$E_8$ and the average size of the 3-Selmer group of the Jacobian of a pointed genus-2 curve

We prove that the average size of the 3-Selmer group of a genus-2 curve with a marked Weierstrass point is 4.


Introduction
In this paper we prove new theorems about the arithmetic statistics of odd genus-2 curves. If f (x) = x 5 + c 12 x 3 + c 18 x 2 + c 24 x + c 30 ∈ Q[x] is a polynomial of non-zero discriminant, then the smooth projective completion of the affine curve C 0 f : y 2 = f (x) is a genus-2 curve with a marked Weierstrass point (the unique point at infinity). Conversely, any pair (C , P), where C is a (smooth, projective, connected) curve of genus 2 and P ∈ C (Q) is a marked Weierstrass point, arises from a unique such polynomial f (x) satisfying the following conditions: 1. The coefficients of f (x) are integers and the discriminant of f (x) is non-zero.
2. No polynomial of the form n −10 f (n 2 x) has integer coefficients, where n ≥ 2 is an integer.
We write E for the set of all polynomials f (x) = x 5 +c 12 x 3 +c 18 x 2 +c 24 x+c 30 ∈ Z[x] of non-zero discriminant, and E min ⊂ E for the subset satisfying condition 2. above. For f (x) ∈ E , we write C f for the corresponding pointed genus-2 curve and J f for the Jacobian of C f , a principally polarized abelian surface over Q. We define the height ht(f ) of a polynomial f (x) ∈ E by the formula ht(f ) = sup i |c i (f )| 120/i . Note that for any a > 0, the set {f ∈ E | ht(f ) < a} is finite. We can now state our first main theorem. Here is one consequence of Theorem 1.1 for rational points, which follows from work of Poonen and Stoll [PS14]: Theorem 1.2. A positive proportion of curves (C , P) ∈ F satisfy C (Q) = {P}. More precisely, we have lim inf a→∞ |{f ∈ E min | ht(f ) < a, |C f (Q)| = 1}| |{f ∈ E min | ht(f ) < a}| > 0.

Method of proof
recovering the previous map in the case that R = k is a field.
This construction is based on changing our point of view from G-orbits in V to isomorphism classes of triples (H ′ , θ ′ , γ ′ ), where H ′ is a reductive group of type E 8 , θ ′ is a stable Z/3Z-grading, and γ ′ ∈ h ′ (1). We give a construction that begins with a Heisenberg group (such as the µ 3 -extension of J f [3] arising from the Mumford theta group of thrice the canonical principal polarization of J f ) and a representation W of this Heisenberg group, and returns a Lie algebra h ′ of type E 8 with a stable Z/3Z-grading θ ′ , together with a representation of g ′ = (h ′ ) θ ′ on the same space W . The existence of this construction, which seems to be related to twisted vertex operator realizations of affine Kac-Moody algebras [Lep85], still seems remarkable to us! The general version of this construction will be described in a future work of the first author [Rom].
The next step in the proof of Theorem 1.1 is to introduce integral structures. All of the objects H, θ, G, V can be defined naturally over Z, and we can require that our polynomials c 12 , . . . , c 30 lie in Z[V ] G . If p is a prime and f (x) = x 5 + c 12 x 3 + c 18 x 2 + c 24 x + c 30 ∈ Z p [x] is a polynomial of non-zero discriminant, then our constructions so far yield a map J f (Q p )/3J f (Q p ) → G(Q p )\V f (Q p ). However, it is essential to be able to show that each G(Q p )-orbit in V f (Q p ) which is in the image of this map admits an integral representative, i.e. intersects V f (Z p ) non-trivially. This has been a sticking point for some time. In our earlier papers [Tho15,RTa], our failure to construct integral representatives in full generality meant we could provide upper bounds only for the average sizes of the Selmer sets, and not full Selmer groups, of the families of curves studied there.
In this paper we introduce a new general technique to construct integral orbit representatives. We describe it briefly here. If f (x) ∈ Z p [x] is a polynomial of non-zero discriminant, we choose a lifting to f (x) ∈ Z p [u] [x] with favourable properties. In particular, the discriminant of f (x) should be non-zero in F p [u] and square-free in Q p [u]. The construction giving rise to the map (1.2) determines a triple (H ′ , θ ′ , γ ′ ) over the complement in Spec Z p [u] of the locus where the discriminant of f vanishes.
