FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 22 definitions, 2 theorems, 8 propositions, 11 lemmas, 3 corollaries, 15 examples, 4 constructions, 6 remarks and 1 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition / construction
The root lattice \(A_n(-1)\) is the negative-definite chain lattice with diagonal entries \(-2\) and adjacent pairings \(1\); its discriminant is \((-1)^n(n+1)\).
Definition / construction / notation
- The lattices \(\widetilde { T } _ { p , q , r }\) . Here \(p , q , r\) are positive integers. These are of rank \(p { + } q { + } r { - } 2\) and are defined by \(T\) -shaped graphs of the form three arms of lengths \(p\), \(q\) and \(r\) meeting in a single node; the discriminant is computed recursively from the shape of the graph.
Lemma 4.1.1
Lemma 4.1.1. Let \(\mathbb { Z } ^ { n } = \mathbb { O } ^ { n } \langle 1 \rangle\) be the standard euclidean lattice. The sublattice consisting of the integral vectors \(( x _ { 1 } , \ldots , x _ { n } ) \in \mathbb { Z } ^ { n } ( - 1 )\) for which \(\textstyle \sum _ { j = 1 } ^ { n } x _ { j } \equiv 0\) mod 2 is isometric to \(D _ { n } ( - 1 )\) .
Observation 4.1.2
Observation 4.1.2. Two 2-roots in each of the indecomposable root lattices \(A _ { n }\) , \(D _ { n }\) , \(E _ { 6 }\) , \(E _ { 7 }\) , \(E _ { 8 }\) are conjugate under the Weyl group of the lattice.
Definition / construction
The semidefinite simply-laced root lattices are the extended \(A\)–\(D\)–\(E\) diagrams; deleting the extending vertex gives the corresponding definite Dynkin diagram, and the extended lattice has one-dimensional nullspace generated by the indicated positive root combination.
Lemma 4.1.3
Lemma 4.1.3. The sublattice of \(E _ { 8 } ( - 1 )\) orthogonal to the root \(\beta = 2 \alpha _ { 1 } + 4 \alpha _ { 2 } + 6 \alpha _ { 3 } +\) \(5 \alpha _ { 4 } + 4 \alpha _ { 5 } + 3 \alpha _ { 6 } + 2 \alpha _ { 7 } + 3 \alpha _ { 8 }\) is isometric to \(E _ { 7 } ( - 1 )\) . Consequently, the orthogonal complement of every root in \(E _ { 8 } ( - 1 )\) is isometric to \(E _ { 7 } ( - 1 )\) .
Proposition 4.1.4
Proposition 4.1.4. Let \(\Gamma\) be as above. 1. If \(b _ { \Gamma }\) is negative semi-definite, there are two possibilities, namely \(b _ { \Gamma }\) is negative definite or \(b _ { \Gamma }\) is negative semi-definite and \(b _ { \Gamma } \otimes \mathbb { Q }\) has a one-dimensional null-space spanned by a vector all of whose coordinates are positive. 2. The form \(b _ { \Gamma }\) is negative definite if and only if it is one of the Dynkin diagrams \(A _ { n } ( - 1 )\) , \(D _ { n } ( - 1 )\) , \({ E } _ { 6 } ( - 1 )\) , \(E _ { 7 } ( - 1 )\) , \(E _ { 8 } ( - 1 )\) . In the negative semi-definite case we have in addition the extended Dynkin diagrams (with the exception of \(\widetilde { A ( - 1 ) }\) where two edges meet in two vertices).
Definition / construction
A Lorentz lattice is an integral lattice of Lorentzian signature.
Lemma 4.1.5
Lemma 4.1.5. The hyperbolic unimodular lattice \(\widetilde { T } _ { 2 , 3 , 7 } = E _ { 1 0 } ( - 1 )\) is isometric to the Enriques lattice \(U \oplus E _ { 8 } ( - 1 )\) . More generally, for \(n \geq 8\) one has \[ E _ { n + 3 } ( - 1 ) \simeq E _ { 8 } ( - 1 ) \oplus U \oplus A _ { n - 7 } ( - 1 ) . \]
Lemma 4.1.6
Lemma 4.1.6. The roots of \(E _ { 8 } ( - 1 )\) are the 240 vectors \(\pm \alpha _ { i j } , \pm \alpha _ { i j k } , \pm \beta _ { i j } , \pm \gamma _ { i } , 1 \le\) \(i < j < k \leq 8\) , where \[ \begin{array} { r } { \begin{array} { r l } & { \alpha _ { i j } = e _ { i } - e _ { j } , } \\ & { \alpha _ { i j k } = e _ { 0 } - e _ { i } - e _ { j } - e _ { k } , } \\ & { \beta _ { i j } = 2 e _ { 0 } - \left( e _ { 1 } + \cdot \cdot \cdot + e _ { 8 } \right) + e _ { i } + e _ { j } , } \\ & { \gamma _ { i } = 3 e _ { 0 } - \left( e _ { 1 } + \cdot \cdot \cdot + e _ { 8 } \right) - e _ { i } . } \end{array} } \end{array} \] All the roots of \(E _ { 8 } ( - 1 )\) are in the same orbit under the Weyl group of \(E _ { 8 } ( - 1 )\) Moreover, the 8 roots \(\alpha _ { i j } , \beta _ { i j }\) , \(i j \in \{ 1 2 , 3 4 , 5 6 , 7 8 \}\) are mutually orthogonal and hence span a sublattice isometric to \(\mathbb { O } ^ { 8 } A _ { 1 } ( - 1 )\) .
Lemma 4.1.7
Lemma 4.1.7. The dual lattice of \(A _ { n } ( - 1 )\) can be identified with the \(\mathbb { Q }\) -valued lattice \(A _ { n } ^ { \prime } : = ( \mathbb { Z } ^ { n } , Q _ { n } )\) where \[ Q _ { n } : = \left( \begin{array} { l l l l l } { { - 1 + s } } & { { s } } & { { \cdots } } & { { s } } & { { } } \\ { { s } } & { { - 1 + s } } & { { s } } & { { \vdots } } \\ { { \vdots } } & { { \ddots } } & { { \ddots } } & { { \vdots } } \\ { { s } } & { { \cdots } } & { { - 1 + s } } & { { s } } \\ { { s } } & { { \cdots } } & { { s } } & { { - 1 + s } } \end{array} \right) \quad s = \frac { 1 } { n + 1 } . \] The discriminant group is cyclic of order \(n + 1\) and generated by any basis vector ????, ?? = 1, . . . , ??, of ℤ??. In particular, ??#???? ( \(\begin{array} { r } { b _ { A _ { n } ( - 1 ) } ^ { \# } ( e _ { 1 } , e _ { 1 } ) = \frac { - n } { n + 1 } } \end{array}\) .
Definition / construction
A weighted graph \(\Gamma\) with the prescribed diagonal and edge weights determines an integral graph lattice \(L_\Gamma\).
