On the arithmetic of simple singularities of type E

An ADE Dynkin diagram gives rise to a family of algebraic curves. In this paper, we use arithmetic invariant theory to study the integral points of the curves associated to the exceptional diagrams \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_6, E_7$$\end{document}E6,E7, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_8$$\end{document}E8. These curves are non-hyperelliptic of genus 3 or 4. We prove that a positive proportion of each family consists of curves with integral points everywhere locally but no integral points globally.


Introduction
Background Consider the following families of affine plane curves over Q: y 3 = x 4 + y(c 2 x 2 + c 5 x + c 8 ) + c 6 x 2 + c 9 x + c 12 (1.1) y 3 = x 3 y + c 10 x 2 + x(c 2 y 2 + c 8 y + c 14 ) + c 6 y 2 + c 12 y + c 18 (1.2) 14 x + c 20 ) + c 12 x 3 + c 18 x 2 + c 24 x + c 30 . (1.3) its smooth projective completion acquires rational points at infinity. Thus it is natural to study the arithmetic of these families of pointed smooth projective curves. The study of these families can be viewed as a variation on a classical theme: if we started instead with the singularity of type A 2 (given by the equation y 2 = x 3 ), then we would be studying the arithmetic of elliptic curves in standard Weierstrass form. We recall that if Y is a smooth projective curve over a global field k and P ∈ Y (k) is a rational point, then one can define the 2-Selmer set Sel 2 Y of the curve Y ; it is a subset of the 2-Selmer group of the Jacobian of Y that serves as a cohomological proxy for the set Y (k) of k-rational points. In the paper [26], the second author studied the behaviour of the 2-Selmer sets of the curves in the family (1.1), proving the following theorem ([26, Moreover, for any > 0, we can find a subset F ⊂ F 0 defined by congruence conditions such that For the definition of a subset defined by congruence conditions, see (1.4) below. This theorem has the following Diophantine consequence ( [26,Theorem 4.8]): Theorem 1.2 Let > 0, and let F 0 be as in the statement of Theorem 1.2. If b ∈ F 0 , let X b denote the affine curve over Z given by Eq. (1.1). Then there exists a subset F ⊂ F 0 defined by congruence conditions that satisfies the following conditions: 1. For every b ∈ F and for every prime p, X b (Z p ) = ∅.

We have
In other words, a positive proportion of curves in the family (1.1) have no Z-points despite having Z p -points for every prime p. (The presence of marked points at infinity implies that for every b ∈ F 0 , the curve X b also has R-points.) The results of this paper The goal of this paper is to generalize these results to the other two families (1.2) and (1.3) described above. The techniques we use are broadly similar to those of [26], and are based around the relation, introduced in [25], between the arithmetic of these families of curves and certain Vinberg representations associated to the corresponding root systems. We study this relation and then employ the orbitcounting techniques of Bhargava to prove our main theorems. We refer the reader to [26,Introduction] for a more detailed discussion of these ideas.
In order to state the main theorems of this paper precisely, we must introduce some more notation. We will find it convenient to state our results in parallel for the two families (1.2) and (1.3). When it is necessary to split into cases, we will say that we are either in Case E 7 or in Case E 8 . We specify the following notation: Case E 7 : We let B denote the affine scheme A 7 Z with coordinates (c 2 , c 6 , c 8 , c 10 , c 12 , c 14 , c 18 ), and let B = B Q . We let X ⊂ A 2 B denote the affine curve over B given by Eq. (1.2), and X = X Q . We let Y → B denote the family of projective curves defined in [25,Lemma 4.9] (this family is a fibre-wise compactification of X that is smooth at infinity. It can be realized as the closure of X in P 2 B ). We let F 0 denote the set of b ∈ B(Z) such that X b is smooth. If b ∈ F 0 , then we define ht(b) = sup i |c i (b)| 126/i . Case E 8 : We let B denote the affine scheme A 8 Z with coordinates (c 2 , c 8 , c 12 , c 14 , c 18 , c 20 , c 24 , c 30 ), and let B = B Q . We let X ⊂ A 2 B denote the affine curve over B given by Eq. (1. 3), and X = X Q . We let Y → B denote the family of projective curves defined in [25,Lemma 4.9] (again, this family is a fibre-wise compactification of X that is smooth at infinity. It can be realized as the closure of X in a suitable weighted projective space over B). We let F 0 denote the set of b ∈ B(Z) such that X b is smooth. If b ∈ F 0 , then we define ht(b) = sup i |c i (b)| 240/i .
In either case, we say that a subset F ⊂ F 0 is defined by congruence conditions if there exist distinct primes p 1 , . . . , p s and a non-empty open compact subset U p i ⊂ B(Z p i ) for each i ∈ {1, . . . , s} such that F = F 0 ∩ (U p 1 × · · · × U p s ), (1.4) where we are identifying F 0 with its image in B(Z p 1 ) × · · · × B(Z p s ) under the diagonal embedding. Our first main result is then as follows. #{b ∈ F | ht(b) < a} < ∞.