Using an explicit construction of integral representatives in the square-free discriminant case, we extend this triple to the complement in Spec Z p [u] of finitely many closed points. Finally, we use the fact that a reductive group on the punctured spectrum of a 2-dimensional regular local ring extends uniquely to the whole spectrum (see [CTS79,Theorem 6.13]) to extend our triple further to the whole of Spec Z p [u]. Specializing to u = 0, we find the desired integral representative.
This argument is inspired by the proof of the fundamental lemma for Lie algebras [Ngô 10]. The problem of constructing integral representatives can be viewed as the problem of showing that a graded analogue of an affine Springer fibre is non-empty. From this point of view, attempting to deform the problem to a case where it can be solved directly is a natural strategy. Although we develop this technique here just in the case of the stable Z/3Z-grading of E 8 and its relation to odd genus-2 curves, it is completely general. In a future work [RTb], we will return to the families of curves studied in our earlier papers [Tho15,RTa] and obtain complete information about the average sizes of the 2-Selmer groups of their Jacobians.
Once integral representatives have been constructed, we can reduce the problem of studying the average size of the 3-Selmer groups of the curves C f to the problem of studying the number of orbits of G(Z) in V (Z) of bounded height (with congruence conditions and local weights imposed). In the final step in the proof of Theorem 1.1, we use Bhargava's techniques and their interpretation in the framework of graded Lie algebras (as in e.g. [BG13], [Tho15]) to carry out this orbit count and finally prove Theorem 1.1.
Remark 1.3. In the second author's thesis [Tho13], simple curve singularities and their deformations played an important role. The same is true here. The family of affine curves given by the equation y 2 = x 5 + c 12 x 3 + c 18 x + c 24 x + c 30 is a versal deformation of a type A 4 singularity. Here, we think of this family instead as being embedded in the family of affine surfaces y 2 = z 3 + x 5 + c 12 x 3 + c 18 x + c 24 x + c 30 .
This is a versal deformation of the E 8 surface singularity y 2 = z 3 + x 5 , together with its action of µ 3 by the formula ζ · (x, y, z) = (x, y, ζ −1 z). This fact plays an important role in §4.4.

Organization of this paper
We now describe the organization of this paper. In §2 we review relevant properties of the E 8 root lattice and its associated Weyl group. In §3, fundamental for the construction of orbits, we give our "Heisenberg group to graded Lie algebra" functor. In §4, we describe the invariant theory of our graded Lie algebra, and use the construction of §3 to parameterize and construct orbits. An important role is played by two special transverse slices to nilpotent elements, namely the Kostant section and the subregular Slodowy slice: we use the first of these to normalize the set of orbits, and the second to normalize our generators for the ring of G-invariant polynomials on V .
In §5 we give our construction of integral orbit representatives. We treat the local case using the ideas described above, and then deduce the existence of integral orbit representations for Selmer elements in the global case as a consequence. In §6, we give the point-counting results we need in order to prove Theorem 1.1. The power of Bhargava's techniques is such that little more than formal verification is required in order to check that they give the desired result here. We have therefore given a compressed treatment, describing only what is new in this particular case; we trust that the interested reader will be able to easily fill in the details, interpolating from e.g. the proof of [BG13,Theorem 25].
Finally, in §7 we combine all of this to prove our main theorems.

Acknowledgments
Both authors received support from EPSRC First Grant EP/N007204/1. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 714405).

Notation
If H is a group scheme, then we will use a gothic letter h = Lie H for its Lie algebra. If θ : µ n → Aut (H) is homomorphism, then we write h = ⊕ i∈Z/nZ h(i) for the corresponding grading; thus h(i) is the isotypic subspace in h corresponding to the character ζ → ζ i of µ n .