Lemma 4.2.1
Lemma 4.2.1. Let \(\alpha , \beta\) be independent roots such that the form is positive definite on the plane spanned by these roots. Assume also that \(\alpha \cdot \alpha \leq \beta \cdot \beta\) and that the angle between the two roots is obtuse. Then we have \[ \theta _ { \alpha , \beta } = \pi - \frac { \pi } { m _ { \alpha , \beta } } , m _ { \alpha , \beta } \geq 2 \] and only the following possibilities occur: 
Remark 4.2.2
Remark 4.2.2. 1. The conditions of Lemma 4.2.1 are equivalent to the rotation \(\tau _ { \alpha \beta } = \sigma _ { \alpha ^ { \circ } } \sigma _ { \beta }\) having finite order. To see this, one first observes that \[ \mathrm { T r } ( \tau _ { \alpha \beta } ) = - 2 + 4 \frac { ( \alpha \cdot \beta ) ^ { 2 } } { ( \alpha \cdot \alpha ) ( \beta \cdot \beta ) } \] and so \(\mathrm { T r } ( \tau _ { \alpha \beta } ) < 2\) is equivalent to \(( \alpha \cdot \alpha ) ( \beta \cdot \beta ) > ( \alpha \cdot \beta ) ^ { 2 }\) which is a necessary and sufficient condition for \(P\) to be a definite sublattice. On the other hand, if \(\tau _ { \alpha \beta } \neq \mathrm { i d }\) has finite order, the two complex conjugate eigenvalues are roots of unity, say \(\exp ( 2 \pi i c )\) and \(\exp ( - 2 \pi i c )\) , \(c \in \mathbb { Q } - \mathbb { Z }\) , with sum \(2 \cos ( 2 \pi c ) < 2\) . So \(P\) is definite and we are in the situation of the lemma. In the more general situation where we assume that \(b | _ { L }\) takes real values, a group generated by reflections \(\sigma _ { \alpha }\) is called a Coxeter group if each rotation \(\tau _ { \alpha \beta } = \sigma _ { \alpha ^ { \circ } } \sigma _ { \beta }\) has finite order \(2 s _ { \alpha \beta } = m _ { \alpha \beta }\) , and the corresponding graph is called a Coxeter graph. 2. Allowing \(m _ { \alpha , \beta } = \infty\) gives \(s _ { \alpha , \beta } = 4\) (angle \(\pi\) ) and \(\beta \cdot \beta = \alpha \cdot \alpha\) or \(\beta \cdot \beta = 4 \alpha \cdot \alpha\) . This is the last graph in Table 4.2.1. It occurs indeed for hyperbolic type lattices. See Example 17.2.10.3.
Definition / construction
The Dynkin diagram of a root lattice is the weighted graph encoding the root pairings.
Example / comparison
The lattice \(G _ { 2 }\) . This is the lattice \(\mathbb { Z } e _ { 1 } \oplus \mathbb { Z } e _ { 2 }\) with symmetric form given by the Gram matrix \[ \left( { \begin{array} { l l } { 2 } & { - 3 } \\ { - 3 } & { 6 } \end{array} } \right) . \] This is a positive definite form and the two reflections \(\sigma _ { e _ { i } }\) , \(i = 1 , 2\) , which turn out to generate the full isometry group also denoted \(G _ { 2 }\) . Observe that \(e _ { 1 }\) is the 2-root associated to \(\sigma _ { e _ { 1 } }\) and \(e _ { 2 }\) is the 6-root associated to \(\sigma _ { e _ { 2 } }\) . The two make an angle of \(5 \pi / 6\) radians and \(G _ { 2 }\) is the symmetry group of a regular 6-gon with its barycenter in the origin. The group is one of the classical finite reflection groups, cf. [47, Table IV]. Here is the graph of \(G _ { 2 } ( - 1 )\) and its Dynkin diagram:
Example / comparison
The lattice \(F _ { 4 }\) . This is the lattice \(\mathbb { Z } e _ { 1 } \oplus \cdots \oplus \mathbb { Z } e _ { 4 }\) with symmetric form given by the Gram matrix \[ \left( { \begin{array} { c c c c } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { - 1 } & { 1 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 1 } \end{array} } \right) \] 
Graph \(G _ { 2 } ( - 1 )\) and Dynkin diagram \(G _ { 2 }\) It is positive definite and has \(k\) -roots for \(k = 1 , 2\) . Its isometry group turns out to be generated by the reflections in roots \(e _ { 1 } , \ldots , e _ { 4 }\) , the four basis vectors. # Graph \(F _ { 4 } ( - 1 )\) and Dynkin diagram \(F _ { 4 }\) The group is the classical reflection group of order 1152 which is the symmetry group of the so-called 24-cell in \(\mathbb { R } ^ { 4 }\) . See for example [47, Table I (ii) and Table IV].
Example / comparison
Two other important classes are the two closely related odd lattices \(B _ { n }\) and \(C _ { n }\) . To the first there corresponds a negative definite lattice \(B _ { n } ( - 1 )\) described by the following graph. The numbers above the vertices \(x\) indicate whether \(2 q ( x ) = - 2\) or \(= - 1\) . All edges indicate intersection 1 between corresponding roots. 
The graph \(B _ { n } ( - 1 )\) and Dynkin diagram \(B _ { n }\) This is in accordance with the convention for the root system1 \(B _ { n }\) which has a basis of roots \(\{ \alpha _ { 1 } , . . . , \alpha _ { n - 1 } , \alpha _ { n } \}\) such that the length of the first √ \(n - 1\) roots is the same, but \(\alpha _ { n }\) has shorter length with ratio \(\sqrt { 2 }\) between the two. If we scale lengths so that \(b ( \alpha _ { 1 } , \alpha _ { 1 } ) = 2\) , then indeed \(b ( \alpha _ { n } , \alpha _ { n } ) = 1\) . This gives the above Dynkin diagram for \(B _ { n }\) . There is a variant, denoted \(C _ { n }\) , where we interchange the role of the \(( - 1 )\) -roots and the \(\left( - 2 \right)\) -roots. The graph is then different but the Dynkin diagram is the same, although one sometimes draws an arrow from the smaller to the larger root to distinguish the two. The Weyl group is by definition generated by reflections in all roots and in all three examples this group turns out to be the full orthogonal group. See [26, Planches II, VIII, IX].
Definition / construction / notation
The Weyl group is by definition generated by reflections in all roots and in all three examples this group turns out to be the full orthogonal group. See [26, Planches II, VIII, IX].
Construction / comparison
Besides the \(A\)–\(D\)–\(E\) families, the \(B_n,C_n,F_4,G_2\) examples exhaust the irreducible definite root lattices with roots of varying lengths, and each has a positive-semidefinite extended Dynkin diagram.
Example / comparison
The extended root lattice \(\widetilde B_n\) has the displayed extended Dynkin diagram and a one-dimensional nullspace generated by the stated positive combination of simple roots.
Example / comparison
The extended root lattice \(\widetilde C_n\) has the displayed extended Dynkin diagram and a one-dimensional nullspace generated by the stated positive combination of simple roots.
Example / comparison
The extended root lattice \(\widetilde F_4\) has the displayed extended Dynkin diagram and nullspace generated by \(\alpha_0+2(\alpha_1+\alpha_4)+3\alpha_2+4\alpha_3\).