2.
For any > 0, we can find a subset F ⊂ F 0 defined by congruence conditions such that (We note that the average in Case E 7 is at least 2, because the family of curves (1.2) has two marked points at infinity; for a generic member of this family, these rational points define distinct elements inside the 2-Selmer set Sel 2 Y b ). In either case, we can apply Theorem 1.3 to deduce the following consequence.
Theorem 1.4 Let > 0. Then there exists a subset F ⊂ F 0 defined by congruence conditions satisfying the following conditions: Informally, we have shown that a positive proportion of each of the families (1.2) and (1.3) consists of curves with Z p -points for every prime p but no Z-points.
Methodology We now describe some new aspects of the proofs of Theorems 1.3 and 1.4. The main steps of our proofs are the same as those of [26]: we combine the parameterization (constructed in [25]) of 2-Selmer elements by rational orbits in a certain representation (G, V ) arising from a graded Lie algebra with a technique of counting integral orbits (i.e. of the group G(Z) in the set V (Z)). We thus gain information about the average size of 2-Selmer sets. Although our proofs are similar in outline to those of [26], we need to introduce several new ideas here. For example, the most challenging technical step in the argument is to eliminate the contribution of integral points which lie 'in the cusp'. (In the notation of Sect. 2.3, these points correspond to vectors v such that v α 0 = 0, where α 0 is the highest root in the ambient Lie algebra h.) For this step we prove an optimized criterion (Proposition 2.15) for when certain vectors are reducible (this implies that they cannot contribute to the non-trivial part of the 2-Selmer set of a smooth curve in our family).This criterion is based in large part on the Hilbert-Mumford stability criterion. Its application in this context is very natural, but seems to be new.
We then use a computer to carry out a formidable computation to bound the contribution of the parts of the cuspidal region that are not eliminated by this criterion (see Proposition 4.5). For comparison, we note that in [26], the cuspidal region was broken up into 68 pieces; here the analogous procedure leads to a decomposition into 1429 (resp. 9437) pieces in Case E 7 (resp. in Case E 8 ). It would be very interesting if one could discover a 'pure thought' way to tackle this problem that does not rely on case-by-case calculations.
The current setting also differs from that of [26] in that the curves of family (1.2) have more than one marked point at infinity. (The geometric reason for this is that the projective tangent line to a flex point P of a plane quartic curve intersects the curve in exactly one other point Q. This implies that the family (1.2), essentially the universal family of plane quartics with a marked flex point, has two canonical sections.) We find that the orbits that parameterize the divisor classes arising from these points match up in a very pleasant way with a certain subgroup of the Weyl group of the ambient Lie algebra h. (More precisely, while the trivial divisor class is represented by the orbit of the Kostant section, the class of the divisor P − Q is represented by the image of this orbit under a certain element of the Weyl group of h. This element is described in Lemma 2.5.) It remains an interesting open problem to generalize the results of this paper and of [26] to study the average size of the 2-Selmer group of the Jacobians of the curves in (1.1)-(1.3) (and not just the size of their 2-Selmer sets). The rational orbits necessary for this study were constructed in [27], but we do not yet understand how to construct integral representatives for these orbits, in other words, how to prove the analogue of Lemma 3.5 below after replacing the set Y b (Q p ) by J b (Q p ). If this can be achieved, then the work we do in this paper to bound the contribution of the cuspidal region will suffice to obtain the expected upper bound on the average size of the 2-Selmer group (namely 6 in Case E 7 and 3 in Case E 8 ).
Notation Given a connected reductive group H and a maximal torus T ⊂ H, we write X * (T ) = Hom(T, G m ) for the character group of T , X * (T ) for the cocharacter group of T , and W (H, T ) for the (absolute) Weyl group of H with respect to T . Similarly, if c is a Cartan subalgebra of h = Lie(H), then we write (h, c) for the roots of c and W (H, c) for the Weyl group of c. If α ∈ (h, c), then we write h α ⊂ h for the root space corresponding to α. We write N H (T ) (resp. N H (c)) for the normalizer of T (resp. c) in H, and Z H (T ) (resp. Z H (c)) for the associated centralizer. Similarly, if V is any subspace of h and x ∈ h, then we write z V (x) for the centralizer of x in V .
We write = R >0 for the multiplicative group of positive reals, and d × λ = dλ/λ for its Haar measure (where dλ is the usual Lebesgue measure on the real line). If G is a group defined over a ring R, V is an representation of G, and A ⊂ V , then we write G(R)\A for the set of equivalence classes of A under the relation a ∼ a if there exists γ ∈ G(R) such that γ a = a .

A stable grading
In this section we establish the algebraic foundation for the proofs of our main theorems: in each of our two cases, we describe the parameterization of certain 2-coverings of Jacobians of algebraic curves by orbits in a representation arising from a Z/2Z-graded Lie algebra. Our set-up parallels that of [26]; however, we must address the complications arising from the presence of an additional point at infinity on the curves in the family (1.2). This point makes its presence known in the disconnectedness of the group H θ defined below and in the fact that the central fibre of the family (1.2) is not irreducible.

Definition of the grading
Let k be a field of characteristic 0 with fixed separable closure k s , and let H be a simple adjoint group over k of rank r that is equipped with a k-split maximal torus T . Let h = Lie(H) and t = Lie(T ). We let H = (h, t) and choose a set of simple roots S H = {α 1 , α 2 , . . . , α r } ⊂ H . We also choose a Chevalley basis for h with root vectors {e α | α ∈ H }. Suppose that −1 is an element of the Weyl group W (H, T ) (this is true, e.g., if H has type E 7 or E 8 , but not if H has type E 6 ). Letρ ∈ X * (T ) be the sum of the fundamental coweights with respect to our choice of simple roots S H . Then, up to conjugation by H(k), the automorphism θ := Ad(ρ(−1)) is the unique involution of H such that h dθ =−1 contains a regular nilpotent element of h ([25, Corollary 2.15]). The grading induced by this involution is stable in the sense of [19,Sect. 5.3].
We define G = (H θ ) • and V = h dθ =−1 . Then G is a split semisimple group, and V is an irreducible representation of G, of the type studied by Kostant-Rallis in the case k = C [13]. The invariant theory of V is closely related to that of the adjoint representation of H. We now summarize some aspects of the invariant theory of the pair (G, V ). Proofs may be found in [13,28], or [16]. We refer the reader to [25,Sect. 2] for a more detailed summary in the present setting. Let us call a vector v ∈ V semisimple (resp. nilpotent, resp. regular) if it has this property when viewed as an element of h. We have the following proposition: 1. The components of the Jordan decomposition v = v s + v n in h in fact lie in V .

The stabilizer of v in G is finite (and hence the G-orbit of v has maximal dimension) if and only if v is regular.
We see in particular that a vector v ∈ V has both a closed orbit and a finite stabilizer (i.e. v is stable in the sense of [15] Before stating the next result, we review some basic definitions from geometric invariant theory. Recall that given a one-parameter subgroup λ : G m → G k s , we may decompose is called the set of weights for v with respect to λ.

Corollary 2.4
Let v ∈ V . Then the following are equivalent: 3. For any non-trivial one-parameter subgroup λ : G m → G k s , the vector v has a positive weight with respect to λ.
Proof What remains to be shown is that the third condition is equivalent to the vector v having a closed orbit and a finite stabilizer in G. This is the Hilbert-Mumford stability criterion (see e.g. [15]).
We now describe G and V more explicitly. By our definition of θ, it is clear that T ⊂ G.

