On the arithmetic of simple singularities of type E
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Abstract
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 \(E_6, E_7\), \(E_8\). These curves are nonhyperelliptic 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.
1 Introduction
We recall that if Y is a smooth projective curve over a global field k and \(P \in Y(k)\) is a rational point, then one can define the 2Selmer set \({{\mathrm{Sel}}}_2 Y\) of the curve Y; it is a subset of the 2Selmer group of the Jacobian of Y that serves as a cohomological proxy for the set Y(k) of krational points. In the paper [26], the second author studied the behaviour of the 2Selmer sets of the curves in the family (1.1), proving the following theorem ([26, Theorem 4.3]):
Theorem 1.1
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
 1.
For every \(b \in {\mathcal {F}}\) and for every prime p, \({\mathcal {X}}_b({\mathbb {Z}}_p) \ne \emptyset \).
 2.We have$$\begin{aligned} \liminf _{a\rightarrow \infty } \frac{\# \left\{ b \in {\mathcal {F}}\mid {{\mathrm{ht}}}(b)< a,\text { } {\mathcal {X}}_b({\mathbb {Z}}) = \emptyset \right\} }{\#\{ b \in {\mathcal {F}}\mid {{\mathrm{ht}}}(b) < a\}} > 1  \epsilon . \end{aligned}$$
In other words, a positive proportion of curves in the family (1.1) have no \({\mathbb {Z}}\)points despite having \({\mathbb {Z}}_p\)points for every prime p. (The presence of marked points at infinity implies that for every \(b \in {\mathcal {F}}_0\), the curve \({\mathcal {X}}_b\) also has \({\mathbb {R}}\)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.

Case \(\mathbf {E_7}\): We let \({\mathcal {B}}\) denote the affine scheme \({\mathbb {A}}^7_{\mathbb {Z}}\) with coordinates \((c_2, c_6, c_8, c_{10}, c_{12}, c_{14}, c_{18})\), and let \(B = {\mathcal {B}}_{\mathbb {Q}}\). We let \({\mathcal {X}}\subset {\mathbb {A}}^2_{\mathcal {B}}\) denote the affine curve over \({\mathcal {B}}\) given by Eq. (1.2), and \(X = {\mathcal {X}}_{\mathbb {Q}}\). We let \(Y \rightarrow B\) denote the family of projective curves defined in [25, Lemma 4.9] (this family is a fibrewise compactification of X that is smooth at infinity. It can be realized as the closure of X in \({\mathbb {P}}^2_B\)). We let \({\mathcal {F}}_0\) denote the set of \(b \in {\mathcal {B}}({\mathbb {Z}})\) such that \(X_b\) is smooth. If \(b \in {\mathcal {F}}_0\), then we define \({{\mathrm{ht}}}(b) = \sup _i  c_i(b) ^{126/i}\).

Case \(\mathbf {E_8}\): We let \({\mathcal {B}}\) denote the affine scheme \({\mathbb {A}}^8_{\mathbb {Z}}\) with coordinates \((c_2, c_8, c_{12}, c_{14}, c_{18}, c_{20}, c_{24}, c_{30})\), and let \(B = {\mathcal {B}}_{\mathbb {Q}}\). We let \({\mathcal {X}}\subset {\mathbb {A}}^2_{\mathcal {B}}\) denote the affine curve over \({\mathcal {B}}\) given by Eq. (1.3), and \(X = {\mathcal {X}}_{\mathbb {Q}}\). We let \(Y \rightarrow B\) denote the family of projective curves defined in [25, Lemma 4.9] (again, this family is a fibrewise 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 \({\mathcal {F}}_0\) denote the set of \(b \in {\mathcal {B}}({\mathbb {Z}})\) such that \(X_b\) is smooth. If \(b \in {\mathcal {F}}_0\), then we define \({{\mathrm{ht}}}(b) = \sup _i  c_i(b) ^{240/i}\).
Theorem 1.3
 1.Let \({\mathcal {F}}_0 \subset {\mathcal {F}}\) be a subset defined by congruence conditions. Then we have$$\begin{aligned} \limsup _{a\rightarrow \infty } \frac{\sum _{\begin{array}{c} b \in {\mathcal {F}}\\ {{\mathrm{ht}}}(b)< a \end{array}} \# {{\mathrm{Sel}}}_2(Y_b) }{\#\{ b \in {\mathcal {F}}\mid {{\mathrm{ht}}}(b)< a\}} < \infty . \end{aligned}$$
 2.For any \(\epsilon > 0\), we can find a subset \({\mathcal {F}}\subset {\mathcal {F}}_0\) defined by congruence conditions such that$$\begin{aligned} \limsup _{a\rightarrow \infty } \frac{\sum _{\begin{array}{c} b \in {\mathcal {F}}\\ {{\mathrm{ht}}}(b)< a \end{array}} \# {{\mathrm{Sel}}}_2(Y_b) }{\#\{ b \in {\mathcal {F}}\mid {{\mathrm{ht}}}(b)< a\}} < \left\{ \begin{array}{cc} 2 + \epsilon &{} \text {Case }\mathbf {E_7}; \\ 1 + \epsilon &{} \text {Case }\mathbf {E_8}. \end{array} \right. \end{aligned}$$
(We note that the average in Case \(\mathbf {E_7}\) 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 2Selmer set \({{\mathrm{Sel}}}_2 Y_b\)). In either case, we can apply Theorem 1.3 to deduce the following consequence.
Theorem 1.4
 1.
For every \(b \in {\mathcal {F}}\) and for every prime p, \({\mathcal {X}}_b({\mathbb {Z}}_p) \ne \emptyset \).
 2.We have$$\begin{aligned} \liminf _{a\rightarrow \infty } \frac{\# \{ b \in {\mathcal {F}}\mid {{\mathrm{ht}}}(b)< a,\text { } {\mathcal {X}}_b({\mathbb {Z}}) = \emptyset \} }{\#\{ b \in {\mathcal {F}}\mid {{\mathrm{ht}}}(b) < a\}} > 1  \epsilon . \end{aligned}$$
Informally, we have shown that a positive proportion of each of the families (1.2) and (1.3) consists of curves with \({\mathbb {Z}}_p\)points for every prime p but no \({\mathbb {Z}}\)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 2Selmer 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 \(G({\mathbb {Z}})\) in the set \(V({\mathbb {Z}})\)). We thus gain information about the average size of 2Selmer 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 \(v_{\alpha _0} = 0\), where \(\alpha _0\) is the highest root in the ambient Lie algebra \({\mathfrak {h}}\).) 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 nontrivial part of the 2Selmer 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 \(\mathbf {E_7}\) (resp. in Case \(\mathbf {E_8}\)). It would be very interesting if one could discover a ‘pure thought’ way to tackle this problem that does not rely on casebycase 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 \({\mathfrak {h}}\). (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 \({\mathfrak {h}}\). 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 2Selmer group of the Jacobians of the curves in (1.1)–(1.3) (and not just the size of their 2Selmer 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 \(Y_b({\mathbb {Q}}_p)\) by \(J_b({\mathbb {Q}}_p)\). 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 2Selmer group (namely 6 in Case \(\mathbf {E_7}\) and 3 in Case \(\mathbf {E_8}\)).
Notation Given a connected reductive group H and a maximal torus \(T \subset H\), we write \(X^*(T) = {{\mathrm{Hom}}}(T, {\mathbb {G}}_m)\) 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 \({\mathfrak {c}}\) is a Cartan subalgebra of \({\mathfrak {h}}= {{\mathrm{Lie}}}(H)\), then we write \(\Phi ({\mathfrak {h}}, {\mathfrak {c}})\) for the roots of \({\mathfrak {c}}\) and \(W(H, {\mathfrak {c}})\) for the Weyl group of \({\mathfrak {c}}\). If \(\alpha \in \Phi ({\mathfrak {h}}, {\mathfrak {c}})\), then we write \({\mathfrak {h}}_\alpha \subset {\mathfrak {h}}\) for the root space corresponding to \(\alpha \). We write \(N_H(T)\) (resp. \(N_H({\mathfrak {c}})\)) for the normalizer of T (resp. \({\mathfrak {c}}\)) in H, and \(Z_H(T)\) (resp. \(Z_H({\mathfrak {c}})\)) for the associated centralizer. Similarly, if V is any subspace of \({\mathfrak {h}}\) and \(x \in {\mathfrak {h}}\), then we write \({\mathfrak {z}}_V(x)\) for the centralizer of x in V.
We write \(\Lambda = {\mathbb {R}}_{>0}\) for the multiplicative group of positive reals, and \(d^\times \lambda = d \lambda / \lambda \) for its Haar measure (where \(d \lambda \) 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 \(A \subset V\), then we write \(G(R) \backslash A\) for the set of equivalence classes of A under the relation \(a \sim a'\) if there exists \(\gamma \in G(R)\) such that \(\gamma a = a'\).