Example / comparison
The extended root lattice \(\widetilde G_2\) has the displayed extended Dynkin diagram and nullspace generated by \(\alpha_0+3\alpha_1+2\alpha_2\).
Proposition 4.2.4
Proposition 4.2.4. Let \(L\) be a positive semi-definite irreducible root lattice of rank ?? with Dynkin diagram \(\Gamma\) . There are two possibilities: 1. \(b _ { \Gamma }\) is positive definite and then \(\Gamma = A _ { n } , B _ { n } , C _ { n } , D _ { n }\) or \(E _ { 6 } , E _ { 7 } , E _ { 8 }\) , \(F _ { 2 }\) , \(G _ { 4 }\) ; 2. ??Γ is positive semi-definite and \(b _ { \Gamma } \otimes \mathbb { Q }\) has a one-dimensional null-space spanned by a vector all of whose coordinates are positive. In that case Γ is the corresponding extended Dynkin diagram.
Definition / construction
The weights in a linear plumbing graph determine the associated continued fraction.
Remark 4.3.1
Remark 4.3.1. Observe that the above procedure with \(t\) odd and \(s\) even leads to odd \(n\) and so this gives graphs \(\Gamma _ { \pmb { a } }\) with an even number of vertices. In case \(t\) is even and \(s\) odd, one gets graphs with an odd number of vertices. In both situations, the resulting even integers \(a _ { i }\) are unique and so the resulting even lattices are unique.
Proposition 4.3.2
Proposition 4.3.2. Let ??, ?? be two co-prime integers with \(1 \ < \ s \ < \ | t |\) . Setting \(d _ { 0 } = t , d _ { 1 } = s\) , the (first) euclidean algorithm inductively gives integers \(d _ { k }\) , \(k =\) \(2 , \ldots , n + 1\) , and matrices \[ A _ { k } : = \left( { \begin{array} { c c } { a _ { k } } & { - 1 } \\ { 1 } & { 0 } \end{array} } \right) , k = 0 , \ldots , n , w i t h A _ { k } \left( { \begin{array} { c } { d _ { k + 1 } } \\ { d _ { k + 2 } } \end{array} } \right) = \left( { \begin{array} { c } { d _ { k } } \\ { d _ { k + 1 } } \end{array} } \right) \] The integers \(a _ { k }\) are such that \(d _ { k } = a _ { k } d _ { k + 1 } - d _ { k + 2 }\) with \(0 < d _ { k + 2 } < d _ { k + 1 }\) and \(d _ { n + 1 } = 1\) Using the second algorithm one produces even \(a _ { k }\) starting with ?? and \(t\) that are not both odd and choosing \(a _ { k }\) such that \(a _ { k } d _ { k + 1 }\) is the even multiple of \(d _ { k + 1 }\) closest to \(d _ { k }\) . In this case \(0 < | d _ { k + 1 } | < | d _ { k } |\) and \(d _ { n + 1 } = \pm 1\) . For practical purposes it is better to use instead the inverses \(A _ { k } ^ { - 1 } = { \binom { 0 } { - 1 } } \quad a _ { k } \biggr ) .\) since then starting from \(( t , s )\) we get the required numbers \(a _ { 0 } , \ldots , a _ { n }\) directly.
Example / comparison
Example 4.3.3. For \((s,t)=(2,5)\) the ordinary continued-fraction algorithm gives \(5/2=[3,2]\), while the even-coefficient variant gives \(5/2=[2,-2]\).
Lemma 4.3.4
Lemma 4.3.4. The products \(A_k\cdots A_n\) have the stated matrix expression in the Euclidean-algorithm data \(d_k,d_k^*\); the reverse data satisfy \(d_{n-k+1}^*=a_kd_{n-k}^*-d_{n-k-1}^*\) and \(a_k=c_k+c_{k+1}^{-1}=c_{n-k+1}^*+(c_{n-k}^*)^{-1}\). For the even-coefficient algorithm, \(0<|d_1^*|<\cdots<|d_{n+1}^*|\).
Corollary 4.3.5
Corollary 4.3.5. Assume we have performed the second algorithm for \(( s , t )\) , where \(1 < s < | t |\) , and where \(t , s\) are not both odd, yielding a continued fraction \(t / s =\) \([ a _ { 0 } , \ldots , a _ { n } ]\) with \(a _ { i }\) even. Suppose \(d _ { n + 1 } = \pm 1\) and write \[ A _ { 0 } \cdot \cdot \cdot A _ { n } = d _ { n + 1 } \left( \begin{array} { c c } { { t } } & { { - s ^ { * } } } \\ { { s } } & { { - t ^ { * } } } \end{array} \right) , \quad s s ^ { * } - t t ^ { * } = 1 . \] Then (4.11) gives the (second variant of ) the euclidean algorithm for \(( t ^ { * } , s ^ { * } )\) in reverse order and yields the continued fraction expansion \(s ^ { * } / t ^ { * } = \left[ a _ { 0 } , \ldots , a _ { n - 1 } \right]\) .
Definition / construction
For the Euclidean-algorithm data \(d_k\), the displayed linear combinations define vectors \(a,c,d\) used to encode the bilinear form associated to the graph \(\Gamma_a\).
Lemma 4.3.6
Lemma 4.3.6. Let \(B _ { n + 1 - k }\) be the matrix obtained from \(B _ { n + 1 } = B _ { \Gamma _ { a } }\) by deleting the first \(k\) rows and columns. Then 1. the coordinates of \(\pmb { c }\) are \(c _ { k } \ : = \ : d _ { k } / d _ { k + 1 }\) , \(k = 0 , \ldots , n\) ; and \(a _ { k } = c _ { k } + \frac { 1 } { c _ { k + 1 } }\) \(k = 0 , \ldots , n - 1\) . Moreover, if the \(a _ { k }\) are even, \(a _ { k }\) and \(c _ { k }\) have the same sign; 2. the coordinates of \(\mathbf { \pmb { d } }\) are \(d _ { k } = \pm \operatorname* { d e t } B _ { n + 1 - k }\) , \(k = 0 , \ldots , n\) , and the sign is the sign of \(d _ { n + 1 }\) . In particular, \(\mathrm { d i s c } ( b _ { \Gamma _ { a } } ) = \pm d _ { 0 }\) ; 3. over \(\mathbb { Q }\) the form given by \(B _ { n + 1 - m }\) is isometric to the diagonal form \(\mathrm { d i a g } ( c _ { m } , \ldots , c _ { n } )\) , \(m = 0 , \ldots , n\) .
Corollary 4.3.7
Corollary 4.3.7. Given two co-prime integers \(s , t\) with \(1 < s < | t |\) , ?? and ?? not both odd. The euclidean algorithm (4.10) yields a unique continued fraction expansion \(t / s = \left[ a _ { 0 } , \ldots , a _ { n } \right]\) with \(a _ { j }\) even. The associated graph \(\Gamma _ { \pmb { a } }\) , \(\pmb { a } = ( a _ { 0 } , \ldots , a _ { n } )\) , defines an even bilinear form \(b _ { \Gamma _ { \pmb { a } } }\) . Let \(n _ { \pm }\) be the number of positive, respectively negative \(a _ { i }\) . Then the signature of \(b _ { \Gamma _ { \pmb { a } } }\) equals \(( n _ { + } , n _ { - } )\) .