The stabilizer of s under the action of W H on T is given by Stab
There is a split short exact sequence of groups More precisely, let S G ⊂ G be a choice of root basis and define Then Stab W H (s) ∼ = W G , and the inclusion N H θ (T ) → H θ induces an isomor- We remark that if H is of type E 7 , then the group H θ /G has order 2; if H is of type E 8 , then H θ /G is trivial.
Proof For the first item, note that since H is adjoint, w · s is completely determined by its action on the root spaces h α . We have that w · s acts trivially on h α if and only if α ∈ w −1 ( G ), and otherwise w · s acts on h α as multiplication by −1. For the second item,

Transverse slices over V G
We continue to use the notation of Sect. 2.1, and now begin our study of the categorical quotient map  1 We consider these affine subspaces for the sl 2 -triples corresponding to two conjugacy classes of nilpotent elements, namely the regular and subregular classes. Proposition 2.6 Let E ∈ V be a regular nilpotent element. Then: 1. There exists a unique normal sl 2 -triple containing E. Let κ be the Kostant section associated to this sl 2 -triple. Then π| κ is an isomorphism.
Consequently, there is a canonical bijection Proof The first part follows from work of Kostant   Next recall that V contains a subregular nilpotent element e (by definition, this means that e is nilpotent and dim Stab G (e) = 1; the existence of subregular nilpotents in V is proved in [25,Proposition 2.27]). We now discuss the sections corresponding to such an element.
Moreover, the elements c 2 , c 6 , c 8 , c 10 This compactification has two sections P 1 and P 2 at infinity, given by the equations 14] the section corresponding to E is P 1 . Then for each b ∈ B(k) such that (b) = 0, the following diagram commutes: where the maps in the diagram are specified as follows. The top arrow ι b is induced by the inclusion X → V . The left arrow η b is the restriction of the Abel-Jacobi map P → [(P)−(P 1 )]. To define γ b , we use Proposition 2.6 to obtain an injective homomorphism Proof In this theorem and the next, the first part (i.e. the explicit determination of the family X) is carried out in [ Moreover, the elements c 2 , c 8 , c 12  2. Let Y → B denote the compactification of X → B described in [25,Lemma 4.9]. Let P : B → Y denote the unique section at infinity (so that Y = X ∪ P). Then for each b ∈ B(k) such that (b) = 0, the following diagram commutes: Proof Let ω ∈ be the non-trivial element, and let E = α∈S H e ω(α) . Then E is a regular nilpotent element of V . Since H θ (k) acts simply transitively on the set of such elements, there is a unique element w ∈ H θ (k) lifting ω such that w(E) = E . Let κ denote the Kostant section corresponding to E . Then wκ = κ and so . The proof is essentially the same as the proof of [24,Theorem 5.3], but for the convenience of the reader, we give the details here. Let X b be the base change of X b to the fixed separable closure k s /k, and define Y b similarly. There is a short exact sequence of étale homology groups: There is a natural symplectic duality on H 1 (X b , F 2 ) which has radical μ 2 , and which descends to the usual Poincaré duality (or Weil) pairing on H 1 (Y b , F 2 ) = J b [2]. Through an explicit calculation, one can see that δ b ([(P 2 ) − (P 1 )]) is the image of the non-trivial element of μ 2 under the connecting homomorphism associated to the dual short exact sequence where we have used the Weil pairing to identify J b [2] with its dual. Let H sc denote the simply connected cover of H with centre A H sc . Note that θ lifts naturally to an automorphism of H sc , which will again denote by θ , and that because H sc is simply connected, the fixed-point subgroup G := (H sc ) θ is connected [22,Theorem 8.1]. Let C = Z H (κ b ) and let C sc = Z H sc (κ b ). Then C ⊂ H and C sc ⊂ H sc are maximal tori, and we have Z G (κ b ) = C sc [2] and Z G (κ b ) = im(C sc [2] → C [2]). It follows from the proof of [25,Theorem 4.10] that the short exact sequence (2.1) is isomorphic to and its dual is isomorphic to where we have used the W H -invariant duality on X * (C) and the isomorphism C [25,Corollary 2.12], which states that this Weyl-invariant duality descends to a non-degenerate symplectic alternating duality on Z G (κ b ). Therefore to prove the claim we must show that ) of the non-trivial element of π 0 (H θ ) under the connecting homomorphism associated with the short exact sequence (2.2). This follows from a computation with cocycles. Indeed, the second part of Proposition 2.6 asserts that there exists g ∈ G(k s ) such that κ b = gκ b . Then the cohomology class γ b (κ b ) is represented by the cocycle σ → g −1 ( σ g).
is a lift of the non-trivial element of π 0 (H θ ), so the claim follows from the fact that σ cc −1 We have established the claim, and the first part of the lemma. To finish the the proof, we note that δ b ([(P 2 ) − (P 1 )]) is non-trivial if and only if the connecting homomorphism π 0 (H θ ) → H 1 (k, Z G (κ b )) is injective. By exactness, this is equivalent to the surjectivity of the map H 0 (k, Z G (κ b )) → H 0 (k, C [2]), which is exactly the criterion given in the statement of the lemma.
Proof By the lemma, it is equivalent to show that the map H 0 (k, [2]) is clearly surjective.

Reducibility conditions
We now define the notion of k-reducibility and study the properties of k-reducible elements of V (k).
The factors of the Cartan decomposition h = t⊕ α∈ H h α are invariant under the action of θ; this leads to a corresponding decomposition Since the β i form a basis for X * (T ) ⊗ Q, each element γ ∈ X * (T ) may be written uniquely as Lemma 2.14 Let v ∈ V and decompose v as α∈ V v α as in (2.3). Suppose one of the following holds: Then v is k-reducible.
(We recall that the subgroup ⊂ W H was defined in Lemma 2.5.) Proof For the first part of the lemma, we will apply the criterion of Corollary 2.4. This corollary implies that if v ∈ V and there exists a non-trivial cocharacter λ ∈ X * (T ) such that v has no (strictly) positive weights with respect to λ, then (v) = 0. Let {ω 1 , . . . ,ω r } ⊂ X * (T ) ⊗ Q be the basis dual to the basis {β 1 , . . . , β r } of X * (T ) ⊗ Q, and let λ = r i=1 a iωi . Then there exists a positive integer m such that mλ ∈ X * (T ). The weights of v with respect to mλ are exactly the values α, mλ = m r i=1 a i n i (α) for those α ∈ V such that n i (α) = 0, so v has no positive weights with respect to mλ.
For the second item, let E = α∈S H e α , where each e α is a root vector of our fixed Chevalley basis (see Sect. 2.1). Then E is a regular nilpotent element of V , and is therefore contained in a unique normal sl 2 -triple, which in turn determines a Kostant section κ ⊂ V (see Proposition 2.6). Suppose that the vector v ∈ V satisfies the condition v α = 0 if α ∈ + V − S H . We may assume that if α ∈ S H , then v α = 0; otherwise v also satisfies the condition in the first part of the lemma. In this case, exactly the same argument as in the proof of [26,Lemma 2.6 We can again assume that v α = 0 if α ∈ w(S H ). Let E = α∈w(S H ) e α , and let κ be the Kostant section corresponding to E . Since the group H θ (k) acts simply transitively on the set of regular nilpotents of V ([25, Lemma 2.14]), there is a unique element x ∈ H θ (k) such that x · E = E. Then x normalizes the torus T , since t = Lie(T ) is the unique Cartan subalgebra of h containing the semisimple parts of the normal sl 2 -triples containing E and E respectively. Thus x corresponds to an element of the Weyl group W H ; since W H acts simply transitively on the set of root bases of H, we see that x is a representative in H θ (k) of w. As in the previous paragraph, the proof of [26,Lemma 2.6] shows that x −1 v is G(k)-conjugate to an element of κ, hence that v is G(k)-conjugate to an element of κ .
Given a subset M ⊂ V , we define the linear subspace