2 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 2coverings of Jacobians of algebraic curves by orbits in a representation arising from a \({\mathbb {Z}}/ 2 {\mathbb {Z}}\)graded Lie algebra. Our setup 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 \(H^\theta \) defined below and in the fact that the central fibre of the family (1.2) is not irreducible.
2.1 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 ksplit maximal torus T. Let \({\mathfrak {h}}= {{{\mathrm{Lie}}}(H)}\) and \({\mathfrak {t}}= {{{\mathrm{Lie}}}(T)}\). We let \(\Phi _H = {\Phi ({\mathfrak {h}}, {\mathfrak {t}})}\) and choose a set of simple roots \(S_H = \{\alpha _1, \alpha _2, \ldots , \alpha _r \} \subset \Phi _H\). We also choose a Chevalley basis for \({\mathfrak {h}}\) with root vectors \(\{e_\alpha \mid \alpha \in \Phi _H\}\). 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 \(E_6\)). Let \(\check{\rho } \in X_*(T)\) 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 \(\theta := {{\mathrm{Ad}}}(\check{\rho }(1))\) is the unique involution of H such that \({\mathfrak {h}}^{d\theta = 1}\) contains a regular nilpotent element of \({\mathfrak {h}}\) ([25, Corollary 2.15]). The grading induced by this involution is stable in the sense of [19, Sect. 5.3].
We define \(G = (H^\theta )^\circ \) and \(V = {\mathfrak {h}}^{d \theta = 1}\). 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 \(k = {\mathbb {C}}\) [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.
Definition 2.1
Let \({\mathfrak {c}}\subset {\mathfrak {h}}\) be a Cartan subalgebra. If \({\mathfrak {c}}\subset V\), then \({\mathfrak {c}}\) is called a Cartan subspace of V.
Theorem 2.2
 1.
Any two Cartan subspaces \({\mathfrak {c}}, {\mathfrak {c}}' \subset V\) are conjugate by an element of \(G(k^s)\).
 2.Let \({\mathfrak {c}}\subset V\) be a Cartan subspace, and define \(W(G, {\mathfrak {c}}) = N_G({\mathfrak {c}}) / Z_G({\mathfrak {c}})\). Then the natural mapsand$$\begin{aligned} W(G, {\mathfrak {c}}) \rightarrow W(H, {\mathfrak {c}}) \end{aligned}$$are isomorphisms. In particular, \(k[V]^G\) is isomorphic to a polynomial algebra on \(r = {{\mathrm{rank}}}H\) generators.$$\begin{aligned} k[{\mathfrak {h}}]^H \rightarrow k[V]^G \rightarrow k[{\mathfrak {c}}]^{W(G,{\mathfrak {c}})} \end{aligned}$$
Let us call a vector \(v \in V\) semisimple (resp. nilpotent, resp. regular) if it has this property when viewed as an element of \({\mathfrak {h}}\). We have the following proposition:
Proposition 2.3
 1.
The components of the Jordan decomposition \(v = v_s + v_n\) in \({\mathfrak {h}}\) in fact lie in V.
 2.
The vector v has a closed Gorbit in V if and only if it is semisimple.
 3.
The stabilizer of v in G is finite (and hence the Gorbit of v has maximal dimension) if and only if v is regular.
We see in particular that a vector \(v \in V\) has both a closed orbit and a finite stabilizer (i.e. v is stable in the sense of [15]) if and only if it is regular semisimple. Let \(\tilde{\Delta }\in k[{\mathfrak {h}}]^H\) be the image under the isomorphism \(k[{\mathfrak {t}}]^{W(H, T)} \rightarrow k[{\mathfrak {h}}]^{H}\) of the product of all roots \(\alpha \in \Phi _H\). Then \(\tilde{\Delta }(v) \ne 0\) if and only if \(v \in {\mathfrak {h}}\) is regular semisimple. We call \(\Delta := \tilde{\Delta }_V\) the discriminant polynomial. Then \(\Delta \) is homogeneous of degree \(\# \Phi _H\). If \(v \in V\) is a vector such that \(\Delta (v) \ne 0\), then \({\mathfrak {z}}_{\mathfrak {h}}(v) \subset V\), and \({\mathfrak {z}}_{\mathfrak {h}}(v)\) is the unique Cartan subspace of V containing v.
Before stating the next result, we review some basic definitions from geometric invariant theory. Recall that given a oneparameter subgroup \(\lambda : \mathbb {G}_m \rightarrow G_{k^s}\), we may decompose \(V(k^s)\) as \(\oplus _{i \in {\mathbb {Z}}} V_i\), where \(V_i = \{v \in V(k^s) \mid \lambda (t)\cdot v = t^i v\}\). If we decompose a vector \(v \in V\) as \(v = \sum v_i\) where \(v_i \in V_i\) for all i, then \(\{ i \mid v_i \ne 0 \}\) is called the set of weights for v with respect to \(\lambda \).
Corollary 2.4
 1.
v is regular semisimple.
 2.
\(\Delta (v) \ne 0\).
 3.
For any nontrivial oneparameter subgroup \(\lambda : {\mathbb {G}}_m \rightarrow G_{k^s}\), the vector v has a positive weight with respect to \(\lambda \).
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]).\(\square \)
We now describe G and V more explicitly. By our definition of \(\theta \), it is clear that \(T \subset G\). Let \(\Phi _G = \Phi (G, T)\); then \(\Phi _G \subset \Phi _H\), and the complement \(\Phi _V := \Phi _H  \Phi _G\) is the set of weights for the action of T on V. The Weyl group \(W_G := W(G, T)\) is the subgroup of \(W_H := W(H, T)\) generated by reflections corresponding to the roots of \(\Phi _G\).
Lemma 2.5
 1.
The stabilizer of s under the action of \(W_H\) on T is given by \({{\mathrm{Stab}}}_{W_H}(s) = \{ w \in W_H \mid w(\Phi _G) = \Phi _G \}\).
 2.There is a split short exact sequence of groups More precisely, let \(S_G \subset \Phi _G\) be a choice of root basis and defineThen \({{\mathrm{Stab}}}_{W_H}(s) \cong W_G \rtimes \Omega \), and the inclusion \(N_{H^\theta }(T) \hookrightarrow H^\theta \) induces an isomorphism \(\Omega \cong H^\theta / G\).$$\begin{aligned} \Omega = \{ w \in W_H \mid w(S_G) = S_G \} \subset {{\mathrm{Stab}}}_{W_H}(s). \end{aligned}$$
We remark that if H is of type \(E_7\), then the group \(H^\theta / G\) has order 2; if H is of type \(E_8\), then \(H^\theta /G\) is trivial.
Proof
For the first item, note that since H is adjoint, \(w\cdot s\) is completely determined by its action on the root spaces \({\mathfrak {h}}_\alpha \). We have that \(w\cdot s\) acts trivially on \({\mathfrak {h}}_\alpha \) if and only if \(\alpha \in w^{1}(\Phi _G)\), and otherwise \(w\cdot s\) acts on \({\mathfrak {h}}_\alpha \) as multiplication by \(1\). For the second item, note that by item 1, the group \({{\mathrm{Stab}}}_{W_H}(s)\) is a subgroup of \({{\mathrm{Aut}}}(\Phi _G) \cong W_G \rtimes D\), where \(D = \{\sigma \in {{\mathrm{Aut}}}(\Phi _G) \mid \sigma (S_G) = S_G\}\). Clearly \(W_G \subset {{\mathrm{Stab}}}_{W_H}(s)\) and \({{\mathrm{Stab}}}_{W_H}(s) \cap D = \Omega \), so \({{\mathrm{Stab}}}_{W_H}(s) \cong W_G \rtimes \Omega \). The isomorphism with \(H^\theta /G\) follows from [11, Sect. 2.2]. \(\square \)
2.2 Transverse slices over Open image in new window
We consider these affine subspaces for the \({\mathfrak {s}}{\mathfrak {l}}_2\)triples corresponding to two conjugacy classes of nilpotent elements, namely the regular and subregular classes.
Proposition 2.6
 1.
There exists a unique normal \({\mathfrak {s}}{\mathfrak {l}}_2\)triple containing E. Let \(\kappa \) be the Kostant section associated to this \({\mathfrak {s}}{\mathfrak {l}}_2\)triple. Then \(\pi _\kappa \) is an isomorphism.