Example / comparison
Example 4.3.8. For the chain \(A_m(-1)\), the continued-fraction data are \(\mathbf a=(-2,\ldots,-2)\), \(\mathbf c=(-(n+2)/(n+1),\ldots,-2)\), and \(\mathbf d=((-1)^{n+1}(n+2),(-1)^n(n+1),\ldots,-2)\), satisfying the relations of Lemma 4.3.6.
Proposition 4.3.9
Proposition 4.3.9. Let \(( s , t )\) be two co-prime integers satisfying \(1 \ < \ s \ < \ | t |\) , which are not both odd and such that \(t / s = \left[ a _ { 0 } , \ldots , a _ { n } \right]\) with \(a _ { j }\) even. Let \[ Q ^ { - 1 } = \left( \begin{array} { l l l l l l l l } { s / t } & { 1 } & { 0 } & { \cdots } & { \cdots } & { \cdots } & { 0 } \\ { 1 } & { a _ { 0 } } & { 1 } & { 0 } & & { \cdots } & { 0 } \\ { 0 } & { 1 } & { a _ { 1 } } & { 1 } & { \ddots } & & & { \vdots } \\ { \vdots } & { \ddots } & { \ddots } & { \ddots } & { \ddots } & { \ddots } & { \vdots } \\ { \vdots } & & { \ddots } & { \ddots } & { \ddots } & { 1 } & { 0 } \\ { 0 } & { \cdots } & & { \ddots } & { 1 } & { a _ { n - 2 } } & { 1 } \\ { 0 } & { \cdots } & { \cdots } & { 0 } & { 1 } & { a _ { n - 1 } } \end{array} \right) . \] Then \(Q\) is the Gram matrix of an even integral bilinear form with \(\mathrm { d i s c } ( Q ) = \pm t\) and with discriminant bilinear form \(( \mathbb { Z } / t \mathbb { Z } , \langle s / t \rangle )\) .
Remark 4.3.10
Remark 4.3.10. Since the lattice \(( \mathbb { Z } ^ { n } , b )\) is even, the form \(^ { b }\) is the polar form of a quadratic form \(q\) . This \(q\) induces the quadratic torsion form \([ ( t / 2 ) / s ]\) if \(t\) is even, and \(\left[ t / ( 2 s ) \right]\) is \(t\) is odd.
Example / comparison
Example 4.3.11. We continue with the algorithm of Example 4.3.3 for \(5 / 2 =\) \(\lfloor 2 , - 2 \rfloor\) . Here \(A _ { 0 } A _ { 1 } = - { \left( \begin{array} { l l } { 5 } & { 2 } \\ { 2 } & { 1 } \end{array} \right) }\) and so \(s ^ { * } = - 2 , t ^ { * } = - 1\) . Since \(a _ { 0 } = 2\) and \(a _ { 1 } = - 2\) we find \(Q ^ { - 1 } = { \left( \begin{array} { l l } { { \frac { 2 } { 5 } } } & { 1 } \\ { 1 } & { 2 } \end{array} \right) }\) and \(Q = - \left( { \begin{array} { c c } { 1 0 } & { - 5 } \\ { - 5 } & { 2 } \end{array} } \right)\) an even matrix with discriminant \(- 5\) and signature \(( 1 , 1 )\) . By construction the corresponding even lattice has the right discriminant form \(\langle 2 \cdot 5 ^ { - 1 } \rangle\) .
Theorem 4.4.1
Theorem 4.4.1 ( [250]). Two lens spaces \(L ( t , s )\) and \(L ( t , s ^ { \prime } )\) have the same oriented homotopy type3 if and only if ????′ is a quadratic residue modulo \(t\) .
Theorem 4.4.2
Theorem 4.4.2. \(L ( t , s )\) and \(L ( t ^ { \prime } , s ^ { \prime } )\) are oriented homeomorphic (diffeomorphic) if and only if \(t = t ^ { \prime }\) and either \(s \equiv s ^ { \prime }\) mod \(t\) , or \(s \equiv ( s ^ { \prime } ) ^ { - 1 }\) mod \(t\) .
Lemma 4.4.3
Lemma 4.4.3 ( [99, §7]). If \(X\) is an ?? \(^ 2\) -homology three-sphere, there exists \(a\) four-manifold \(Y\) with boundary \(X\) such that 1. \(H _ { 1 } ( Y , \mathbb { Z } )\) has no 2-torsion; # 2. ???? is an even form.
Construction / comparison
For a simply connected oriented four-manifold \(Y\) with boundary \(X\), Lefschetz duality and the universal coefficient theorem identify \(H_2(Y,X;\mathbf Z)\simeq H^2(Y;\mathbf Z)\simeq\operatorname{Hom}(H_2(Y;\mathbf Z),\mathbf Z)\) when \(H_1(Y;\mathbf Z)\) is torsion-free; the boundary exact sequence then identifies the boundary torsion with the cokernel of the intersection correlation map.
Definition / construction
The boundary of a compact oriented \(4\)-manifold carries a linking pairing on its torsion first homology induced from the intersection lattice.
Definition / construction / notation
This turns out to be the linking pairing which is defined as follows. Take two classes \(c = \lfloor \gamma \rfloor\) and \(c ^ { \prime } = [ \gamma ^ { \prime } ]\) in \(H _ { 1 } ( X ) = \mathbb { Z } / t \mathbb { Z }\) where \(\gamma , \gamma ^ { \prime }\) are representative oriented cycles. Then \(t c = 0\) so there is an oriented 2-chain \(\Sigma\) which bounds \(t \gamma\) . Assume that \(\gamma ^ { \prime }\) is chosen general enough to meet \(\Sigma\) transversely. The intersection number \(\Sigma \cdot \gamma ^ { \prime }\) leads to the rational number \(t ^ { - 1 } ( \Sigma \cdot \gamma ^ { \prime } )\) which modulo integers turns out to depend only on the two torsion classes \(c , c ^ { \prime }\) . See [203, §77]. It is called the linking number \[ t ^ { - 1 } ( \Sigma \cdot \gamma ^ { \prime } ) = \ln \mathrm { k } ( c , c ^ { \prime } ) \in \mathbb { Q } / \mathbb { Z } . \]
Construction / comparison
If two simply connected bounding four-manifolds for the same homology three-sphere have even intersection forms, gluing them along the boundary gives a closed simply connected four-manifold; Rohlin divisibility implies that their signatures agree modulo \(16\), making the boundary index modulo \(16\) well defined.
Definition 4.4.4
Definition 4.4.4. The index mod 16 of \(X\) is defined as the index mod 16 of the intersection form of any bounding fourfold \(Y\) as in Lemma 4.4.3: \[ \tau _ { 1 6 } ( X ) : = \tau ( S _ { Y } ) \bmod 1 6 . \]
Definition / construction
The \(\mu\)-invariant of a suitable oriented \(3\)-manifold is the index modulo \(16\) of an even bounding intersection form.