Proposition 2.15
Let M be a subset of V , and suppose that one of the following three conditions is satisfied: 3. There exist β ∈ S G , α ∈ V − M, and integers a 1 , . . . , a r not all equal to zero such that the following conditions hold: Proof If either of the first two conditions is satisfied, then the desired reducibility follows from Lemma 2.14. We now show that if the third condition is satisfied, then every element , and so v is k-reducible by the second part of the proposition. We can therefore assume that v α = 0.
the subgroup of G generated by the root groups corresponding to β and −β. Condition (a) implies that the decomposition V = V (M) ⊕ V M is G β -invariant. Since the ambient group H is simply laced, the β-root string through α has length two, and thus h α ⊕ h α−β is an irreducible G β -submodule of V . The existence of an irreducible representation of degree two implies that G β ∼ = SL 2 .
Since SL 2 (k) acts transitively on the non-zero vectors in the unique two-dimensional irreducible representation of SL 2 , we can find g ∈ G β (k) ⊂ G(k) such that (gv) α = 0. This shows that gv ∈ V (M ∪ {α}), hence that v is k-reducible, as required.

Roots and weights
We conclude Sect. 2 by fixing coordinates in H and G. From now on we assume H has type E 7 or type E 8 . As above we let + H be the set of positive roots corresponding to our choice of root basis S H . Similarly, we define − H ⊂ H to be the subset of negative roots. We note that there exists a unique choice of root basis S G of G such that the positive roots + G determined by S G are given by + G = G ∩ + H . Indeed, this follows from a consideration of Weyl chambers: the Weyl chambers for H (resp. G) are in bijection with the root bases of H (resp. G ), and each Weyl chamber for H is contained in a unique Weyl chamber for G. If C H is the fundamental Weyl chamber of H corresponding to S H , and C G is the unique Weyl chamber for G containing C H , then defining S G to be the root basis corresponding to C G yields the desired property. We note that the set of negative roots − G determined by S G is given by − G = G ∩ − H .
We will later need to carry out explicit calculations, so we now define S G in terms of the simple roots of S H in each case E 7 and E 8 . We number the simple roots of H and G as in Bourbaki [7,Planches].

Case E 7
We have S H = {α 1 , . . . , α 7 }, where the Dynkin diagram of H is as follows: The root basis S G = {β 1 , . . . , β 7 } described above consists of the roots where the Dynkin diagram is as follows: We note that the existence of a diagram automorphism for G implies that there are two possible choices of numbering of the roots in S G consistent with the conventions of Bourbaki; we keep the above choice for the rest of this paper.

Case E 8
We have S H = {α 1 , . . . , α 8 }, where the Dynkin diagram of H is as follows: The root basis S G = {β 1 , . . . , β 8 } described above consists of the roots where the Dynkin diagram is as follows: Once again the existence of a diagram automorphism for G means that there are two possible choices of numbering of the roots in S G consistent with Bourbaki; we keep the above choice for the rest of this paper.

Integral structures, measures, and orbits
In Sect. 2, we introduced the following data: From now on, we also fix the regular nilpotent element E = α∈S H e α ∈ V . We now assume that k = Q and study integral structures on these objects.

Integral structures and measures
Our choice of Chevalley basis of h with root vectors {e α | α ∈ H } determines a Chevalley basis of g, with root vectors {e α | α ∈ G }. It hence determines Z-forms h Z ⊂ h and g Z ⊂ g (in the sense of [4]). Moreover, V = V ∩ h Z is an admissible Z-lattice that contains E.
We extend G to a group scheme over Z given by the Zariski closure of the group G in GL(V). By abuse of notation, we also refer to this Z-group scheme as G. Then the group G(Z) acts on the lattice V(Z) ⊂ V (Q). The Cartan decomposition V = ⊕ α∈ V h α is defined over Z, so extends to a decomposition V = ⊕ α∈ V V α . Since there exists a subregular nilpotent element in V = V(Q), we may choose a subregular nilpotent element e ∈ V(Z). In Case E 7 , we impose the additional condition that E corresponds to P 1 in the sense described in Theorem 2.9.
Fix a maximal compact subgroup K ⊂ G(R). Let P = TN ⊂ G be the Borel subgroup corresponding to the root basis S G , and let P = T N ⊂ G be the opposite Borel subgroup.
We also define measures on V and B as in [26,Sect. 2.8] by fixing an invariant differential top form ω V on V and by defining ω B = dc 2 ∧ dc 6 ∧ · · · ∧ dc 18 in Case E 7 (resp. ω B = dc 2 ∧ dc 8 ∧ · · · ∧ dc 30 in Case E 8 ). If v is a place of Q, then the formulae db = |ω B | v and dv = |ω V | v define measures on B(Q v ) and V (Q v ) respectively. Fixing these choices, we have the following useful result.

Lemma 3.2
There exists a rational number W 0 ∈ Q × with the following property: let k /Q be any field extension, and let c ⊂ V (k ) be a Cartan subspace. Let μ c : G k × c → V k be the natural action map. Then μ * Proof The proof is identical to that of [ is of compact support and locally constant (resp. continuous), and we have the formula

Define a function m p
Then m p is locally constant.
Then we have the formula Proof The first part follows from Lemma 3.2 and the p-adic formula for integration in fibres; see [12,Sect. 7.6]. To prove the second part, we note that the function v → # Stab G(Q p ) (v) is locally constant, because the universal stabilizer Z → V reg.ss. is finite étale. It therefore suffices to show that the function be the restriction of the natural action map. Then μ is proper, and so μ −1 (V(Z p ) ∩ π −1 (B v )) is compact. It follows that the characteristic function χ of the set μ(μ −1 (V(Z p ) ∩ π −1 (B v ))) ⊂ V(Z p ) reg. ss. is locally constant and of compact support. For v ∈ U , we have n p (v ) = F χ (π(v )), where F χ is as defined in the statement of the first part of the proposition. Thus by the first part of the proposition n p is locally constant. The third part of the proposition follows from the first two.