 2.Let \(b \in B(k)\), and let \(\kappa _b = (\pi _\kappa )^{1}(b)\). If \(\Delta (b) \ne 0\), then \(V_b\) forms a single \(G(k^s)\)orbit. Consequently, there is a canonical bijectionwhere the G(k)orbit of \(\kappa _b \in V_b(k)\) corresponds to the neutral element of \(H^1(k, Z_G(\kappa _b))\).$$\begin{aligned} G(k) \backslash V_b(k) \cong \ker [ H^1(k, Z_G(\kappa _b)) \rightarrow H^1(k, G) ], \end{aligned}$$
Proof
The first part follows from work of Kostant and Rallis as applied in [25]: see especially lemmas 2.17 and 3.5. The second part follows from [2, Proposition 1] as applied in [25, Proposition 4.13]. \(\square \)
For \(b \in B(k)\), we continue to write \(\kappa _b\) for the fibre over b. We observe that if H has type \(E_7\), then there are two Gconjugacy classes of regular nilpotent elements in V. If H has type \(E_8\), then there is a single Gconjugacy class of regular nilpotent elements (see [25, Corollary 2.25]). In either case, two regular nilpotent elements \(E, E' \in V(k)\) are G(k)conjugate if and only if they are \(G(k^s)\)conjugate (see e.g. [25, Lemma 2.14]). Combined with the first part of Proposition 2.6, this implies a strong uniqueness property for the sections \(\kappa \rightarrow B\):
Corollary 2.7
 1.
We have \(\kappa = \kappa '\) if and only if \(\kappa _0 = \kappa '_0\).
 2.
The sections \(\kappa \) and \(\kappa '\) are G(k)conjugate if and only if \(\kappa _0\) and \(\kappa '_0\) lie in the same \(G(k^s)\)orbit in V.
Next recall that V contains a subregular nilpotent element e (by definition, this means that e is nilpotent and \(\dim {{\mathrm{Stab}}}_G(e) = 1\); the existence of subregular nilpotents in V is proved in [25, Proposition 2.27]). We now discuss the sections corresponding to such an element.
Theorem 2.8
 1.
The fibres of \(X \rightarrow B\) are reduced connected affine curves. If \(b \in B(k)\), then \(X_b\) is smooth if and only if \(\Delta (b) \ne 0\).
 2.
Let \(b \in B(k)\), and suppose that \(\Delta (b) \ne 0\). Let \(Y_b\) denote the smooth projective completion of \(X_b\), and let \(J_b = {{\mathrm{Pic}}}^0 Y_b\) be the Jacobian of \(Y_b\). There is a canonical isomorphism \(J_b[2] \cong Z_{G}(\kappa _b)\) of finite étale kgroups, where \(\kappa \) is any choice of Kostant section.
Proof
For the first part, see [25, Theorem 3.8] and [25, Corollary 3.16]. For the second part, see [25, Corollary 4.12]. \(\square \)
The next two theorems identify the fibres of the morphism \(X \rightarrow B\) in Theorem 2.8 when H has type \(E_7\) or \(E_8\). We find it convenient to split into cases.
Theorem 2.9
 1.We may choose homogeneous generators \(c_2, c_6, c_8, c_{10}, c_{12}, c_{14}, c_{18}\) of \(k[V]^G\) and functions \(x, y \in k[X]\) so that k[X] is isomorphic to a polynomial ring in the elements \(c_2, {c_6, c_8, c_{10}, c_{12}}, c_{14}, x, y\), and the morphism \(X \rightarrow B\) is determined by the relation (1.2):Moreover, the elements \(c_2, c_6, c_8, c_{10}, c_{12}, c_{14}, c_{18}, x, y \in k[X]\) are eigenvectors for the action of \({\mathbb {G}}_m\) on X mentioned above, with weights as in the following table:$$\begin{aligned} y^3 = x^3 y + c_{10} x^2 + x(c_2 y^2 + c_8 y + c_{14} ) + c_6 y^2 + c_{12} y + c_{18}. \end{aligned}$$
 2.Let \(Y \rightarrow B\) denote the natural compactification of \(X \rightarrow B\) as a family of plane quartic curves, given in homogeneous coordinates asThis compactification has two sections \(P_1\) and \(P_2\) at infinity, given by the equations \([x_0 : y_0 : z_0] = [0 : 1 : 0]\) and \([x_0 : y_0 : z_0] = [1 : 0 : 0]\) respectively (note that \(P_1\) is a flex point). Assume that under the bijection of [25, Lemma 4.14] the section corresponding to E is \(P_1\). Then for each \(b \in B(k)\) such that \(\Delta (b) \ne 0\), the following diagram commutes: where the maps in the diagram are specified as follows. The top arrow \(\iota _b\) is induced by the inclusion \(X \hookrightarrow V\). The left arrow \(\eta _b\) is the restriction of the Abel–Jacobi map \(P \mapsto [(P)  (P_1)]\). To define \(\gamma _b\), we use Proposition 2.6 to obtain an injective homomorphism to \(G(k) \backslash V_b(k) \rightarrow H^1(k, Z_G(\kappa _b))\), and then compose with the identification \(Z_G(\kappa _b) \cong J_b[2]\) of Theorem 2.8. The bottom arrow \(\delta _b\) is the connecting homomorphism associated to the Kummer exact sequence$$\begin{aligned} y_0^3 z_0= & {} x_0^3 y_0 + c_{10} x_0^2 z_0^2 + x_0\left( c_2 y_0^2 z_0 + c_8 y_0 z_0^2 + c_{14} z_0^3\right) \\&\quad + c_6 y_0^2 z_0^2 + c_{12} y_0 z_0^3 + c_{18}z_0^4. \end{aligned}$$
Proof
In this theorem and the next, the first part (i.e. the explicit determination of the family X) is carried out in [25, Theorem 3.8], the weights for the \({\mathbb {G}}_m\) action are given in [25, Proposition 3.6], and the second part is the content of [25, Theorem 4.15]. \(\square \)
We note that, having fixed a choice of regular nilpotent E, we can always assume, after possibly replacing e by a \(H^\theta (k)\)conjugate, that E corresponds to \(P_1\) under the bijection of [25, Lemma 4.14] referred to in the second part of Theorem 2.9.
Theorem 2.10
 1.We may choose homogeneous generators \(c_2, c_8, c_{12}, c_{14}, c_{18}, c_{20}, c_{24}, c_{30}\) of \(k[V]^G\) and functions \(x, y \in k[X]\) so that k[X] is isomorphic to a polynomial ring in the elements \(c_2, {c_8, c_{12}, c_{14}, c_{18}, c_{20}}, c_{24}, x, y\), and the morphism \(X \rightarrow B\) is determined by the relation (1.3):Moreover, the elements \(c_2, c_8, c_{12}, c_{14}, c_{18}, c_{20}, c_{24}, c_{30}, x, y \in k[X]\) are eigenvectors for the action of \({\mathbb {G}}_m\) on X mentioned above, with weights as in the following table:$$\begin{aligned} y^3 = x^5 + y(c_2 x^3 + c_8 x^2 + c_{14} x + c_{20} ) + c_{12} x^3 + c_{18} x^2 + c_{24} x + c_{30}. \end{aligned}$$
 2.Let \(Y \rightarrow B\) denote the compactification of \(X \rightarrow B\) described in [25, Lemma 4.9]. Let \(P : B \rightarrow Y\) denote the unique section at infinity (so that \(Y = X \cup P\)). Then for each \(b \in B(k)\) such that \(\Delta (b) \ne 0\), the following diagram commutes: where the maps in the diagram are specified as follows. The top arrow \(\iota _b\) is induced by the inclusion \(X \hookrightarrow V\). The left arrow \(\eta _b\) is the restriction of the Abel–Jacobi map \(Q \mapsto [(Q)  (P)]\). To define \(\gamma _b\), we use Proposition 2.6 to obtain an injective homomorphism to \(G(k) \backslash V_b(k) \rightarrow H^1(k, Z_G(\kappa _b))\), and then compose with the identification \(Z_G(\kappa _b) \cong J_b[2]\) of Theorem 2.8. The bottom arrow \(\delta _b\) is the connecting homomorphism associated to the Kummer exact sequence
Lemma 2.11
In Case \(\mathbf {E_7}\), suppose \(b \in B(k)\) is such that \(\Delta (b) \ne 0\). Then \(\delta _b([(P_2)  (P_1)])\) is in the image of \(G(k)\backslash V_b(k)\) under \(\gamma _b\), and \(\delta _b([(P_2)  (P_1)])\) is nontrivial if and only if \(H^0(k, Z_G(\kappa _b)) = H^0(k, Z_H(\kappa _b)[2])\).
Proof
Therefore to prove the claim we must show that \(\gamma _b(\kappa _b')\) is equal to the image in \(H^1(k, Z_G(\kappa _b))\) of the nontrivial element of \(\pi _0(H^\theta )\) 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 \(g \in G(k^s)\) such that \(\kappa '_b = g\kappa _b\). Then the cohomology class \(\gamma _b(\kappa _b')\) is represented by the cocycle \(\sigma \mapsto g^{1}( {}^\sigma g)\). But \(c := g^{1}w \in Z_{H^\theta }(\kappa _b) = C[2]\) is a lift of the nontrivial element of \(\pi _0(H^\theta )\), so the claim follows from the fact that \({}^\sigma c c^{1} = ({}^\sigma c c^{1})^{1}= g^{1}({}^\sigma g)\) for all \(\sigma \in {{\mathrm{Gal}}}(k^s / k)\).