Definition / construction / notation
Start with the tautological line bundle \(L\) on the complex projective line \(\mathbb { P } ^ { 1 } \simeq S ^ { 2 }\) . Endow \(\mathbb { P } ^ { 1 }\) with homogeneous coordinates \(z _ { \mathrm { 0 } } , z _ { \mathrm { 1 } }\) . By definition, the fiber of \(L\) over the point \(\left( z _ { 0 } : z _ { 1 } \right)\) is the line in \(\mathbb { C } ^ { 2 }\) spanned by the vector \(( z _ { 0 } , z _ { 1 } )\) . In algebraic geometry language, this line bundle is the dual of the ample line bundle \({ \mathfrak { O } } _ { \mathbb { P } ^ { 1 } } ( 1 )\) . The unit circles in the fibers of this bundle trace out an \(S ^ { 1 }\) bundle over \(S ^ { 2 }\) . The total space \(X ( - 1 )\) of the resulting bundle is the three sphere \(S ^ { 3 }\) as one easily sees.
Definition / construction / notation
Any lens space is obtained by means of a procedure which is called plumbing disc bundles along a tree which we now explain. The starting point are the lens spaces \(X ( a ) = L ( - a , 1 )\) which correspond to a graph with one vertex weighted by \(a\) . A graph with two vertices with weights \(( a _ { 1 } , a _ { 2 } )\) leads to plumbing of \(X _ { 1 } = X ( a _ { 1 } )\) and \(X _ { 2 } ~ = ~ X ( a _ { 2 } )\) given by the following procedure. Let \(Y _ { i }\) be the disc bundle corresponding to \(X _ { i }\) , \(i = 1 , 2\) . Pick a small disc \(D _ { \mathrm { b a s e } } ^ { i }\) in the base sphere \(S ^ { 2 }\) of \(Y _ { i }\) so that the disc bundle over it is homeomorphic to a product \(D _ { \mathrm { b a s e } } ^ { i } \times D _ { \mathrm { f i b e r } } ^ { i }\) . Now glue \(D _ { \mathrm { b a s e } } ^ { 1 } \times D _ { \mathrm { f i b e r } } ^ { 1 }\) to \(D _ { \mathrm { b a s e } } ^ { 2 } \times D _ { \mathrm { f i b e r } } ^ { 2 }\) by flipping the two discs. The plumbing procedure is available in all dimensions.
Definition / construction / notation
From relation (4.12) we see that equation (4.14) is related to the euclidean algorithm for the pair \(( t , s ^ { * } )\) . It yields the graph \(\Gamma _ { a }\) , \(\pmb { a } = ( a _ { 0 } , \ldots , a _ { n } ) = ( - m _ { 0 } , \ldots , - m _ { n } )\) . If we reverse the order of the plumbing in equation (4.13) we must reverse the order of the \(m _ { j }\) and the modified equation (4.14) is associated to the euclidean algorithm for \(( t , - s )\) . Of course this procedure neither changes the non-oriented graph \(\Gamma _ { a }\) nor the oriented homeomorphism type of \(L ( t , s )\) and we say that \(L ( t , s )\) is obtained by plumbing circle bundles over \(S ^ { 2 }\) along the graph \(\Gamma _ { a }\) . The lens space \(L ( t , s )\) is the boundary of a disc bundle, say \(Y ( s , t )\) . It is simply connected because, by construction, the base of the fibration, a connected string of \(n + 1\) two spheres – a simply connected topological space – is a deformation retract of \(Y ( s , t )\) . The 2-homology of \(Y ( s , t )\) is a free group with generators \(\gamma _ { i }\) the zero-sections of the disc bundle over each copy \(S ^ { 2 }\) in the string of spheres. Two consecutive generators meet in a point and one can show that \(\gamma _ { i } \cdot \gamma _ { i } = - m _ { i }\) .
Proposition 4.4.5
Proposition 4.4.5. The lens space \(L ( t , s )\) is homeomorphic to the threefold obtained by plumbing \(S ^ { 1 }\) -bundles over \(S ^ { 2 }\) along the graph \(\Gamma _ { ( - m _ { 0 } , . . . , - m _ { n } ) }\) where \[ - t / s = [ - m _ { n } , . . . , - m _ { 0 } ] . \] The lens space \(L ( t , s )\) bounds an oriented simply connected 4-dimensional manifold \(Y ( t , s )\) which is a disc bundle over a string of \(n { + 1 }\) one-point connected two-spheres. Its homology classes give a basis of \(H _ { 2 } ( Y ( t , s ) , \mathbb { Z } )\) . The Gram matrix of the intersection pairing with respect to this basis is the matrix \(B _ { \Gamma }\) , \(\Gamma = \Gamma _ { ( - m _ { 0 } , . . . , - m _ { n } ) }\) . If ?? and ?? have different parity, one may assume that the \(m _ { i }\) are even and then the index mod 16 of the lens space \(L ( t , s )\) equals \(\tau ( B _ { \Gamma } )\) mod 16.
Construction / comparison
Complex conjugation in the second coordinate induces an orientation-reversing diffeomorphism \(L(t,s)\simeq-L(t,t-s)\). Thus when \(s\) and \(t\) are both odd one may replace \(L(t,s)\) by \(-L(t,t-s)\) to use the even-coefficient Euclidean algorithm.
Example / comparison
Example 4.4.6. For lens spaces with fundamental group of order \(7\), the mod-\(16\) index separates the three oriented homeomorphism types inside the two oriented homotopy types; in particular \(L(7,1),L(7,2),L(7,4)\) have one oriented homotopy type but are not all oriented-homeomorphic.
Remark: Remark
Remark. The linking pairing on \(L ( s , t )\) gives the torsion quadratic form \(\scriptstyle \left. { \frac { \Delta } { t } } \right.\) . In fact the isometry class of the torsion form classifies lens spaces up to homotopy. In this example the two non-isometric torsion quadratic forms are \(\left. { \scriptstyle { \frac { 1 } { 7 } } } \right.\) and for example \(\big < \frac { 6 } { 7 } \big >\) , since \(^ { 1 }\) is a quadratic residue modulo 7, but 6 is not. The example exhibits forms with different \(\mu\) -invariants but isometric linking forms such as the ones for \(L ( 7 , 1 )\) and \(L ( 7 , 2 )\) .
Definition / construction / notation
has a singularity at the origin. It is called the Hirzebruch–Jung singularity \(A _ { t , s }\) .