If G is a group scheme over a base S, X an S-scheme on which G acts, T an S-scheme, and x ∈ X(T ), then we write Z G (x) for the scheme-theoretic stabilizer of x, which is a T -scheme. By a Lie algebra over S, we mean a coherent sheaf of O S -modules g together with an alternating bilinear form [·, ·] : g × g → g that satisfies the Jacobi identity. Similarly, if g is a Lie algebra that is equipped with a Lie algebra homomorphism g → End OS (W ), for some locally free sheaf W of O S -modules, and x ∈ W ⊗ OS O T , then we define z g (x) to be the Lie centralizer of x, which is a Lie algebra over T .
If G is reductive and A ⊂ G is a maximal torus, we write X * (A) = Hom(A, G m ) for its character group, Φ(G, A) ⊂ X * (A) for its set of roots, and X * (A) for its cocharacter group. We write N G (A) for the normalizer of A and W (G, A) = N G (A)/A for the Weyl group of A in G.
If G is a smooth group scheme over a scheme S, then we write H 1 (S, G) for the set of isomorphism classes of G-torsors over S, which we think of as a non-abelianétale cohomology set. If S = Spec R is affine then we will write H 1 (R, G) for the same object.
If G is a smooth linear algebraic group over a field k which acts on an integral affine variety X, and G 0 is reductive, then we write X G = Spec k[X] G for the categorical quotient, which is again an integral affine variety.
2 The E 8 root lattice Throughout this paper, we will constantly make use of the properties of a certain conjugacy class of automorphisms of the E 8 root lattice. We therefore record some of these properties here. For us, an E 8 root lattice Λ is a finite free Z-module, equipped with a symmetric bilinear pairing (·, ·) : Λ × Λ → Z with the following property: let Φ ⊂ Λ be the set of elements α ∈ Λ with (α, α) = 2. Then Φ forms a root system in Λ R of Dynkin type E 8 (elements of Φ are called roots). Any two E 8 root lattices are isomorphic.
We write Aut(Λ) for the group of automorphisms of Λ that preserve the pairing (·, ·). Since E 8 has no diagram automorphisms, Aut(Λ) equals the Weyl group W (Λ), which is generated by the reflections in the root hyperplanes α ⊥ (for α ∈ Φ). We recall that an element w ∈ W (Λ) is said to be elliptic if Λ w = 0.
Lemma 2.1. W (Λ) contains a unique conjugacy class of elliptic elements of order 3. Let w be such an element, and let Λ w = Λ/(w − 1)Λ be the group of w-coinvariants in Λ. Then: 1. There is an isomorphism Λ w ∼ = F 4 3 .
2. Any choice of orbit representatives for the action of w on Φ gives a complete set of coset representatives for the non-zero elements of Λ w . The centralizer of w in W (Λ) acts transitively on Λ w − {0}.
Proof. See [Ree11, Table 1 We also note for later use that if w ∈ W (Λ) is elliptic of order 3, then w 2 γ + wγ + γ = 0 for all γ ∈ Λ. In [Ree11], for any elliptic element w ∈ W (Λ), Reeder defines a symplectic pairing on Λ w that is invariant under the action of the centralizer of w in W (Λ). We now describe a slight variant of this pairing.
Let S be a Z[1/3]-scheme. We now let Λ be anétale sheaf of E 8 root lattices on S. By this we mean that Λ is a locally constantétale sheaf of finite free Z-modules, which is equipped with a pairing Λ × Λ → Z making each stalk Λ s above a geometric point s → S into an E 8 root lattice. Then Aut(Λ) is a finiteétale S-group.
In this setting we define an elliptic µ 3 -action on Λ to be a homomorphism θ : µ 3 → Aut(Λ) such that for any geometric point s → S and any primitive 3rd root of unity ζ ∈ µ 3 (s), θ(ζ) ∈ Aut(Λ s ) is an elliptic element of order 3.
If θ is an elliptic µ 3 -action on Λ, then we write Λ θ for the sheaf of θ-coinvariants; by Lemma 2.1, it is a locally constantétale sheaf of F 3 -vector spaces of rank 4. We define a pairing ·, · : Λ θ × Λ θ → µ 3 by the formula λ, µ = ζ ((1−θ(ζ))λ,µ) , we just need to show that T → S is surjective. Since the formation of T commutes with base change, we are therefore free to assume that S = Spec k is the spectrum of an algebraically closed field.