Selmer elements and integral orbits
We now discuss the construction of elements of V(Z p ) and V(Z) from rational points of algebraic curves. The idea behind this construction is as follows. In Theorems 2.9 and 2.10, we have described how a transverse slice X to a subregular nilpotent in V can be identified with an explicit family of curves over B. The embedding X → V is defined over Q. After we fix integral structures, this means that a point of X (Z) (resp. X (Z p )) defines an element of V(Z) (resp. V(Z p )), after possibly clearing a bounded denominator. The main problem in this section is therefore to show if b ∈ B(Z p ) is of non-zero discriminant, then a class in J b (Q p )/2J b (Q p ) which is represented by a point of Y b (Q p ) is in fact represented either by a point X b (Z p ), or by a point at infinity.

Lemma 3.4
There exists an integer N 0 ≥ 1 with the following properties: 1. For any prime p and any b ∈ B(Z p ), we have N 0 · κ b ∈ V(Z p ).

For any prime p and any x
In the first three items N 0 is acting via the G m -action discussed in Sect. 2.2. In the third item N 0 is acting via the natural G m -action on B.
Proof This follows from the existence of the contracting G m -actions on κ, κ , and X , cf.
Lemma 3.5 There exists an integer N 1 ≥ 1 with the following property: for any prime p and is contained in the image of the composite map: (where γ b is as in Theorems 2.9 and 2.10 for the case when k = Q p ).
Proof We just treat the case when H is of type E 7 ; the E 8 case is more straightforward, since there is only one point at infinity. We will show that we can take N 1 = 2 4 N 2 0 , where N 0 is as in Lemma 3.4. We recall that the curve Y b is given by the equation and has two sections P 1 = [0 : 1 : 0] and P 2 = [1 : 0 : 0] at infinity; the map Y b (Q p ) → J b (Q p )/2J b (Q p ) sends a point P to the class of the divisor (P) − (P 1 ). We define Y to be the closed subscheme of P 2 B defined by the same equation; then the complement in Y of its sections at infinity is naturally identified with X by Theorem 2.9. For b ∈ B(Q p ), Y b is smooth in an open neighbourhood of these sections at infinity. If t ∈ Q × p , then the isomorphism X b → X t 2 b induced by the action of G m on X extends to an isomorphism Y b → Y t 2 b that maps [x 0 : y 0 : z 0 ] to [t 8 x 0 : t 12 y 0 : z 0 ].
We first claim that if b ∈ 2 4 B(Z p ), then every divisor class in the image of the map is represented by either the zero divisor, the divisor P 2 − P 1 , or a divisor of the form P − P 1 for some P ∈ X b (Z p ).
If P ∈ Y b (Q p ), then we write P for the image of P in Y b (F p ). The special fibre Y b,F p is reduced, and has at most two irreducible components, which are geometrically irreducible. Moreover, if there are two irreducible components, then P 1 and P 2 lie on distinct irreducible components. Indeed, due to the presence of the contracting G m -action, any property of the morphism Y → B which is open on the base can be checked in the central fibre. Thus [23,Tag 0C0E] implies that all of the fibres of Y are geometrically reduced; and then [23, Tag 055R] implies that the two sections P 1 , P 2 together meet all irreducible components in every geometric fibre. In particular, every irreducible component of Y b,F p is geometrically irreducible. Let be the open subscheme of Pic Y b /Z p corresponding to those invertible sheaves that are fibrewise of degree 0 on each irreducible component (see [5,Sect. 8.4]). Then J b is a smooth and separated scheme over Z p (see [5,Sect. 9.4,Theorem 2]). We note that if Q ∈ J b (Z p ) has trivial image in J b (Z p /2 3 pZ p ), then Q is divisible by 2 in J b (Z p ) (this follows from [21, Theorem 6.1] and its generalization [8, Proposition 3.1]).
To prove the claim, it suffices to show that if P / ∈ X b (Z p ), then one of the divisor classes [(P) − (P 1 )] or [(P) − (P 2 )] is divisible by 2 in J b (Q p ). We can assume that xy = 0. We note that if P / ∈ X b (Z p ), then (at least) one of x, y must be non-integral. If x is integral then the defining equation of Y b shows that y is integral too. We can therefore write x = p m u, y = p n v, with u, v ∈ Z × p and m < 0. We note that if n < 0, then we must have 2n = 3m, hence we can write n = 3k, m = 2k for some k < 0.
We first treat the case where p is odd. If n < 0, then we have We now show how the claim implies the lemma. We drop our assumption on the parity of p, and take b = N 2 0 c, where c ∈ 2 4 B(Z p ). Given a class φ in H 1 (Q p , J c [2]), if φ is in the image of Y c (Q p ), then φ is represented by either P 1 , P 2 , or an element of X c (Z p ). Let φ denote the corresponding class in H 1 (Q p , J b [2]). If P 1 is a representative, then κ b ∈ V b (Q p ) represents the corresponding rational orbit. By Lemma 3.4, we have κ b = N 0 · κ c ∈ V(Z p ), so κ b is even an integral representative for this rational orbit. If P 2 is a representative, then κ b ∈ V(Z p ) is an integral representative, by the same argument.
Suppose instead that φ is represented by a divisor (P) − (P 1 ), where P ∈ X c (Z p ). Then φ is represented by the divisor (N 0 · P) − (P 1 ), where now N 0 · P ∈ N 0 · X (Z p ). By Lemma 3.4, is an integral representative for the rational orbit corresponding to the class φ. This completes the proof.

Consequently, for any b ∈ B(Z) such that
Proof Suppose c ∈ Sel 2 (Y b ). We first show that c ∈ γ b (G(Q)\V (Q)); by Proposition 2.6 this is the case exactly when the image c of c under the map We now consider V (Q p ).