Definition / construction / notation
The disk bundle \(Y ( t , s )\) considered in Proposition 4.4.5 constructed using the non-modified euclidean algorithm (4.9) has a natural complex structure, as shown in [15, Ch. III.5]. Moreover, away from the zero section this structure is compatible with the complex structure on \([ \mathbb { C } ^ { 2 } - \{ 0 \} ] / \rho _ { t , s } ( \mu _ { t } )\) . The zero section of the disc bundle is a string of complex projective lines \(E ^ { ( n + 1 ) } : = E _ { 1 } \cup \cdot \cdot \cdot \cup E _ { n + 1 }\) whose intersection behaviour is given by the graph \(\Gamma _ { ( - m _ { 0 } , . . . , - m _ { n } ) }\) where \(- t / s = [ - m _ { n } , \ldots , - m _ { 0 } ]\) . Hence \(E _ { j } \cdot E _ { j } = - m _ { j }\) , and consecutively numbered \(E _ { j }\) meet transversely in a point and there are no other intersections between the curves \(E _ { j }\) . The graph \(\Gamma _ { ( - m _ { 0 } , . . . , - m _ { n } ) }\) is called the dual graph of the configuration consisting of the \(E _ { j }\) . Glueing \(Y ( t , s )\) and \(\mathbb { C } ^ { 2 } - \{ 0 \}\) along \(Y ( t , s ) - E ^ { ( n + 1 ) }\) results in a complex surface \(\widetilde { W _ { t , s } }\) together with a holomorphic map \(\pi : \widetilde { W _ { t , s } } \longrightarrow W _ { t , s }\) which is biholomorphic outside \(E ^ { ( n + 1 ) }\) and contracts \(E ^ { ( n + 1 ) }\) to the singularity at \(0\) . The subvariety \(E ^ { ( n + 1 ) }\) is called the exceptional divisor. A holomorphic map such as \(\pi\) from a smooth surface to a surface with an isolated singularity \(p\) and which is biholomorphic outside \(\pi ^ { - 1 } p\) is a resolution of the singularity \(p\) . The simplest example is the singularity \(A _ { 2 , 1 }\) , an ordinary double point with local equation \(u v = w ^ { 2 }\) . The exceptional subvariety consists of a single \(\mathbb { P } ^ { 1 }\) with self-intersection \(- 2\) . More generally, we may consider \(A _ { n + 1 , n }\) with local equation \(u v = w ^ { n + 1 }\) . This is also a double point, usually denoted \(A _ { n }\) . By Eqn. (4.8) one has \(\textstyle { \frac { n + 1 } { n } } = [ 2 , 2 , \dotsc , 2 ]\) ( \(n\) times), and so the exceptional set consists of a string of \(n\) curves with self-intersection \(- 2\) . In other words, its dual graph is \(A _ { n } ( - 1 )\) .
Proposition 4.5.1
Proposition 4.5.1 (Mumford [163]). The intersection matrix for the components of the exceptional divisor of an isolated singularity is negative definite.
Proposition 4.5.2
Proposition 4.5.2. Let \(x \in X\) be an isolated surface singularity and let \(X\) be the minimal resolution of singularities of \(X\) . The singularity \(x\) is rational if one of the following equivalent conditions is true: 1. \(K _ { X }\) is trivial in the neighborhood of the exceptional locus; 2. ?? is a Du Val singularity; 3. ?? is an isolated quotient singularity, that is, there is a neighborhood of \(x\) in \(\bar { X }\) which is biholomorphic to \(U / G\) where \(U\) is a neighborhood of the origin in \(\mathbb { C } ^ { 2 }\) and \(G\) is a finite subgroup of SU(2).
Definition / construction / notation
A surface fibration is a holomorphic map \(f:X\to C\) from a surface to a smooth curve with connected curve fibers.
Lemma 4.5.3
Lemma 4.5.3 (Zariski’s Lemma). Denote the span of the classes of the \(D _ { j }\) by \(\langle \Gamma \rangle \subset \mathsf { N S } ( X )\) . Then the intersection form restricted to ⟨Γ⟩ is negative semi-definite. Its null-space is one-dimensional and spanned by the class \([ F ]\) of a generic fiber.
Remark: Remark
Remark. It is possible that \(f\) admits multiple fibers with multiplicity \(> 1\) (this cannot occur if the fibration has a section). For a multiple fiber there are local coordinates \(( u , v )\) in \(X\) such that the fibration is given by \(f ( u , v ) = v ^ { k }\) . In that case the fiber \(F _ { 0 }\) over \(v = 0\) has multiplicity \(k\) and thus \(\left[ F _ { 0 } \right] = k \cdot \left[ F \right]\) .
Definition / construction / notation
Next, we investigate a special case, that of a fibration in genus 1 curves, also called an elliptic fibration. A surface equipped with an elliptic fibration is called an elliptic surface. We assume in addition that there is a section, and that \(f\) is relatively minimal , that is, \(X\) does not contain \(( - 1 )\) -curves as component of a fiber of \(f\) . The canonical divisor of \(X\) is an integral multiple of the class of a fiber, as expressed by the canonical bundle formula \[ [ K _ { X } ] = r \cdot [ F ] , ~ r = 2 g ( C ) - 2 + \frac { 1 } { 1 2 } e ( X ) . \]
Corollary 4.5.4
Corollary 4.5.4. Suppose that \(f : X \to C\) is a fibration in elliptic curves with a section. Then the dual graph of a reducible singular fiber is an extended Dynkin diagram of type \(\widetilde { A }\) -??- \(\widetilde { E }\) . 4
Remark 4.5.5
Remark 4.5.5. In Table 4.5.2 below we give Kodaira’s list of singular fibers (cf. [15, Ch. V.7]). The multiple fibers are either multiple non-singular elliptic curves or multiple \(A _ { n }\) -fibers. The irreducible singular fibers are either nodal ( \(\stackrel { \triangledown } { \hat { A } _ { 0 } }\) ) or cuspidal (type \(I I\) ). There are two \(\stackrel { \triangledown } { A } _ { 1 }\) -fibers: the two components meet in two points transversally or are tangent in one point. There are also two type \({ \widetilde { A } } _ { 2 }\) -fibers: the three components either meet transversally and form a cycle or they meet transversally in one point. The different types have specific Kodaira symbols. We collect these as well as the values of the Euler numbers of the singular fibers in a table. Here \(T\) is an irreducible Dynkin diagram and \(\widetilde { T }\) the corresponding extended Dynkin diagram.
Proposition 4.5.6
Proposition 4.5.6. Let \(f : X \to C\) be a (relatively minimal) elliptic fibration with a section ?? and singular fibers over \(\Sigma \subset C\) . Then 1. \(\begin{array} { r } { e ( X ) = \sum _ { c \in \Sigma } e ( F _ { c } ) \geq 0 } \end{array}\) and > 0 as soon as there is at least one singular fiber. 2. \([ s ] \cdot [ s ] = - { \textstyle \frac { 1 } { 1 2 } } e ( X )\) . 3. \(\begin{array} { r } { \tau ( X ) = - \frac { 2 } { 3 } e ( X ) } \end{array}\) .
Example / comparison
Example 4.5.7. An elliptic K3 surface with seven \(I_1\) fibers, seven \(I_2\) fibers and one type-III fiber has trivial lattice \(U\oplus A_1(-1)^{\oplus8}\) of discriminant \(-2^8\). If its Mordell–Weil group is torsion, it is either trivial or \(\mathbf Z/2\); in the non-trivial case the Picard lattice is \(U(2)\oplus D_4(-1)^{\oplus2}\). A degree-\(14\) weighted-projective K3 family realizes the latter case.
