and on the other hand, pairs \( \left( {\beta ,\left\lbrack \mathfrak{a}\right\rbrack }\right) \) where \( \beta \in \mathbb{Z}/{... | Proof. This consists in a series of easy verifications, which are essentially identical to those made in checking that the ideal class group of \( K \) is isomorphic to the group of classes of positive definite primitive quadratic forms of discriminant \( D \), and the details are left to the reader. Note that if \( b ... | No |
Theorem 8.6.7. Let \( \tau = \left( {\beta ,\left\lbrack \mathfrak{a}\right\rbrack }\right) \) be a Heegner point of level \( N \) and discriminant \( D \), let \( K = \mathbb{Q}\left( \sqrt{D}\right) \), and denote by \( H \) the Hilbert class field of \( K \) . Then \( \varphi \left( \tau \right) \in E\left( H\right)... | Thus we see that using the analytic function \( \varphi \), we can obtain a point with coordinates in \( H \), hence with algebraic coordinates. This is the \ | No |
Lemma 8.6.8. If \( \varepsilon = - 1 \), then in fact \( P \in E\left( \mathbb{Q}\right) \) . | Proof. Indeed, it is easy to see that \( \varepsilon = - 1 \) is equivalent to saying that \( \varphi \circ W = \varphi \), so that\n\n\[ \overline{\varphi \left( \left( {\beta ,\left\lbrack \mathfrak{b}\right\rbrack }\right) \right) } = \overline{\varphi \left( {W\left( {\beta ,\left\lbrack \mathfrak{b}\right\rbrack }... | Yes |
Theorem 8.6.9 (Gross-Zagier). If \( \gcd \left( {D,{2N}}\right) = 1 \) and \( D \neq - 3 \) the point \( P \) computed above satisfies\n\n\[ \widehat{h}\left( P\right) = \frac{\sqrt{\left| D\right| }}{4\operatorname{Vol}\left( E\right) }{L}^{\prime }\left( {E,1}\right) L\left( {{E}_{D},1}\right) \]\n\n(for \( D = - 3 \... | Since \( L\left( {{E}_{D},1}\right) \) can easily be computed using the exponentially convergent series given above, this allows us to check whether \( \widehat{h}\left( P\right) \) is close to 0, hence whether we will obtain a torsion point. But it is especially interesting to combine it with the BSD formula: indeed, ... | Yes |
Lemma 8.7.1. Keep the above notation, let \( {c}_{2} = \mathop{\min }\limits_{{1 \leq i \leq r}}{\lambda }_{i} \) be the smallest eigenvalue of \( Q \), and set \( {c}_{1} = \exp \left( {\mu \left( E\right) + {2.14}}\right) \), where \( \mu \left( E\right) \) is defined in Theorem 8.1.18. If \( P = \left( {x\left( P\ri... | Proof. Since \( x\left( P\right) \in \mathbb{Z} \smallsetminus \{ 0\} \) we have \( h\left( P\right) = \max \left( {\log \left( \left| {x\left( P\right) }\right| \right) ,0}\right) = \) \( \log \left( \left| {x\left( P\right) }\right| \right) \), so by Theorem 8.1.18, \( \log \left( \left| {x\left( P\right) }\right| \r... | Yes |
Lemma 8.7.2. Let \( P = \left( {x\left( P\right), y\left( P\right) }\right) \in {E}^{0} \) be a real point, and assume that \( \left| {x\left( P\right) + {b}_{2}/{12}}\right| \geq 2\max \left( {\left| {e}_{1}\right| ,\left| {e}_{2}\right| ,\left| {e}_{3}\right| }\right) \) . If we choose the (essentially) unique determ... | Proof. Write \( P = \left( {X\left( P\right), Y\left( P\right) }\right) \) . Since \( P \in {E}^{0} \) we have \( X\left( P\right) \geq {e}_{3} \) . Since \( X\left( P\right) = x\left( P\right) + {b}_{2}/{12} \) this implies that \( X\left( P\right) > 0 \), since otherwise \( {e}_{3} \leq X\left( P\right) \leq 0 \), so... | Yes |
Corollary 8.7.3. Let \( {c}_{i} \) be the constants defined in the above two lemmas and set \( {c}_{5} = \sqrt{{c}_{1}{c}_{3}} \) . If \( P = \left( {x\left( P\right), y\left( P\right) }\right) \) is an integral point in \( {E}^{0} \) with \( x\left( P\right) \neq 0 \) and if \( \left| {x\left( P\right) + {b}_{2}/{12}}... | Proof. Clear by combining the two lemmas. | No |
Corollary 1. Let \( M \) and \( N \) be closed subspaces of the Hausdorff linear topological space \( X \) with \( N \) of finite dimension. Then \( M + N \) is a closed subspace of \( X \) . | Proof. Let \( {Q}_{M} : X \rightarrow X/M \) be the quotient map. The subspace \( {Q}_{M}\left( N\right) \) is a finite dimensional subspace of \( X/M \), hence closed in \( X/M \) . Consequently, its inverse image \( {Q}_{M}^{-1}\left( {{Q}_{M}\left( N\right) }\right) = M + N \) is closed. | Yes |
Corollary 2. Let \( X \) and \( Y \) be linear topological spaces with \( X \) Hausdorff and finite dimensional. Then any linear map \( R : X \rightarrow Y \) is necessarily continuous. | Proof. Let \( T : {\mathbb{F}}^{n} \rightarrow X \) be an isomorphism. Then the map \( R \circ T : {\mathbb{F}}^{n} \rightarrow Y \) must be of the form \( \left( {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right) \rightarrow {\lambda }_{1}{y}_{1} + \cdots + {\lambda }_{n}{y}_{n} \) for suitable vectors \( {y}_{1},\ldots ,... | Yes |
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