Lemma 3.8
There exists a constant ε ∈ (0, 1) with the following property: let p be a prime congruent to 1 mod 6. Then there exists a non-empty open compact subset U p ⊂ B(Z p ) such that for all b ∈ U p , we have (b) = 0, X b (Z p ) = ∅, and Proof Let p be a prime with p ≡ 1 mod 6. It suffices to show that we can find a single By continuity considerations of the type in [18,Sect. 8], we can then take U p to be any sufficiently small open compact neighbourhood of b in B(Z p ). We will in fact exhibit b ∈ U p such that (b) = 0, X b (Z p ) = ∅, the component group of the Néron model of J b is isomorphic to (Z/2Z) 2 , and the image of Y b (Q p ) in is the identity. This will imply that the lemma holds with ε = 1 4 . We first return to the E 6 family of curves (1.1): 9 x + c 12 described in Sect. 1 of this paper. In this case the existence of such a point b is asserted in [26,Proposition 2.15]. The proof given there is incorrect; more precisely, the description of the special fibre of a regular model of the curve y 3 = x 4 − p 2 is incorrect. We will first remedy this error. The calculation in this case will also play a role in the proof of the lemma in Cases E 7 and E 8 .
We consider instead the curve given by the equation y 3 = (x − 1)(x 3 − p 2 ). (This curve can be put into the canonical form (1.1) by a linear change of variable in x.) Let Y be the curve inside P 2 Z p given by the projective closure of this equation, and let Z ⊂ A 2 Z p denote the complement of the unique point at infinity. It is clear that Z(Z p ) = ∅. Moreover, Y has a unique point that is not regular, namely the point corresponding to (x, y) = (0, 0) in the special fibre Z F p .
This singularity can be resolved by blowing up. Let Y → Y denote the blow-up at the unique non-regular point of Y. Then Y has exactly 3 non-regular points. The special fibre of Y has two irreducible components, namely the strict transform of Y F p and a smooth exceptional divisor. Let Y → Y denote the blow-up of the 3 non-regular points. Then Y is regular, and the special fibre Y F p has 5 irreducible components: the strict transform C 1 of Y F p , the strict transform C 5 of the exceptional divisor in Y F p , and the smooth exceptional divisors C 2 , C 3 , C 4 of the blow-up Y → Y.
We note that blow-up commutes with flat base change, so to verify our claims about the component group it suffices to perform these blow-ups in the completed local ring of Y at the maximal ideal (p, x, y), which is in turn isomorphic to Z p x, w /(w 3 − x 3 + p 2 ).
Here we find that all the irreducible components in the special fibre of Y F p are smooth and geometrically irreducible, and their intersection graph is given as follows: All intersections are transverse, and the multiplicities of C 1 , C 2 , C 3 , C 4 and C 5 are respectively 1, 2, 2, 2, and 3. The intersection matrix of the special fibre of Y is therefore Let We now turn to Case E 7 . Consider a perturbation where λ ∈ Z p − {0}. Using the procedure of Proposition 5.1, we can make a change of variable to put this curve in the form (1.2): the perturbation causes the point [0 : 1 : 0] at infinity to be a flex point, but no longer a hyperflex point. One may check that the curve obtained in this way has non-trivial integral points. For λ close enough to 0, this curve will also satisfy the condition Finally, we turn to Case E 8 . We now let Z be the curve given by the equation y 3 = (x 2 − 1)(x 3 − p 2 ), and let Y denote the projective curve over Z p containing Z and given by the multihomogeneous equation . Then Y is smooth along the unique section at infinity. We see that Y has a unique non-regular point, namely the point inside Z corresponding to the maximal ideal (p, x, y). The completed local ring of Z at this point is isomorphic to Z p x, w /(w 3 − x 3 + p 2 ). It follows that the singularities of Y can be resolved by two blow-ups, exactly as in the E 6 case described above. Moreover, the intersection matrix is equal to M as defined above, and the isomorphism class of the component group of the Néron model of the Jacobian of Y Q p is also (Z/2Z) 2 . This concludes the proof.

Lemma 3.9
There exists an open subset U 2 ⊂ B(Z 2 ) such that for all b ∈ U 2 , we have (b) = 0, X b (Z 2 ) = ∅, and the image of the map X b (Z 2 ) → J b (Q 2 )/2J b (Q 2 ) does not intersect the subgroup generated by the divisor class [(P 1 ) − (P 2 )] in Case E 7 (resp. does not contain the identity in Case E 8 ).
Proof If c ∈ B(F 2 ) is such that X c is smooth, let us write Y c for the smooth projective 4 Counting points In Sect. 3 we have defined an algebraic group over Z and a representation V, as well as various associated structures. In Sect. 4, we continue with the same notation and now show how to estimate the number of points in G(Z)\V(Z) of bounded height.
We first prove a simplified result, Theorem 4.1. The more refined version (Theorem 4.7), which is needed for applications, will be given at the end of this section. Let L ⊂ B(R) be one of the subsets L k described in Lemma 3.7, and let s : L → V (R) be the corresponding section. Then L is a connected semialgebraic subset of B(R); s is a semialgebraic map; and s(L) has compact closure in V (R). The map × L → B(R), (λ, ) → λ · given by the G m -action on B is an open immersion, and ht(λ · ) = λ deg .
For any subset A ⊂ V(Z), we write A irr for the subset of points a ∈ A that are Qirreducible, in the sense of Sect. 2.3. We recall that r is the rank of H. Our first result is as follows.

Theorem 4.1
There exist constants C, δ > 0, not dependent on choice of L, such that Our proof is very similar to that of [26, Theorem 3.1], except that a significant amount of case-by-case computation is required in order to control the contribution of elements that are 'in the cusp' (i.e. elements that lie in the codimension-one subspace of V where the coordinate corresponding to the highest root of H vanishes; see Proposition 4.5 below).
To avoid repetition, we omit the details of proofs that are essentially the same as proofs appearing in [26,Sect. 3]. First we introduce some notation. Recall that we have fixed a choice of S = ωT c K ⊂ G(R) as in Proposition 3.1, where ω ⊂ N (R) is a compact subset and T c ⊂ T (R) • is open. As in [26,Sect. 3.1], we fix a compact semialgebraic set G 0 ⊂ G(R) × of non-empty interior with the property that K · G 0 = G 0 . We assume that the projection of G 0 to is contained in [1, C 0 ] for some constant C 0 and that vol(G 0 ) = 1. Given a subset A ⊂ V(Z) we let The following two lemmas are the analogues in our situation of [26,Lemma 3.3] and [26,Lemma 3.4]; the proofs are the same. a} ≤ N (A, a) and N  *  (A, a).