In this case, there exist (by assumption) maximal tori A, A ′ ⊂ H on which θ, θ ′ act through elliptic automorphisms of order 3. Using the conjugacy of maximal tori, we can therefore assume that A = A ′ . Using Lemma 2.1, we can assume that θ, θ ′ define the same element of the Weyl group of A.
We have therefore reduced the problem to the statement that if w ∈ W (H, A) is an elliptic element of order 3, then any two lifts n, n ′ to the normalizer N H (A)(k) are H(k)-conjugate. In fact, they are even A(k)-conjugate, as follows from the fact that the morphism 1 − w : A → A isétale (and surjective). This completes the proof.
Note that if R → R ′ is a homomorphism of Z[1/3]-algebras, then pullback determines functors Heis R → Heis R ′ and GrLieT R → GrLieT R ′ . We could define stacks Heis and GrLieT over Z[1/3], and even try to represent them as quotient stacks using the objects introduced in §4 (as in [HLHN14]). We have chosen not to do this here in order to avoid obscuring the main point of our constructions, which are based on relatively explicit calculations.
We will introduce variants of these categories in §4.3. In particular, we will introduce the category GrLie R of pairs (H, θ) (we forget the torus).

Definition of the functor
The main goal of §3 is to prove the following theorem.
We now define the functor Heis R → GrLieT R . Let (Λ, θ, H ) ∈ Heis R . We can assume Spec R is connected. Let R → R ′ be a Galois finiteétale extension 1 over which Λ and H become constant. Let Γ = Aut R (R ′ ). We will first define a Lie algebra over R ′ and then recover a Lie algebra over R byétale descent. (For this naive form ofétale descent, see [Sta18,Tag 0D1V].) For the remainder of the section we fix a choice of primitive 3rd root of unity ζ ∈ µ 3 (R ′ ). For ease of notation, we write w = θ(ζ).
To construct an element of GrLieT R ′ , we first let A be the torus over R with X * (A) = Λ. We next construct a Lie algebra over R ′ of type E 8 . Let a = Lie A. Note that a R ′ ∼ = Hom(Λ, R ′ ), and so for any α ∈ Φ, the corootα corresponds to an element of a R ′ . In fact a R ′ is generated by {α | α ∈ Φ}. Let h ′ be the quotient of the free R ′ -module with basis elements X α , α ∈ Φ, by the relations X ζ α = ζX α (for any α ∈ Φ). Finally, Thus h R ′ is a free R ′ -module of rank 248 generated by {α, X β | α ∈ Φ, β ∈ Φ} (by abuse of notation we also write X β for the image of this vector in h ′ ).
We define a bilinear map [·, ·] : h R ′ × h R ′ → h R ′ as follows. We set [x, y] = 0 for any x, y ∈ a. We let for any α ∈ Φ, β ∈ Λ. Finally, the bracket of vectors X α , X β is defined by the formula We observe that the map is well-defined, i.e. it respects the defining relation X ζ α = ζX α .
Proposition 3.2. With the above definition, h R ′ is a Lie algebra (i.e. the bracket [·, ·] is antisymmetric and satisfies the Jacobi identity).
In order to prove the proposition, we first make the following observation.
We also point out the useful fact that because the pairing ·, · is alternating, we have α β = β α whenever α + β = 0. for generators x, y, z. If any of x, y, z are in a R ′ , then it follows easily from the definition of the bracket that (3.1) is zero.
Thus we restrict our attention to the case when x = X α , y = X β and z = X γ for some α, β, γ ∈ Λ.
For the rest of the proof we assume α + β + γ = 0. For (3.1) to be nonzero, at least one term in (3.1) must be nonzero. Without loss of generality, we may assume the first term is nonzero, and so either β + γ = 0 or β + γ ∈ Φ. We deal with each of these cases separately.
Recall that Γ = Aut R (R ′ ). Theétale sheaves Λ and H become constant over R ′ , and the group Γ acts on their sections over Spec R ′ . This group therefore also acts on the sections over Spec R ′ of Λ. We define a semi-linear action of Γ on h R ′ by giving it its canonical action on a R ′ and by defining Lemma 3.4. The action of Γ on h R ′ just defined leaves the Lie bracket invariant.