Definition / construction / notation
If \(K\) is the function field of the base curve \(c\) of an elliptic fibration \(\pi : X C\) with a section, one can view \(X\) as an elliptic curve \(E\) over \(K\) . The section gives a \(K\) -rational point, say \(O _ { K }\) . If we view \(X\) in this way we shall write \(E _ { K }\) . The point \(o _ { K }\) serves as the origin of an additive group on \(E _ { K }\) , the Mordell-Weil group of \(E _ { K }\) , written \(\mathrm { M W L } ( E _ { K } )\) . In geometric language \(\mathrm { M W L } ( E _ { K } )\) is the group of sections for the fibration \(f\) . In the previous subsection we wrote \(\mathrm { M W L } ( ( ) X )\) to stress the surface. In particular, there is a natural map \(\mathrm { M W L } ( E _ { K } ) \to \mathsf { N S } ( X )\) sending a section \(s\) to its class \([ s ]\) in \(\mathsf { N S } ( X )\) . However, this map is in general not injective and of course never surjective since it misses the class of a fiber. Surprisingly, by [210] one does obtain an isomorphism of groups \[ \begin{array} { r } { \psi : \mathrm { M W L } ( E _ { K } ) \stackrel { \sim } { } \mathsf { N S } ( X ) / \mathsf { N S } ( X ) _ { \mathrm { t r i v } } } \\ { s \mapsto [ s ] \bmod \mathsf { N S } ( S ) _ { \mathrm { t r i v } } . } \end{array} \]
Lemma 4.5.8
Lemma 4.5.8. The quotient map \(\psi:\operatorname{MWL}(E_K)\to\operatorname{NS}(X)/\operatorname{NS}(X)_{\rm triv}\) lifts to a homomorphism \(\phi:\operatorname{MWL}(E_K)\to\operatorname{NS}(X)_{\mathbf Q}\) whose kernel is the torsion subgroup. Hence \(\operatorname{MWL}(f)=\operatorname{MWL}(E_K)/\operatorname{tors}\) injects into \(\operatorname{NS}(X)_{\rm triv,\mathbf Q}^{\perp}\) and becomes an isomorphism over \(\mathbf Q\).
Definition / construction
The negative intersection form on the orthogonal complement of the trivial lattice induces a positive-definite rational height pairing on \(\operatorname{MWL}(f)\). The resulting pair is the Mordell–Weil lattice, and its rank is the Mordell–Weil rank.
Example / comparison
The Hesse pencil. See [202, Example 3.13, 6.36]. This is the pencil of plane cubics given by \[ x ^ { 3 } + y ^ { 3 } + z ^ { 3 } + t x y z = 0 . \] The curve is singular exactly for \(t = 3 , 3 \rho , 3 \rho ^ { 2 }\) , where \(\rho\) is a primitive cube root of unity. Also for \(t = \infty\) one has a singular fiber. Each singular fiber is a union of three distinct lines, and so is of type \({ \overset { \sim } { A } } _ { 2 }\) . The base points of the pencil are the nine points \(( 1 , - 1 , 0 )\) , \(( 1 , - \rho , 0 )\) , \(( 1 , - \rho ^ { 2 } , 0 )\) and their cyclic permutations. These are the 9 inflection points for each of the smooth members of the pencil and so the addition law of the elliptic curve shows that they correspond to 3-torsion. Replacing \(t\) with \(t _ { \mathrm { 1 } } / t _ { \mathrm { 0 } }\) gives a surface \(X \subset \mathbb { P } ^ { 1 } \times \mathbb { P } ^ { 2 }\) which is smooth as one easily verifies. The projection of \(\mathbb { P } ^ { 1 } \times \mathbb { P } ^ { 2 }\) onto the first factor induces an elliptic fibration \(\pi : X \to \mathbb { P } ^ { 1 }\) . Since \(\infty\) corresponds to \(( 1 , 0 ) \in \mathbb { P } ^ { 1 }\) the fiber at infinity corresponds to the curve \(x y z = 0\) , a union of three lines. The base points of the pencil become 9 sections, e.g. \(( 1 , - 1 , 0 )\) becomes \(t \mapsto t \times ( 1 , - 1 , 0 )\) . We may take the latter as the zero section. The 9 sections give a full 3-torsion group of the Mordell–Weil group. Adding the Euler numbers of the singular fibers, Proposition 4.5.6.1 gives \(e ( X ) = 4 \cdot 3 = 1 2\) , in accordance with the rationality of \(X\) . Indeed, \(X\) is the projective plane blown up at the 9 base points of the pencil and so \(e ( X ) = 3 + 9 = 1 2\) . The N´eron–Severi lattice is the full intersection lattice \(H ^ { 2 } ( X , \mathbb { Z } )\) and hence is unimodular. Indeed, it is easy to see that \({ \mathsf { N S } } ( X ) \simeq \langle 1 \rangle \oplus \oplus ^ { 9 } \langle - 1 \rangle\) with basis given by the class of a line and the classes of the 9 disjoint exceptional curves coming from blowing up the base points of the pencil. The trivial lattice has rank \(2 + 4 \cdot 2 = 1 0\) since it is isometric to \(U \odot \odot ^ { 4 } A _ { 2 } ( - 1 )\) and so the Mordell–Weil group is torsion, more precisely \(\mathrm { M W L } ( \pi ) \simeq \oplus ^ { 2 } \mathbb { Z } / 3 \mathbb { Z }\) . This shows that the class map \(\mathrm { M W L } ( \pi ) \to \mathsf { N S } ( X )\) need not be injective, since as in the present example, \(\mathsf { N S } ( X )\) can be torsion free while \(\operatorname { M W L } ( \pi )\) contains torsion. For this example the Mordell-Weil lattice is the trivial lattice \(0\) .
Example / comparison
We consider in detail the hexagonal lattice which is the leading example in the book [202]. See in particular [202, Example 5.9, 6.26, 6.41]. One starts with an elliptic pencil in the plane given in inhomogeneous coordinates \(x , y\) by \[ y ^ { 2 } + t y = ( x - a _ { 1 } ) ( x - a _ { 2 } ) ( x - a _ { 3 } ) , a _ { i } \in \mathbb { C } , \] where the \(a _ { i }\) are pairwise distinct. Making this homogeneous by setting \(x =\) \(z _ { 1 } / z _ { 0 } , y = z _ { 2 } / z _ { 0 }\) and \(t = t _ { 1 } / t _ { 0 }\) , one obtains \[ t _ { 0 } z _ { 0 } z _ { 2 } ^ { 2 } + t _ { 1 } z _ { 0 } ^ { 2 } z _ { 2 } = t _ { 0 } ( z _ { 1 } - a _ { 1 } z _ { 0 } ) ( z _ { 1 } - a _ { 2 } z _ { 0 } ) ( z _ { 1 } - a _ { 3 } z _ { 0 } ) , \] a surface in \(\mathbb { P } ^ { 2 } \times \mathbb { P } ^ { 1 }\) which has a single singularity at \(\infty \times ( 0 , 0 , 1 )\) , where \(\infty = ( 0 , 1 ) \in\) \(\mathbb { P } ^ { 1 }\) . A local calculation shows it to be a double point of type \(A _ { 5 }\) . Let \(X\) be its minimal desingularisation. The projection of \(\mathbb { P } ^ { 1 } \times \mathbb { P } ^ { 2 }\) onto the first factor induces an elliptic fibration \[ f : X \to \mathbb { P } ^ { 1 } \] which has 4 singular fibers of type \(I _ { 1 } ^ { * }\) (irreducible with one node) and a single reducible fiber over \(\infty\) which is of type \({ \widetilde { E } } _ { 6 }\) (matching the \(A _ { 5 }\) -singularity). We reproduce the corresponding Dynkin diagram with multiplicities which now correspond to multiplicities of the fiber components. 