We have established the claim, and the first part of the lemma. To finish the the proof, we note that \(\delta _b([(P_2)  (P_1)])\) is nontrivial if and only if the connecting homomorphism \(\pi _0(H^\theta ) \rightarrow H^1(k, Z_G(\kappa _b))\) is injective. By exactness, this is equivalent to the surjectivity of the map \(H^0(k, Z_G(\kappa _b)) \rightarrow H^0(k, C[2])\), which is exactly the criterion given in the statement of the lemma. \(\square \)
Corollary 2.12
In Case \(\mathbf {E_7}\), let \(b \in B(k)\) be such that \(\Delta (b) \ne 0\), and let \(C = Z_H(\kappa _b)\). Suppose that the map \({{\mathrm{Gal}}}(k^s / k) \rightarrow W(H_{k^s}, C_{k^s})\) induced by the action of \({{\mathrm{Gal}}}(k^s/k)\) on \(C_{k^s}\) is surjective. Then \(\delta _b([(P_2)(P_1)])\) is nontrivial in \(H^1(k, J_b[2])\).
Proof
By the lemma, it is equivalent to show that the map \(H^0(k, Z_G(\kappa _b)) \rightarrow H^0(k, C[2])\) is surjective. We have \(H^0(k, C[2]) = C^{W(H, C)}[2](k) = Z_H[2](k)\). Since the group H is adjoint, the centre \(Z_H\) is trivial, so the map \(H^0(k, Z_G(\kappa _b)) \rightarrow H^0(k, C[2])\) is clearly surjective. \(\square \)
2.3 Reducibility conditions
We now define the notion of kreducibility and study the properties of kreducible elements of V(k).
Definition 2.13
Let \(v \in V\). We say that v is kreducible if \(\Delta (v) = 0\) or if v is G(k)conjugate to an element of a Kostant section. Otherwise, we say that v is kirreducible.
Lemma 2.14
 1.
There exist rational numbers \(a_1, \dots , a_r\) not all equal to zero such that if \(\alpha \in \Phi _V\) and \(v_\alpha \ne 0\), then \(\sum a_i n_i(\alpha ) \le 0\).
 2.
There exists \(w \in \Omega \) such that \(v_\alpha = 0\) if \(\alpha \in w(\Phi _V^+  S_H)\).
(We recall that the subgroup \(\Omega \subset W_H\) 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 \(v \in V\) and there exists a nontrivial cocharacter \(\lambda \in X_*(T)\) such that v has no (strictly) positive weights with respect to \(\lambda \), then \(\Delta (v) = 0\). Let \(\{\check{\omega }_1, \ldots , \check{\omega }_r\} \subset X_*(T) \otimes {\mathbb {Q}}\) be the basis dual to the basis \(\{\beta _1, \dots , \beta _r\}\) of \(X^*(T) \otimes {\mathbb {Q}}\), and let \(\lambda = \sum _{i=1}^r a_i \check{\omega }_i\). Then there exists a positive integer m such that \(m\lambda \in X_*(T)\). The weights of v with respect to \(m\lambda \) are exactly the values \(\langle \alpha , m\lambda \rangle = m\sum _{i=1}^r a_i n_i(\alpha )\) for those \(\alpha \in \Phi _V\) such that \(n_i(\alpha ) \ne 0\), so v has no positive weights with respect to \(m\lambda \).
For the second item, let \(E = \sum _{\alpha \in S_H} e_\alpha \), where each \(e_\alpha \) 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 \({\mathfrak {s}}{\mathfrak {l}}_2\)triple, which in turn determines a Kostant section \(\kappa \subset V\) (see Proposition 2.6). Suppose that the vector \(v \in V\) satisfies the condition \(v_\alpha = 0\) if \(\alpha \in \Phi _V^+  S_H\). We may assume that if \(\alpha \in S_H\), then \(v_\alpha \ne 0\); 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] shows that v is G(k)conjugate to an element of \(\kappa \), hence is kreducible.
Now suppose that there is a nontrivial element \(w \in \Omega \) such that the vector \(v \in V\) satisfies the condition \(v_\alpha = 0\) if \(\alpha \in w(\Phi _V^+  S_H)\). We can again assume that \(v_\alpha \ne 0\) if \(\alpha \in w(S_H)\). Let \(E' = \sum _{\alpha \in w(S_H)} e_\alpha \), and let \(\kappa '\) be the Kostant section corresponding to \(E'\). Since the group \(H^\theta (k)\) acts simply transitively on the set of regular nilpotents of V ([25, Lemma 2.14]), there is a unique element \(x \in H^\theta (k)\) such that \(x \cdot E' = E\). Then x normalizes the torus T, since \({\mathfrak {t}}= {{\mathrm{Lie}}}(T)\) is the unique Cartan subalgebra of \({\mathfrak {h}}\) containing the semisimple parts of the normal \({\mathfrak {s}}{\mathfrak {l}}_2\)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 \(H^\theta (k)\) 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 \(\kappa \), hence that v is G(k)conjugate to an element of \(\kappa '\). \(\square \)
Proposition 2.15
 1.
There exists \(w \in \Omega \) such that \(w(\Phi _V^+  S_H) \subset M\).
 2.
There exist integers \(a_1, \dots , a_r\) not all equal to zero such that if \(\alpha \in \Phi _V\) and \(\sum _{i=1}^r a_i n_i(\alpha ) > 0\), then \(\alpha \in M\).
 3.There exist \(\beta \in S_G\), \(\alpha \in \Phi _V  M\), and integers \(a_1, \dots , a_r\) not all equal to zero such that the following conditions hold:
 (a)
We have \(\{\gamma \pm \beta \mid \gamma \in M\} \cap \Phi _V \subset M\).
 (b)
\(\alpha  \beta \in \Phi _V  M\).
 (c)
If \(\gamma \in \Phi _V\) and \(\sum _{i=1}^r a_i n_i(\gamma ) > 0\), then \(\gamma \in M \cup \{ \alpha \}\).
 (a)
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 of V(M)(k) is kreducible. Let \(v \in V(M)(k)\). If \(v_\alpha = 0\), then \(v \in V(M \cup \{ \alpha \})(k)\), and so v is kreducible by the second part of the proposition. We can therefore assume that \(v_\alpha \ne 0\).
Let \(V_{M} = \{ v \in V \mid v_\gamma = 0 \text { for all } \gamma \in \Phi _V  M \}\). Then there is a Tinvariant direct sum decomposition \(V = V(M) \oplus V_{M}\). Fix a homomorphism \(\mathrm {SL}_2 \rightarrow G_\beta \) where \(G_\beta \) is the subgroup of G generated by the root groups corresponding to \(\beta \) and \(\beta \). Condition (a) implies that the decomposition \(V = V(M) \oplus V_{M}\) is \(G_\beta \)invariant. Since the ambient group H is simply laced, the \(\beta \)root string through \(\alpha \) has length two, and thus \({\mathfrak {h}}_\alpha \oplus {\mathfrak {h}}_{\alpha \beta }\) is an irreducible \(G_\beta \)submodule of V. The existence of an irreducible representation of degree two implies that \(G_\beta \cong \mathrm {SL}_2\).
Since \(\mathrm {SL}_2(k)\) acts transitively on the nonzero vectors in the unique twodimensional irreducible representation of \(\mathrm {SL}_2\), we can find \(g \in G_\beta (k) \subset G(k)\) such that \((g v)_\alpha = 0\). This shows that \(gv \in V(M \cup \{ \alpha \})\), hence that v is kreducible, as required. \(\square \)
2.4 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 \(\Phi _H^+\) be the set of positive roots corresponding to our choice of root basis \(S_H\). Similarly, we define \(\Phi _H^ \subset \Phi _H\) to be the subset of negative roots. We note that there exists a unique choice of root basis \(S_G\) of \(\Phi _G\) such that the positive roots \(\Phi _G^+\) determined by \(S_G\) are given by \(\Phi _G^+ = \Phi _G \cap \Phi _H^+\). Indeed, this follows from a consideration of Weyl chambers: the Weyl chambers for H (resp. G) are in bijection with the root bases of \(\Phi _H\) (resp. \(\Phi _G\)), 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 \(\Phi _G^\) determined by \(S_G\) is given by \(\Phi _G^ = \Phi _G \cap \Phi _H^\).
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 \(\mathbf {E_7}\) and \(\mathbf {E_8}\). We number the simple roots of H and G as in Bourbaki [7, Planches].