We let h = h Γ R ′ , H = Aut R (h). Our assumption that N = 2 × 3 × 5 × 7 is a unit in R says exactly that the Killing form of h is non-degenerate, and therefore (by the main theorem of [Vas16]) that H is a reductive group over R with geometric fibres of Dynkin type E 8 . Moreover, we have Lie H = h and we can identify Z H (a) = A.
Proof. Recall that we have defined w = θ(ζ). In order to prove the lemma, it suffices to show that for any generators x, y of h R ′ . If x, y ∈ a R ′ , then both sides of (3.3) are zero. If x =α and y = X β for some α ∈ Φ, β ∈ Λ, then equality in (3.3) follows from the fact that (wα, wβ) = (α, β). Suppose x = X α and y = X β for some α, β ∈ Λ. If (α, β) ≥ 0, then both sides of (3.3) are zero, and if α + β ∈ Φ, then equality in (3.3) follows from the fact that wα, wβ = α, β . If α + β = 0, then where we are using that α β ∈ µ 3 . Thus we again have equality in (3.3). It is immediate from the definition that θ as defined above is equivariant for the action of Γ.
We write again θ : µ 3 → Aut(h) = H for the induced homomorphism. This completes the construction of the triple (H, θ, A). It is (up to canonical isomorphism) independent of the choice of primitive 3rd root of unity ζ ∈ µ 3 (R ′ ); indeed, this choice entered only in the definition of the Lie bracket via the formula [X α , X β ] = (−1) (α,θ(ζ)(β)) α, β X α β (when α, β ∈ Φ and α + β is a root). The other choice would give [X α , X β ] = (−1) (α,θ(ζ −1 )(β)) α, β X α β . If h ′ R ′ denotes the Lie algebra over R ′ defined using the other primitive 3rd root of unity ζ −1 , then the map h R ′ → h ′ R ′ which is the identity on the summand a R ′ and which sends X α ∈ h R ′ to −X α ∈ h ′ R ′ is an isomorphism which intertwines the two actions of Aut R (R ′ ), and therefore defines an isomorphism between the Lie algebras over R corresponding to the two possible choices of root of unity.
This completes the definition of the functor Heis R → GrLieT R , and therefore the proof of Theorem 3.1.
We observe that if (Λ, θ, H ) ∈ Heis R , then there is a morphism f R : H 0 (R, Λ θ ) → Aut HeisR (Λ, θ, H ) defined as follows: if λ ∈ H 0 (R, Λ θ ), then f R (λ) acts as the identity on Λ and as µ → λ, µ µ on H . Let (H, θ, A) ∈ GrLieT R be the tuple corresponding to (Λ, θ, H ) under the construction of Theorem 3.1. Varying R and taking into the account the functorial nature of our construction, we obtain a morphism of group schemes over R: We can describe this explicitly: Lemma 3.6. Let notation be as in the above discussion. Then there is a canonical isomorphism Λ θ ∼ = A θ , under which the morphism (3.4) corresponds to the adjoint action of Proof. By definition, we have X * (A) = Λ, hence X * (A) ∼ = Λ ∨ = Hom(Λ, Z). There is a canonical isomor- There is an isomorphism Λ θ ∼ = (Λ ∨ ⊗ µ 3 ) θ , given by the formula λ → (1 − θ(ζ))λ ⊗ ζ for λ ∈ Λ; this does not depend on the choice of ζ and also depends only on the image of λ in Λ θ .
Composing the above two isomorphisms gives the desired isomorphism Λ θ ∼ = A θ . Now fix λ ∈ Λ θ , and let R → R ′ be a Galois finiteétale cover over which Λ and H become constant.
We will give an explicit description of λ as an automorphism of h R ′ = a R ′ ⊕ h ′ . By definition, it acts as the identity on a R ′ and sends the vector X α to X λ,α α . In other words, it leaves invariant the α-root space and acts by the scalar λ, α there.
On the other hand, the element (1 − w)λ(ζ) in A θ also acts as the identity on A and leaves invariant each root space, acting on the α-root space by the scalar This completes the proof.