Lemma 4.2 Let A ⊂ V (Z) be a G-invariant subset. Then
where δ G is as defined in Sect. 3.1.
In order to actually count points, we will use the following result, which follows from [1,Theorem 1.3]. This replaces the use of [26,Proposition 3.5], itself based on a result of Davenport [9]. We prefer to cite [1] since the possibility of applying [9] to a general semialgebraic set rests implicitly on the Tarski-Seidenberg principle (see [10]). . To do so, by the same logic as in [26,Sect. 5], it suffices to find a function f : M 1 → R ≥0 satisfying the following two conditions: One can program a computer to generate the list of cusp data in C, after inputting the root datum of h and the description of its 2-grading, and then to verify that there exists such a function f for each (M 0 , M 1 ) ∈ C. We have carried out this verification process. Our code is available in the Mathematica notebooks E7CuspData.nb and E8CuspData.nb. 2 (In the name of efficiency, we actually follow a slightly different procedure, since it is timeconsuming to check the condition in part 3 of Proposition 2.15. Namely, we generate a list of cusp data by eliminating only those pairs (M 0 , M 1 ) such that M 0 satisfies the condition in part 2 of Proposition 2.15. For the cusp data on this list, we check that either a function f as above exists, or that one of the remaining conditions, i.e. part 1 or part 3 of Proposition 2.15, holds. When verifying the condition in part 3, we restrict our search to α ∈ M 1 . The end result is a collection of cusp data satisfying items 1 and 2 above, which suffices to prove the proposition.)
We can now finish the proof of Theorem 4.1. By Lemma 4.2, we have The result now follows on combining Propositions 4.5 and 4.6.
We now state the more refined version of Theorem 4.1 mentioned at the beginning of this section.
under the diagonal embedding). Then there exist constants C, δ > 0 not depending on s or the sets V p 1 , . . . , V p s such that Proof We recall that for each prime p we have defined in the statement of Proposition 3.3 a locally constant function m p : V (Q p ) reg.ss. → R by the formula The same argument as in the proof of [26,Corollary 3.9] leads to an estimate .
Combining Lemma 4.2, Proposition 4.6, and Proposition 4.5, and summing over all choices of L as in Lemma 3.7, yields absolute constants C, δ > 0 such that By the third part of Proposition 3.3, this expression is equal to The products s i=1 |W 0 | p i vol(G(Z p i )) can be bounded independently of s and the primes p 1 , . . . , p s . They can therefore be absorbed into the constant, giving the estimate in the statement of the theorem.

Applications to 2-Selmer sets
In this final section, we prove our main theorems, including the results stated in Sect. 1, by combining all the theory developed so far. In order to avoid confusion, we treat each of the two families of curves (corresponding to Case E 7 and Case E 8 ) in turn.

Applications in Case E 7
As above, we write B = Spec Z[c 2 , c 6 , c 8 , c 10 , c 12 , c 14 , c 18 ] for affine space over Z in 7 variables, and write X → B for the family of affine plane curves given by Eq. (1.2): This family has the following interpretation: Proposition 5.1 Let k/Q be a field. Then: 1. The locus inside B k above which the morphism X k → B k is smooth is the complement of an irreducible closed subset of B k of codimension 1. 2. The set of points b ∈ B(k) for which X b is smooth is in bijection with the set of equivalence classes of triples (C, P 1 , t), where: (a) C is a smooth, non-hyperelliptic curve of genus 3 over k.
(b) P 1 ∈ C(k) is a flex point in the canonical embedding, i.e. the projective tangent line to C at P 1 intersects C with multiplicity 3 at the point P 1 . (c) t ∈ T P 1 C is a non-zero Zariski tangent vector at the point P 1 .
If b corresponds to (C, P 1 , t), then X b is isomorphic to C − {P 1 , P 2 }, where P 2 ∈ C(k) is the unique point such that 3P 1 + P 2 is a canonical divisor. For λ ∈ k × , the coefficients c i satisfy the equality Proof Part 1 follows from the fact that X b is smooth if and only if (b) = 0. The proof of the second part is very similar to the proof of [26,Lemma 4.1], although here we cannot appeal to Pinkham's Theorem. Let (C, P 1 , t) be a tuple of the type described in the proposition, and let P 2 ∈ C(k) be the point such that 3P 1 + P 2 is a canonical divisor. The Riemann-Roch Theorem shows that h 0 (C, O C (3P 1 )) = 2 and h 0 (C, O C (2P 1 + P 2 )) = 2. We can therefore find functions y, x ∈ k(C) × , uniquely determined up to addition of constants, such that the polar divisor of y is 3P 1 and the polar divisor of x is 2P 1 + P 2 , and such that y = z −3 + · · ·, x = z −2 + · · · locally at the point P 1 , where z is a local parameter at P 1 such that dz(t) = 1. We can also assume that y vanishes at the point P 2 .
The 10 monomials 1, x, x 2 , y, yx, yx 2 , yx 3 , y 2 , y 2 x, y 3 all lie in the 9-dimensional space H 0 (C, O C (9P 1 + 2P 2 )). Moreover, the two sets of 9 monomials obtained by removing either y 3 or yx 3 from this list are linearly independent, as can be seen by considering polar divisors. It follows that there is a unique linear relation of the form y 3 = x 3 y + x 2 (c 4 y + c 10 ) + x c 2 y 2 + c 8 y + c 14 + c 6 y 2 + c 12 y + c 18 . (5.1) The function y is uniquely determined by the above data. We also see that there is a unique translate x + a (a ∈ k) such that, after replacing x by x + a, we have c 4 = 0 in Eq. (5.1). The homogenization of Eq. (5.1) then describes the canonical embedding of the curve C.
If k/Q is a field extension and b ∈ B(k) is such that X b is smooth, then we write Y b for the unique smooth projective completion of X b . As in Sect. 1, we define F 0 = {b ∈ B(Z) | X b,Q is smooth}. We say that a subset F ⊂ F 0 is defined by congruence conditions if there exist distinct primes p 1 , . . . , p s and a non-empty open compact subset U p i ⊂ B(Z p i ) for each i ∈ {1, . . . , s} such that where we are identifying F 0 with its image in B(Z p 1 ) × · · · × B(Z p s ) under the diagonal embedding.
We recall that for b ∈ B(R) we have defined ht(b) = sup i |c i (b)| 126/i . This function is homogeneous of degree 126, in the sense that for λ ∈ R × , we have ht(λ · b) = |λ| 126 ht(b). (We note that 126 is the number of roots in the root system of type E 7 , and so also the degree of the discrimimant polynomial considered in Sect. 2.1.) Lemma 5.2 There exists a constant δ > 0 such that if F ⊂ F 0 is a subset defined by congruence conditions as above, then Proof This is an easy consequence of Theorem 4.4.
Our main theorems are now as follows.