2.4.1 Case \({\mathbf {E_7}}\)
2.4.2 Case \({\mathbf {E_8}}\)
3 Integral structures, measures, and orbits

the group H over k, together with split maximal torus \(T \subset H\), root basis \(S_H \subset X^*(T)\), involution \(\theta = {{\mathrm{Ad}}}\check{\rho }(1)\), and Lie algebra \({\mathfrak {h}}= {{\mathrm{Lie}}}H\);

the group \(G = (H^\theta )^\circ \) and its representation on \(V = {\mathfrak {h}}^{d \theta = 1}\), together with a root basis \(S_G \subset X^*(T)\) and Lie algebra \({\mathfrak {g}}= {{\mathrm{Lie}}}G\);

the categorical quotient Open image in new window and quotient map \(\pi : V \rightarrow B\);

the discriminant polynomial \(\Delta \in k[B]\).
3.1 Integral structures and measures
Our choice of Chevalley basis of \({\mathfrak {h}}\) with root vectors \(\{e_\alpha \mid \alpha \in \Phi _H\}\) determines a Chevalley basis of \({\mathfrak {g}}\), with root vectors \(\{e_\alpha \mid \alpha \in \Phi _G\}\). It hence determines \({\mathbb {Z}}\)forms \({\mathfrak {h}}_{\mathbb {Z}}\subset {\mathfrak {h}}\) and \({\mathfrak {g}}_{\mathbb {Z}}\subset {\mathfrak {g}}\) (in the sense of [4]). Moreover, \({\mathcal {V}}= V \cap {\mathfrak {h}}_{\mathbb {Z}}\) is an admissible \({\mathbb {Z}}\)lattice that contains E.
We extend G to a group scheme over \({\mathbb {Z}}\) given by the Zariski closure of the group G in \(\mathrm {GL}({\mathcal {V}})\). By abuse of notation, we also refer to this \({\mathbb {Z}}\)group scheme as G. Then the group \(G({\mathbb {Z}})\) acts on the lattice \({\mathcal {V}}({\mathbb {Z}}) \subset V({\mathbb {Q}})\). The Cartan decomposition \(V = \oplus _{\alpha \in \Phi _V} {\mathfrak {h}}_\alpha \) is defined over \({\mathbb {Z}}\), so extends to a decomposition \({\mathcal {V}}= \oplus _{\alpha \in \Phi _V} {\mathcal {V}}_\alpha \). Since there exists a subregular nilpotent element in \(V = {\mathcal {V}}({\mathbb {Q}})\), we may choose a subregular nilpotent element \(e \in {\mathcal {V}}({\mathbb {Z}})\). In Case \(\mathbf {E_7}\), we impose the additional condition that E corresponds to \(P_1\) in the sense described in Theorem 2.9.
Fix a maximal compact subgroup \(K \subset G({\mathbb {R}})\). Let \(P = TN \subset G\) be the Borel subgroup corresponding to the root basis \(S_G\), and let \(\overline{P} = T\overline{N} \subset G\) be the opposite Borel subgroup. Given \(c \in {\mathbb {R}}\), we define \(T_c = \{ t \in T({\mathbb {R}})^\circ \mid \beta (t) \le c \text { for all } \beta \in S_G \}\).
Proposition 3.1
We can find a compact subset \(\omega \subset \overline{N}({\mathbb {R}})\) and a constant \(c > 0\) such that \(G({\mathbb {A}}) = G({\mathbb {Q}}) \cdot (G(\widehat{{\mathbb {Z}}}) \times \mathfrak {S})\), where \(\mathfrak {S}= \omega T_cK\) and \({\mathbb {A}}\) denotes the adeles of \({\mathbb {Q}}\).
Proof
It suffices to show that \(G({\mathbb {A}}^\infty ) = G({\mathbb {Q}}) \cdot G(\widehat{{\mathbb {Z}}})\) (where \({\mathbb {A}}^\infty = \prod '_p {\mathbb {Q}}_p\) denotes the ring of finite adeles of \({\mathbb {Q}}\)) and that we can choose \(\mathfrak {S}\) so that \(G({\mathbb {Z}}) \cdot \mathfrak {S}= G({\mathbb {R}})\). This is true: see [3, Sect. 6], [17, Theorem 4.15], and [17, Theorem 8.11, Corollary 2]. \(\square \)
Henceforth we fix a choice of \(\mathfrak {S}= \omega T_cK\) as in Proposition 3.1.
We define \({\mathcal {X}}= {{{\mathrm{Spec}}}{\mathbb {Z}}[x, y, c_2, c_6, \dots , c_{14}]}\) in Case \(\mathbf {E_7}\) (resp. \({{\mathrm{Spec}}}{\mathbb {Z}}[x, y, c_2, c_8, \dots , c_{24}]\) in Case \(\mathbf {E_8}\)). Thus \({\mathcal {X}}\) is isomorphic to affine space \({\mathbb {A}}_{\mathbb {Z}}^{r+1}\), and the morphism \(X \rightarrow B\) naturally extends to a morphism \({\mathcal {X}}\rightarrow {\mathcal {B}}\), still given in coordinates by Eq. (1.2) in Case \(\mathbf {E_7}\) (resp. (1.3) in Case \(\mathbf {E_8}\)). For any ring R and any subset \(A \subset {\mathcal {V}}(R)\), we write \(A^\text {reg.ss.}\) for \(\{a \in A \mid \Delta (a) \ne 0\}\). Similarly if \(A' \subset {\mathcal {B}}(R)\) then we write \((A')^\text {reg.ss.}\) for the set \(\{ a \in A' \mid \Delta (a) \ne 0 \}\).
Fix a leftinvariant top form \(\omega _G\) on G; it is determined uniquely up to multiplication by \({\mathbb {Z}}^\times = \{ \pm 1 \}\). For any place v of \({\mathbb {Q}}\), we define a Haar integral on \(G({\mathbb {Q}}_v)\) using the volume element \(dg = \omega _G _v\).
Lemma 3.2
There exists a rational number \(W_0 \in {\mathbb {Q}}^\times \) with the following property: let \(k' / {\mathbb {Q}}\) be any field extension, and let \({\mathfrak {c}}\subset V(k')\) be a Cartan subspace. Let \(\mu _{\mathfrak {c}}: G_{k'} \times {\mathfrak {c}}\rightarrow V_{k'}\) be the natural action map. Then \(\mu _{\mathfrak {c}}^*\omega _V = W_0 \omega _G \wedge \pi _{\mathfrak {c}}^*\omega _B\).
Proof
The proof is identical to that of [26, Proposition 2.13]. \(\square \)
Proposition 3.3
 1.Let \(\phi : {\mathcal {V}}({\mathbb {Z}}_p)^\text {reg.ss.} \rightarrow {\mathbb {R}}\) be a function of compact support that is locally constant (resp. continuous) and invariant under the action of \(G({\mathbb {Z}}_p)\). Then the function \(F_\phi : B({\mathbb {Q}}_p)^\text {reg.ss.} \rightarrow {\mathbb {R}}\) defined by the formulais of compact support and locally constant (resp. continuous), and we have the formula$$\begin{aligned} F_\phi (b) = \sum _{v \in G({\mathbb {Z}}_p) \backslash {\mathcal {V}}_b({\mathbb {Z}}_p)} \frac{\phi (v)}{\# {{\mathrm{Stab}}}_{G({\mathbb {Z}}_p)}(v)} \end{aligned}$$$$\begin{aligned} \int _{v \in {\mathcal {V}}({\mathbb {Z}}_p)} \phi (v) \, dv = W_0_p {{\mathrm{vol}}}(G({\mathbb {Z}}_p)) \int _{b \in {\mathcal {B}}({\mathbb {Z}}_p)} F_\phi (b) \, db. \end{aligned}$$
 2.Define a function \(m_p : {\mathcal {V}}({\mathbb {Z}}_p)^\text {reg.ss.} \rightarrow {\mathbb {R}}\) by the formulaThen \(m_p\) is locally constant.$$\begin{aligned} m_p(v) = \sum _{v' \in G({\mathbb {Z}}_p) \backslash (G({\mathbb {Q}}_p) \cdot v \cap {\mathcal {V}}({\mathbb {Z}}_p))} \frac{ \# {{\mathrm{Stab}}}_{G({\mathbb {Q}}_p)}(v) }{ \# {{\mathrm{Stab}}}_{G({\mathbb {Z}}_p)}(v')}. \end{aligned}$$
 3.Let \(\psi : {\mathcal {V}}({\mathbb {Z}}_p)^\text {reg.ss.} \rightarrow {\mathbb {R}}\) be a continuous function of compact support that is \(G({\mathbb {Q}}_p)\)invariant, in the sense that if \(v, v' \in {\mathcal {V}}({\mathbb {Z}}_p)\), \(g \in G({\mathbb {Q}}_p)\), and \(gv = v'\), then \(\psi (v) = \psi (v')\). Then we have the formula$$\begin{aligned} \int _{v \in {\mathcal {V}}({\mathbb {Z}}_p)} \psi (v) \, dv =  W_0 _p {{\mathrm{vol}}}(G({\mathbb {Z}}_p)) \int _{b \in {\mathcal {B}}({\mathbb {Z}}_p)} \sum _{v \in G({\mathbb {Q}}_p) \backslash {\mathcal {V}}_b({\mathbb {Z}}_p)} \frac{m_p(v)\psi (v)}{\# {{\mathrm{Stab}}}_{G({\mathbb {Q}}_p)}(v)} \, db. \end{aligned}$$
Proof
3.2 Selmer elements and integral orbits
We now discuss the construction of elements of \({\mathcal {V}}({\mathbb {Z}}_p)\) and \({\mathcal {V}}({\mathbb {Z}})\) 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 \(X \rightarrow V\) is defined over \({\mathbb {Q}}\). After we fix integral structures, this means that a point of \({\mathcal {X}}({\mathbb {Z}})\) (resp. \({\mathcal {X}}({\mathbb {Z}}_p)\)) defines an element of \({\mathcal {V}}({\mathbb {Z}})\) (resp. \({\mathcal {V}}({\mathbb {Z}}_p)\)), after possibly clearing a bounded denominator. The main problem in this section is therefore to show if \(b \in {\mathcal {B}}({\mathbb {Z}}_p)\) is of nonzero discriminant, then a class in \(J_b({\mathbb {Q}}_p) / 2 J_b({\mathbb {Q}}_p)\) which is represented by a point of \(Y_b({\mathbb {Q}}_p)\) is in fact represented either by a point \({\mathcal {X}}_b({\mathbb {Z}}_p)\), or by a point at infinity.