The zero section is given by \(t \mapsto ( 1 , t ) \times ( 0 , 1 , 0 )\) . We consider three more sections \(P _ { i }\) given by \(t \mapsto ( 1 , t ) \times ( 1 , a _ { i } , 0 )\) , \(i = { 1 , 2 , 3 }\) . The addition law on the cubic curve shows that \(P _ { 1 } + P _ { 2 } + P _ { 3 } = 0\) . All sections meet the \({ \widetilde { E } } _ { 6 }\) -fiber at \(\infty\) in components with multiplicity 1, i.e., the components corresponding to either one of the three extremal edges. We assume that the zero section meets the \(\alpha _ { 0 }\) -component. We claim that the four sections \(\ 0 , P _ { 1 } , P _ { 2 } , P _ { 3 }\) are mutually disjoint and meet the same fiber component of the \({ \widetilde { E } } _ { 6 }\) -fiber, which we may assume to correspond to \(\alpha _ { 1 }\) . This is easy to see away from the fiber at \(\infty\) and requires a careful local computation at the point \(\infty \times ( 0 , 0 , 1 )\) . Let us compute the Mordell–Weil rank. Adding the Euler numbers of the singular fibers gives \(4 + 8 \ = \ 1 2\) in accordance with the rationality of \(X\) . Indeed, \(p _ { g } ( X ) \ = \ 0 \ = \ q ( X )\) and so \(b _ { 2 } ( X ) ~ = ~ 1 0\) . The N´eron–Severi lattice is the full intersection lattice \(H ^ { 2 } ( X , \mathbb { Z } )\) and hence is unimodular. It is odd, since by Lemma 4.5.6, sub 2, three sections \(P _ { i }\) as well as the zero section all have selfintersection \(- { \textstyle \frac { 1 } { 1 2 } } e ( X ) = - 1\) . As in the case of the Hesse pencil, we find that \(\mathsf { N S } ( X )\) is isometric to \(\langle 1 \rangle \oplus \oplus ^ { \mathrm { y } } \langle - 1 \rangle\) , but this time we use Lemma 4.5.6, sub 3 to see that the index equals \(- { \textstyle \frac { 2 } { 3 } } e ( X ) = - 8\) . The trivial lattice is isometric to \(U \mathbb { \oplus \oplus E _ { 6 } ( - 1 ) }\) and so the Mordell–Weil rank is 2. One can show that there is no torsion (cf. [202, Example 6.32]) and that \(P _ { 1 } , P _ { 2 }\) give a basis for MWL \(( f )\) . Then an easy calculation using Lemma 4.5.8 shows: \[ \phi ( P _ { i } ) = [ P _ { i } ] - [ 0 ] - F - R , \quad R = \sum _ { j = 1 } ^ { 6 } m _ { j } \alpha _ { j } , \] where the \(m _ { i }\) are given in the diagram below, just underneath the roots. Indeed the equations \(0 = \phi ( P _ { i } ) \cdot [ 0 ] = \phi ( P _ { i } ) \cdot F = \phi ( P _ { i } ) \cdot \alpha _ { j }\) , \(j = 1 , \dotsc , 6\) , together with \(\phi ( P _ { i } ) \equiv [ P _ { i } ]\) mod \(\mathbb { N S } ( S ) _ { \mathrm { t r i v } }\) give the (unique) solution we just described. 
Finally, let us compute the height pairing. We give a direct computation, although there is a general recipe for which we refer to [202, Theorem 6.24]. Observe first that \(\begin{array} { r } { R \cdot R = R \cdot P _ { i } = - \frac { 4 } { 3 } } \end{array}\) and \(( \left[ 0 \right] + F + R ) ^ { 2 } = ( \left[ 0 \right] + F + R ) \cdot P _ { i } = - \frac { 1 } { 3 }\) and hence \[ \begin{array} { l } { - \langle P _ { i } , P _ { j } \rangle = \phi ( P _ { i } ) \cdot \phi ( P _ { j } ) = [ P _ { i } ] \cdot [ P _ { j } ] - 2 ( [ 0 ] + F + R ) \cdot [ P _ { i } ] + ( [ 0 ] + F + R ) ^ { 2 } } \\ { \displaystyle = [ P _ { i } ] \cdot [ P _ { j } ] + \frac { 2 } { 3 } - \frac { 1 } { 3 } } \\ { \displaystyle = - \delta _ { i j } + \frac { 1 } { 3 } . } \end{array} \] Taking \(P _ { 1 } , P _ { 2 }\) for the generators of \(\operatorname { M W L } ( f )\) , we conclude that the Gram matrix for the height pairing is \(\left( { \begin{array} { c c } { { \frac { 2 } { 3 } } } & { { - \frac { 1 } { 3 } } } \\ { { - \frac { 1 } { 3 } } } & { { \frac { 2 } { 3 } } } \end{array} } \right)\) This is the inverse of the Gram matrix for \(A _ { 2 }\) with respect to the basic roots \(\alpha _ { 1 } , \alpha _ { 2 }\) of \(A _ { 2 }\) (cf. Section 4.1.A), and so \(\mathrm { M W L } ( f ) \simeq A _ { 2 } ^ { * }\) . This lattice is the hexagonal lattice in the euclidean plane as pictured as [202, Fig. 1.3].
Example / comparison
Large Mordell–Weil rank. In [202] many examples are given. Let us describe just one example due to Shioda [202, Theorem 13.26]. Consider the complex surface \[ X _ { m } : \quad y ^ { 2 } = x ^ { 3 } + t ^ { m } + 1 , \quad m \in \mathbb { N } . \] The map \(( x , y , t ) \mapsto t\) gives it the structure of an elliptic fibration. When \(m = 6 d\) all singular fibers are irreducible of type \(I I\) at the zeroes of \(t ^ { m } + 1\) . By Table 4.5.2 this implies that \(e ( X _ { m } ) ~ = ~ 2 m = 1 2 d\) and thus \(b _ { 2 } ( X _ { m } ) \ = \ 2 m - 2\) . The Hodge decomposition shows that the rank of the N´eron–Severi group is potentially as large as \(1 0 d - 4\) which implies that the Mordell-Weil rank of \(X _ { m }\) can be at most \(1 0 d - 6\) . It turns out that, however, it is much smaller: it has rank at most 68 with equality if and only if \(m\) is divisible by 360.