Theorem 5.3
Let F ⊂ F 0 be a subset defined by congruence conditions. Then In order to state the next theorem, we observe that if b ∈ B(Q) is such that X b is smooth, then the 2-Selmer set Sel 2 (Y b ) always contains the 'trivial' classes arising from divisors supported on the points P 1 , P 2 at infinity (as in the statement of Proposition 5.1). We write Sel 2 (Y b ) triv for the subset of Sel 2 (Y b ) consisting of these classes, and note that # Sel 2 (Y b ) triv ≤ 2, with equality if and only if the divisor class [(P 2 ) − (P 1 )] is not divisible by 2 in J b (Q).
Consequently, for any such choice of F we have The proof of Theorem 5.4 is essentially a refined version of the proof of Theorem 5.3, so we just give the proof of Theorem 5.4.
Proof of Theorem 5.4 Let p 1 , . . . , p s be primes congruent to 1 modulo 6. Let ε ∈ (0, 1) be as in Lemma 3.8, and for each i ∈ {1, . . . , s}, let U p i ⊂ B(Z p i ) be the set described in the statement of Lemma 3.8. These sets have the following property: define where κ is any Kostant section that is not G-conjugate to κ. Then V p i is an open compact subset of V(Z p i ) reg.ss. , and for any b ∈ U p i we have (b) = 0 and We let F = F 0 ∩ (U p 1 × · · · × U p s ). For any b ∈ F, let Sel 2 (Y b ) irr ⊂ Sel 2 (Y b ) denote the subset of 'nontrivial' elements, i.e. the complement of Sel 2 (Y b ) triv in Sel 2 (Y b ). Let A = V(Z) ∩ (V p 1 × · · · × V p s ). Then by Proposition 3.6, for any a > 0 we have b∈F ht(b)<a # Sel 2 (Y b ) irr ≤ G(Q)\{v ∈ A irr | ht(v) < N deg 1 a}.
By combining Theorem 4.7, Lemma 5.2, and the inequality (5.2), we see that there exist constants C, δ > 0, not depending on s or the choice of primes p 1 , . . . , p s , such that b∈F Since # Sel 2 (Y b ) ≤ 2 + # Sel 2 (Y b ) irr , the first sentence in the statement of the theorem now follows on choosing s sufficiently large and letting a → ∞. The second sentence follows from the first on combining it with the following lemma.
Proof Let b ∈ F, and let C b = Z H (κ b ), a maximal torus of H. The Galois action on C b induces an associated homomorphism Gal(Q s /Q) → W (H, C b ). Corollary 2.12 shows that if this homomorphism is surjective, then # Sel 2 (Y b ) triv = 2. It therefore suffices to show that the limit lim a→∞ #{b ∈ F | ht(b) < a, Gal(Q s /Q) → W (H, C b ) surjective} #{b ∈ F | ht(b) < a} exists and equals 1. This is a variant of the Hilbert Irreducibility Theorem and can be proved along similar lines to the arguments in [20,Sect. 13.2].
Theorem 5.6 For any > 0, there exists a subset F ⊂ F 0 defined by congruence conditions such that the following conditions are satisfied: 1. For every b ∈ F and every prime p, we have X b (Z p ) = ∅.

We have
Here Z (2) ⊂ Q denotes the subring of rational numbers of denominator prime to 2. For the sets F constructed in Theorem 5.6, we may say that a positive proportion of the curves X b (b ∈ F) have integral points everywhere locally, but no integral points globally.
Proof By Lemma 3.9 and Lemma 3.10, we can choose for every prime p an open compact subset U p ⊂ B(Z p ) such that the following conditions are satisfied: 1. For each b ∈ U 2 , (b) = 0 and the image of the map X b (Z 2 ) → J b (Q 2 )/2J b (Q 2 ) does not intersect the subgroup generated by [(P 1 ) − (P 2 )]. 2. For every prime p and for every b ∈ U p such that (b) = 0, the set X b (Z p ) is non-empty. 3. For every sufficiently large prime p, U p = B(Z p ).
Let F ⊂ F 0 be the corresponding subset defined by congruence conditions. Fix > 0. By modifying U p at sufficiently many primes congruent to 1 modulo 6, as in the proof of Theorem 5.4, we can assume moreover that the following condition is satisfied: To complete the proof of the theorem, we just need to show that if b ∈ F is such that Sel 2 (Y b ) = Sel 2 (Y b ) triv , then X (Z (2) ) = ∅. To this end, we consider the commutative diagram where the maps are the natural ones. By construction of U 2 , the image of the right-hand vertical map is contained in the complement of the subgroup generated by the divisor class [(P 1 ) − (P 2 )]. By assumption, the image of the bottom horizontal map is contained in the subgroup generated by the divisor class [(P 1 ) − (P 2 )]. This forces X b (Z (2) ) to be empty, as desired.

Applications in Case E 8
We now forget the notation of Sect. 5 This family has the following interpretation: Proposition 5.7 Let k/Q be a field. Then: 1. The locus inside B k above which the morphism X k → B k is smooth is the complement of an irreducible closed subset of B k of codimension 1. 2. The set of points b ∈ B(k) for which X b is smooth is in bijection with the set of equivalence classes of triples (C, P, t), where: (a) C is a smooth, non-hyperelliptic curve of genus 4 over k.
(b) P ∈ C(k) is a point such that 6P is a canonical divisor and h 0 (C, O C (3P)) = 2.
(c) t ∈ T P C is a non-zero Zariski tangent vector at the point P.
If b corresponds to (C, P, t), then X b is isomorphic to C −{P}. For λ ∈ k × , the coefficients c i satisfy the equality c i (C, P, λt) = λ i c i (C, P, t).
The proof is very similar to the proof of [26,Lemma 4.1] and to the proof of Proposition 5.1, so we omit it. If k/Q is a field extension and b ∈ B(k) is such that X b is smooth, then we write Y b for the unique smooth projective completion of X b . As in Case E 7 , we define F 0 = {b ∈ B(Z) | X b,Q is smooth}, and we say that a subset F ⊂ F 0 is defined by congruence conditions if there exist distinct primes p 1 , . . . , p s and a non-empty open compact subset U p i ⊂ B(Z p i ) for each i ∈ {1, . . . , s} such that F = F 0 ∩ (U p 1 × · · · × U p s ).
If b ∈ B(R), then we have ht(b) = sup i |c i (b)| 240/i . This function is homogeneous of degree 240, in the sense that for λ ∈ R × , we have ht(λb) = |λ| 240 ht(b). As in Case E 7 , an application of Theorem 4.4 shows that there exists a constant δ > 0 such that if F ⊂ F 0 is a subset defined by congruence conditions as above, then #{b ∈ F | ht(b) < a} = s i=1 vol(U p i ) a as a → ∞.
Our main theorems in Case E 8 are as follows. We omit the proofs since they are similar, and simpler, than those in Case E 7 in the previous section. #{b ∈ F | ht(b) < a} < 1 + .

Theorem 5.10
For any > 0, there exists a subset F ⊂ F 0 defined by congruence conditions such that the following conditions are satisfied: 1. For every b ∈ F and every prime p, we have X b (Z p ) = ∅.