Lemma 3.4
 1.
For any prime p and any \(b \in {\mathcal {B}}({\mathbb {Z}}_p)\), we have \(N_0\cdot \kappa _b \in {\mathcal {V}}({\mathbb {Z}}_p)\).
 2.
In Case \(\mathbf {E_7}\), let \(w \in \Omega \) be the nontrivial element and let \(\kappa '\) denote the Kostant section corresponding to the regular nilpotent element \(E' = \sum _{\alpha \in S_H} e_{w \alpha }\). Then for any prime p and for any \(b \in {\mathcal {B}}({\mathbb {Z}}_p)\), we have \(N_0 \cdot \kappa '_b \in {\mathcal {V}}({\mathbb {Z}}_p)\).
 3.
For any prime p and any \(x \in {\mathcal {X}}({\mathbb {Z}}_p)\), we have \(N_0\cdot x \in {\mathcal {V}}({\mathbb {Z}}_p)\).
 4.
If \(b \in N_0^2\cdot {\mathcal {B}}({\mathbb {Z}})\), then \(b \in \pi ({\mathcal {V}}({\mathbb {Z}}))\).
Proof
This follows from the existence of the contracting \({\mathbb {G}}_m\)actions on \(\kappa \), \(\kappa '\), and \({\mathcal {X}}\), cf. [26, Lemma 2.8].\(\square \)
Lemma 3.5
Proof
We first claim that if \(b \in 2^4 {\mathcal {B}}({\mathbb {Z}}_p)\), then every divisor class in the image of the map \(Y_b({\mathbb {Q}}_p) \rightarrow J_b({\mathbb {Q}}_p) / 2 J_b({\mathbb {Q}}_p)\) is represented by either the zero divisor, the divisor \(P_2  P_1\), or a divisor of the form \(P  P_1\) for some \(P \in {\mathcal {X}}_b({\mathbb {Z}}_p)\).
If \(P \in Y_b({\mathbb {Q}}_p)\), then we write \(\overline{P}\) for the image of P in \({\mathcal {Y}}_b({\mathbb {F}}_p)\). The special fibre \({\mathcal {Y}}_{b, {\mathbb {F}}_p}\) is reduced, and has at most two irreducible components, which are geometrically irreducible. Moreover, if there are two irreducible components, then \(\overline{P}_1\) and \(\overline{P}_2\) lie on distinct irreducible components. Indeed, due to the presence of the contracting \({\mathbb {G}}_m\)action, any property of the morphism \({\mathcal {Y}}\rightarrow {\mathcal {B}}\) which is open on the base can be checked in the central fibre. Thus [23, Tag 0C0E] implies that all of the fibres of \({\mathcal {Y}}\) 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 \({\mathcal {Y}}_{b, {\mathbb {F}}_p}\) is geometrically irreducible.
Let \({\mathcal {J}}_b = {{\mathrm{Pic}}}^0_{{\mathcal {Y}}_b / {\mathbb {Z}}_p}\) be the open subscheme of \({{\mathrm{Pic}}}_{{\mathcal {Y}}_b / {\mathbb {Z}}_p}\) corresponding to those invertible sheaves that are fibrewise of degree 0 on each irreducible component (see [5, Sect. 8.4]). Then \({\mathcal {J}}_b\) is a smooth and separated scheme over \({\mathbb {Z}}_p\) (see [5, Sect. 9.4, Theorem 2]). We note that if \(Q \in {\mathcal {J}}_b({\mathbb {Z}}_p)\) has trivial image in \({\mathcal {J}}_b({\mathbb {Z}}_p / 2^3 p {\mathbb {Z}}_p)\), then Q is divisible by 2 in \({\mathcal {J}}_b({\mathbb {Z}}_p)\) (this follows from [21, Theorem 6.1] and its generalization [8, Proposition 3.1]).
Let \(P = (x, y) \in Y_b({\mathbb {Q}}_p)\). To prove the claim, it suffices to show that if \(P \not \in {\mathcal {X}}_b({\mathbb {Z}}_p)\), then one of the divisor classes \([(P)  (P_1)]\) or \([(P)  (P_2)]\) is divisible by 2 in \(J_b({\mathbb {Q}}_p)\). We can assume that \(xy \ne 0\). We note that if \(P \not \in {\mathcal {X}}_b({\mathbb {Z}}_p)\), then (at least) one of x, y must be nonintegral. 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 \(u, v \in {\mathbb {Z}}_p^\times \) 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 now show how the claim implies the lemma. We drop our assumption on the parity of p, and take \(b = N_0^2 c\), where \(c \in 2^4 {\mathcal {B}}({\mathbb {Z}}_p)\). Given a class \(\phi \) in \(H^1({\mathbb {Q}}_p, J_c[2])\), if \(\phi \) is in the image of \(Y_c({\mathbb {Q}}_p)\), then \(\phi \) is represented by either \(P_1\), \(P_2\), or an element of \({\mathcal {X}}_c({\mathbb {Z}}_p)\). Let \(\phi '\) denote the corresponding class in \(H^1({\mathbb {Q}}_p, J_b[2])\). If \(P_1\) is a representative, then \(\kappa _b \in V_b({\mathbb {Q}}_p)\) represents the corresponding rational orbit. By Lemma 3.4, we have \(\kappa _b = N_0 \cdot \kappa _c \in {\mathcal {V}}({\mathbb {Z}}_p)\), so \(\kappa _b\) is even an integral representative for this rational orbit. If \(P_2\) is a representative, then \(\kappa '_b \in {\mathcal {V}}({\mathbb {Z}}_p)\) is an integral representative, by the same argument.
Suppose instead that \(\phi \) is represented by a divisor \((P)  (P_1)\), where \(P \in {\mathcal {X}}_c({\mathbb {Z}}_p)\). Then \(\phi '\) is represented by the divisor \((N_0 \cdot P)  (P_1)\), where now \(N_0 \cdot P \in N_0 \cdot {\mathcal {X}}({\mathbb {Z}}_p)\). By Lemma 3.4, we have \(N_0 \cdot {\mathcal {X}}({\mathbb {Z}}_p) \subset {\mathcal {V}}({\mathbb {Z}}_p)\), showing that \(N_0 \cdot P \in {\mathcal {V}}_b({\mathbb {Z}}_p)\) is an integral representative for the rational orbit corresponding to the class \(\phi \). This completes the proof. \(\square \)
Proposition 3.6
Proof
We can therefore choose a vector \(v \in V_b({\mathbb {Q}})\) representing our class c. By Lemma 3.5, for each prime p there exists an element \(g_p \in G({\mathbb {Q}}_p)\) such that \(g_p \cdot v \in {\mathcal {V}}_b({\mathbb {Z}}_p)\). By Proposition 3.1, there is an element \(g \in G({\mathbb {Q}})\) such that \(g_p \in G({\mathbb {Z}}_p) g\) for every prime p. It follows that \(g\cdot v \in {\mathcal {V}}_b({\mathbb {Z}})\), as required. \(\square \)
3.3 Subsets of \(V({\mathbb {R}})\) and \(V({\mathbb {Q}}_p)\)
We conclude this section by constructing some useful subsets of \(V({\mathbb {R}})\) and \(V({\mathbb {Q}}_p)\). We first consider \(V({\mathbb {R}})\). Let \({\mathfrak {c}}_1, \dots , {\mathfrak {c}}_n\) denote representatives of the distinct \(G({\mathbb {R}})\)conjugacy classes of Cartan subspaces of \(V({\mathbb {R}})\). For each \(i \in \{ 1, \dots , n\}\), let \({\mathfrak {c}}_i'\) denote the closed subset of \({\mathfrak {c}}_i^\text {reg.ss.}\) given by \({\mathfrak {c}}_i' = \{ v \in {\mathfrak {c}}_i^{\text {reg.ss.}} \mid {{\mathrm{ht}}}(v) = 1\}\). Arguing as in [26, Sect. 2.9], we can find a cover of \({\mathfrak {c}}_i'\) by finitely many connected semialgebraic open subsets \(U_{ij}\) such that each map \(\pi _{U_{ij}} : U_{ij} \rightarrow \{ b \in B({\mathbb {R}})^\text {reg.ss.} \mid {{\mathrm{ht}}}(b) = 1 \}\) is a homeomorphism onto its image. We write \(L_1, \dots , L_m\) for the sets \(\pi (U_{ij})\) for all ij in any order, and for \(L_k = \pi (U_{ij})\) we set \(s_k := (\pi _{U_{ij}})^{1} : L_k \rightarrow V({\mathbb {R}})^\text {reg.ss.}\). We can extend \(s_k\) to a map \(s_k : \Lambda \cdot L_k \rightarrow V({\mathbb {R}})^\text {reg.ss.}\) by the formula \(s_k (\lambda b) = \lambda s_k(b)\) for any \(\lambda \in \Lambda \), \(b \in L_k\).
Lemma 3.7
Proof
Let \(\mu _k : G({\mathbb {R}}) \times (\Lambda \cdot L_k) \rightarrow V({\mathbb {R}})^\text {reg.ss.}\) be given by \((g, b) \mapsto g \cdot s_k(b)\). Then \(\mu _k\) is a local diffeomorphism onto its image, with fibres of cardinality \(r_k\). By Lemma 3.2 we have \(\mu _k^*\omega _V = W_0 \omega _G \wedge \omega _B\). The displayed formulae follow from this identity. \(\square \)
We now consider \(V({\mathbb {Q}}_p)\).
Lemma 3.8
Proof
We consider instead the curve given by the equation \(y^3 = (x1)(x^3  p^2)\). (This curve can be put into the canonical form (1.1) by a linear change of variable in x.) Let \({\mathcal {Y}}\) be the curve inside \({\mathbb {P}}^2_{{\mathbb {Z}}_p}\) given by the projective closure of this equation, and let \({\mathcal {Z}}\subset {\mathbb {A}}^2_{{\mathbb {Z}}_p}\) denote the complement of the unique point at infinity. It is clear that \({\mathcal {Z}}({\mathbb {Z}}_p) \ne \emptyset \). Moreover, \({\mathcal {Y}}\) has a unique point that is not regular, namely the point corresponding to \((x, y) = (0, 0)\) in the special fibre \({\mathcal {Z}}_{{\mathbb {F}}_p}\).
This singularity can be resolved by blowing up. Let \({\mathcal {Y}}' \rightarrow {\mathcal {Y}}\) denote the blowup at the unique nonregular point of \({\mathcal {Y}}\). Then \({\mathcal {Y}}'\) has exactly 3 nonregular points. The special fibre of \({\mathcal {Y}}'\) has two irreducible components, namely the strict transform of \({\mathcal {Y}}_{{\mathbb {F}}_p}\) and a smooth exceptional divisor. Let \({\mathcal {Y}}'' \rightarrow {\mathcal {Y}}'\) denote the blowup of the 3 nonregular points. Then \({\mathcal {Y}}''\) is regular, and the special fibre \({\mathcal {Y}}''_{{\mathbb {F}}_p}\) has 5 irreducible components: the strict transform \(C_1\) of \({\mathcal {Y}}_{{\mathbb {F}}_p}\), the strict transform \(C_5\) of the exceptional divisor in \({\mathcal {Y}}'_{{\mathbb {F}}_p}\), and the smooth exceptional divisors \(C_2, C_3, C_4\) of the blowup \({\mathcal {Y}}'' \rightarrow {\mathcal {Y}}\).
Lemma 3.9
There exists an open subset \(U_2 \subset {\mathcal {B}}({\mathbb {Z}}_2)\) such that for all \(b \in U_2\), we have \(\Delta (b) \ne 0\), \({\mathcal {X}}_b({\mathbb {Z}}_2) \ne \emptyset \), and the image of the map \({\mathcal {X}}_b({\mathbb {Z}}_2) \rightarrow J_b({\mathbb {Q}}_2) / 2 J_b({\mathbb {Q}}_2)\) does not intersect the subgroup generated by the divisor class \([(P_1)  (P_2)]\) in Case \(\mathbf {E_7}\) (resp. does not contain the identity in Case \(\mathbf {E_8}\)).
Proof
We verified all these properties of the given curves \({\mathcal {X}}_c\) using the ClassGroup functionality in magma [6]. \(\square \)
Lemma 3.10
 1.
For every prime p, there exists an open compact subset \(U_p \subset {\mathcal {B}}({\mathbb {Z}}_p)\) such that for every \(b \in U_p\), \(\Delta (b) \ne 0\) and \({\mathcal {X}}_b({\mathbb {Z}}_p) \ne \emptyset \).
 2.
There exists an integer \(N_3 \ge 1\) such that for every prime \(p > N_3\) and for every \(b \in {\mathcal {B}}({\mathbb {Z}}_p)\) such that \(\Delta (b) \ne 0\), we have \({\mathcal {X}}_b({\mathbb {Z}}_p) \ne \emptyset \).
Proof
For each prime p, it is not difficult to find a point \(c \in {\mathcal {B}}({\mathbb {F}}_p)\) such that \({\mathcal {X}}_{c}\) is smooth and \({\mathcal {X}}_c({\mathbb {F}}_p)\) is nonempty. Taking \(U_p\) to be the preimage of c in \({\mathcal {B}}({\mathbb {Z}}_p)\) establishes the first part of the lemma. The second part follows from Hensel’s Lemma and the Weil bounds; here we are implicitly using the fact, already established in the proof of Lemma 3.5, that for any \(c \in {\mathcal {B}}({\mathbb {F}}_p)\), the irreducible components of \({\mathcal {X}}_{c}\) are geometrically irreducible. \(\square \)
4 Counting points
In Sect. 3 we have defined an algebraic group over \({\mathbb {Z}}\) and a representation \({\mathcal {V}}\), 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 \(G({\mathbb {Z}}) \backslash {\mathcal {V}}({\mathbb {Z}})\) 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 \(L \subset B({\mathbb {R}})\) be one of the subsets \(L_k\) described in Lemma 3.7, and let \(s : L \rightarrow V({\mathbb {R}})\) be the corresponding section. Then L is a connected semialgebraic subset of \(B({\mathbb {R}})\); s is a semialgebraic map; and s(L) has compact closure in \(V({\mathbb {R}})\). The map \(\Lambda \times L \rightarrow B({\mathbb {R}}), (\lambda , \ell ) \mapsto \lambda \cdot \ell \) given by the \({\mathbb {G}}_m\)action on B is an open immersion, and \({{\mathrm{ht}}}(\lambda \cdot \ell ) = \lambda ^{\deg \Delta }\).
For any subset \(A \subset {\mathcal {V}}({\mathbb {Z}})\), we write \(A^\text {irr}\) for the subset of points \(a \in A\) that are \({\mathbb {Q}}\)irreducible, 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
Our proof is very similar to that of [26, Theorem 3.1], except that a significant amount of casebycase computation is required in order to control the contribution of elements that are ‘in the cusp’ (i.e. elements that lie in the codimensionone 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].
Lemma 4.2
Lemma 4.3
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]).
Theorem 4.4
Proposition 4.5
Let \(\alpha _0 \in \Phi _V\) denote the highest root of H with respect to the root basis \(S_H\). Then there exists \(\delta > 0\) such that \(N( S( \alpha _0 ), a) = O(a^{\frac{1}{2} + r / \deg \Delta  \delta })\).
Proof
 1.
\(S(\alpha _0)^{{{\mathrm{irr}}}} \subset \bigcup _{(M_0, M_1) \in \mathcal {C}} S(M_0, M_1)\)
 2.
If \((M_0, M_1) \in \mathcal {C}\), then \(N^*(S(M_0, M_1), a) = O(a^{\frac{1}{2} + r / \deg \Delta  \delta })\).

\(\sum _{\alpha \in M_1} f(a) < \# M_0\)

For each \(1 \le i \le r\), we have \(\sum _{\alpha \in \Phi _G^+} n_i(\alpha )  \sum _{\alpha \in M_0} n_i(\alpha ) + \sum _{\alpha \in M_1} f(\alpha )n_i(\alpha ) > 0\).
Proposition 4.6
Proof
We now state the more refined version of Theorem 4.1 mentioned at the beginning of this section.
Theorem 4.7
Proof
5 Applications to 2Selmer 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 \(\mathbf {E_7}\) and Case \(\mathbf {E_8}\)) in turn.
5.1 Applications in Case \(\mathbf {E_7}\)
Proposition 5.1
 1.
The locus inside \({\mathcal {B}}_k\) above which the morphism \({\mathcal {X}}_k \rightarrow {\mathcal {B}}_k\) is smooth is the complement of an irreducible closed subset of \({\mathcal {B}}_k\) of codimension 1.
 2.The set of points \(b \in {\mathcal {B}}(k)\) for which \({\mathcal {X}}_b\) is smooth is in bijection with the set of equivalence classes of triples \((C, P_1, t)\), where:If b corresponds to \((C, P_1, t)\), then \({\mathcal {X}}_b\) is isomorphic to \(C  \{ P_1, P_2 \}\), where \(P_2 \in C(k)\) is the unique point such that \(3 P_1 + P_2\) is a canonical divisor. For \(\lambda \in k^\times \), the coefficients \(c_i\) satisfy the equality
 (a)
C is a smooth, nonhyperelliptic curve of genus 3 over k.
 (b)
\(P_1 \in C(k)\) 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)
\(t \in T_{P_1} C\) is a nonzero Zariski tangent vector at the point \(P_1\).
$$\begin{aligned} c_i (C, P_1, \lambda t) = \lambda ^{i/2} c_i(C, P_1, t). \end{aligned}$$  (a)
Proof
Part 1 follows from the fact that \({\mathcal {X}}_b\) is smooth if and only if \(\Delta (b) \ne 0\). 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 \(P_2 \in C(k)\) be the point such that \(3 P_1 + P_2\) is a canonical divisor. The Riemann–Roch Theorem shows that \(h^0(C, {\mathcal {O}}_C(3P_1)) = 2\) and \(h^0(C, {\mathcal {O}}_C(2P_1 + P_2)) = 2\). We can therefore find functions \(y, x \in k(C)^\times \), uniquely determined up to addition of constants, such that the polar divisor of y is \(3 P_1\) and the polar divisor of x is \(2 P_1 + P_2\), and such that \(y = z^{3} + \cdots \), \(x = z^{2} + \cdots \) 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 \(P_2\).
If \(k / {\mathbb {Q}}\) is a field extension and \(b \in {\mathcal {B}}(k)\) is such that \({\mathcal {X}}_b\) is smooth, then we write \(Y_b\) for the unique smooth projective completion of \({\mathcal {X}}_b\).
We recall that for \(b \in {\mathcal {B}}({\mathbb {R}})\) we have defined \({{\mathrm{ht}}}(b) = \sup _i  c_i(b) ^{{126} / i}\). This function is homogeneous of degree 126, in the sense that for \(\lambda \in {\mathbb {R}}^\times \), we have \({{\mathrm{ht}}}(\lambda \cdot b) =  \lambda ^{126} {{\mathrm{ht}}}(b)\). (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 \(\Delta \) considered in Sect. 2.1.)
Lemma 5.2
Proof
This is an easy consequence of Theorem 4.4. \(\square \)
Our main theorems are now as follows.
Theorem 5.3
In order to state the next theorem, we observe that if \(b \in {\mathcal {B}}({\mathbb {Q}})\) is such that \({\mathcal {X}}_b\) is smooth, then the 2Selmer set \({{\mathrm{Sel}}}_2(Y_b)\) 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 \({{\mathrm{Sel}}}_2(Y_b)^{\text {triv}}\) for the subset of \({{\mathrm{Sel}}}_2(Y_b)\) consisting of these classes, and note that \(\#{{\mathrm{Sel}}}_2(Y_b)^{\text {triv}} \le 2\), with equality if and only if the divisor class \([(P_2)  (P_1)]\) is not divisible by 2 in \(J_b({\mathbb {Q}})\).
Theorem 5.4
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
Lemma 5.5
Proof
Theorem 5.6
 1.
For every \(b \in {\mathcal {F}}\) and every prime p, we have \({\mathcal {X}}_b({\mathbb {Z}}_p) \ne \emptyset \).
 2.We have$$\begin{aligned} \liminf _{a\rightarrow \infty } \frac{ \# \{ b \in {\mathcal {F}}\mid {\mathcal {X}}_b({\mathbb {Z}}_{(2)}) = \emptyset \} }{ \# \{ b \in {\mathcal {F}}\mid {{\mathrm{ht}}}(b) < a\} } > 1  \epsilon . \end{aligned}$$
Here \({\mathbb {Z}}_{(2)} \subset {\mathbb {Q}}\) denotes the subring of rational numbers of denominator prime to 2. For the sets \({\mathcal {F}}\) constructed in Theorem 5.6, we may say that a positive proportion of the curves \({\mathcal {X}}_b\) (\(b \in {\mathcal {F}}\)) have integral points everywhere locally, but no integral points globally.
Proof
 1.
For each \(b \in U_2\), \(\Delta (b) \ne 0\) and the image of the map \({\mathcal {X}}_b({\mathbb {Z}}_2) \rightarrow J_b({\mathbb {Q}}_2) / 2 J_b({\mathbb {Q}}_2)\) does not intersect the subgroup generated by \([(P_1)  (P_2)]\).
 2.
For every prime p and for every \(b \in U_p\) such that \(\Delta (b) \ne 0\), the set \({\mathcal {X}}_b({\mathbb {Z}}_p)\) is nonempty.
 3.
For every sufficiently large prime p, \(U_p = {\mathcal {B}}({\mathbb {Z}}_p)\).
 4.We have$$\begin{aligned} \liminf _{X \rightarrow \infty } \frac{ \# \{ b \in {\mathcal {F}}\mid {{\mathrm{ht}}}(b)< a\text { and } {{\mathrm{Sel}}}_2(Y_b) = {{\mathrm{Sel}}}_2(Y_b)^{\text {triv}}\}}{\# \{ b \in {\mathcal {F}}\mid {{\mathrm{ht}}}(b) < a\}} > 1  \epsilon . \end{aligned}$$
5.2 Applications in Case \(\mathbf {E_8}\)
Proposition 5.7
 1.
The locus inside \({\mathcal {B}}_k\) above which the morphism \({\mathcal {X}}_k \rightarrow {\mathcal {B}}_k\) is smooth is the complement of an irreducible closed subset of \({\mathcal {B}}_k\) of codimension 1.
 2.The set of points \(b \in {\mathcal {B}}(k)\) for which \({\mathcal {X}}_b\) is smooth is in bijection with the set of equivalence classes of triples (C, P, t), where:If b corresponds to (C, P, t), then \({\mathcal {X}}_b\) is isomorphic to \(C  \{ P \}\). For \(\lambda \in k^\times \), the coefficients \(c_i\) satisfy the equality
 (a)
C is a smooth, nonhyperelliptic curve of genus 4 over k.
 (b)
\(P \in C(k)\) is a point such that 6P is a canonical divisor and \(h^0(C, {\mathcal {O}}_C(3P)) = 2\).
 (c)
\(t \in T_{P} C\) is a nonzero Zariski tangent vector at the point P.
$$\begin{aligned} c_i (C, P, \lambda t) = \lambda ^i c_i(C, P, t). \end{aligned}$$  (a)
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.
Our main theorems in Case \(\mathbf {E_8}\) are as follows. We omit the proofs since they are similar, and simpler, than those in Case \(\mathbf {E_7}\) in the previous section.
Theorem 5.8
Theorem 5.9
Theorem 5.10
 1.
For every \(b \in {\mathcal {F}}\) and every prime p, we have \({\mathcal {X}}_b({\mathbb {Z}}_p) \ne \emptyset \).
 2.We have$$\begin{aligned} \liminf _{a\rightarrow \infty } \frac{ \# \{ b \in {\mathcal {F}}\mid {\mathcal {X}}_b({\mathbb {Z}}_{(2)}) = \emptyset \} }{ \# \{ b \in {\mathcal {F}}\mid {{\mathrm{ht}}}(b) < a\} } > 1  \epsilon . \end{aligned}$$
Footnotes
 1.
We note that the definition of a Kostant section is often more general than the one stated here, but in this paper we restrict our attention to sections of this form.
 2.
These Mathematica notebooks may be found at https://www.dpmms.cam.ac.uk/~jat58/E7CuspData.nb and https://www.dpmms.cam.ac.uk/~jat58/E8CuspData.nb respectively.
Notes
Acknowlegements
During the period in which this research was conducted, Jack Thorne served as a Clay Research Fellow. Both of the authors were supported in part by EPSRC First Grant EP/N007204/1. We thank Fabrizio Barroero for useful conversations.
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