Identifying h(0)
Theorem 3.1 associates to any triple (Λ, θ, H ) ∈ Heis R a triple (H, θ, A) ∈ GrLieT R . In the proof of our next result, we show that if W is an irreducible representation of H on which the central µ 3 acts by its tautological (scalar) character, then we can identify h(0) with sl(W ) and H with a certain subgroup of SL(W ). This result will play an essential role in the construction of orbits in §4.3.
Theorem 3.7. Let (Λ, θ, H ) ∈ Heis R . Let W be a locally free R-module of rank 9, and suppose ρ : is a homomorphism such that the central µ 3 acts on W through its tautological character. Let (H, θ, A) denote the image of (Λ, θ, H ) under the functor of Theorem 3.1 and let G = H θ . Then: 1. G is a semisimple reductive group. Let G sc denote its simply connected cover.
2. There is a commutative diagram of R-groups with exact rows: To prove the theorem, we can again assume that Spec R is connected, and choose a Galois finiteétale extension R → R ′ over which Λ and H become constant. The group G is reductive with geometric fibres of type SL 9 /µ 3 , so its simply connected cover G sc → G exists, and has a kernel which is a group of multiplicative type over R of order 3. Let g = Lie G.
The first step in the proof of Theorem 3.7 is to define an action of the Lie algebra g on W ; equivalently, to define a Γ-equivariant map Let π : Λ → H denote the canonical projection. We define a map ρ ′ : Proposition 3.8. With the above definition, ρ ′ is a well-defined Lie algebra homomorphism that commutes with the action of Γ = Aut R (R ′ ).
Proof. We see that ρ ′ is well defined exactly because ρ(ζ) = ζ · 1 W . The key point is to check that the Lie bracket is preserved, or in other words that the relation holds for all α, β ∈ Φ. We give a case-by-case-proof depending on the value of (α, β). Before beginning, we note again the useful fact that if α ∈ Φ, then α + w(α) + w 2 (α) = 0. In particular, α + w(α) and α + w 2 (α) are roots.
Case 1. If (α, β) = ±2, then α = ±β. If α = β then both sides of (3.5) vanish. If α = −β, then the right-hand side vanishes because π( α) and π( β) commute in H . On the other hand, (3.6) The first line of (3.6) is zero because it is an element of g R ′ ∩ a R ′ = 0. The second line vanishes because α − wα is not a root. The third line vanishes because α − w 2 α is not a root. Therefore both sides of (3.5) are zero in this case.
It remains to check that ρ ′ is equivariant for the action of Γ. We just need to note the formulae The induced map g → sl(W ) is an isomorphism, inducing an isomorphism G ad → PGL(W ) (by [Vas16] once again), hence a unique isomorphism G sc → SL(W ) which is compatible with the given map g → sl(W ) ([Con14, Exercise 6.5.2]). Let H ′ denote the pre-image of A θ ∼ = Λ θ in G sc . Identifying the centre of G sc with µ 3 via its action on W , we find that H ′ fits into a diagram To complete the proof of Theorem 3.7, we must show that there is an isomorphism H ∼ = H ′ of central extensions of Λ θ by µ 3 . We will show that in fact the images of H and H ′ in SL(W ) coincide. This can be checked over the extension R → R ′ .
Let p : G sc → G ad denote the projection to the adjoint group. We can characterize H as Similarly, we can characterize H ′ as To prove the theorem, it is therefore enough to show that the two homomorphisms Λ θ → PGL(W ), one derived from Ad •ρ, the other derived from Ad | H ′ , are the same. Since g R ′ is spanned by the elements Z β , and the non-trivial elements of Λ θ are all of the form α mod(θ − 1), for some α ∈ Φ, it is enough to show that the two possible actions of α on Z β are the same for all α ∈ Φ, β ∈ Φ.
For the first action, we see that, by definition, is a scalar multiple of ρ(π( β)). Therefore we have For the second action, we use the fact that defined by Lemma 3.6, sends a root α to the element (1 − w)α(ζ). We calculate the corresponding action on Z β as The equality of the expressions (3.7) and (3.8) concludes the proof of Theorem 3.7.

A stable grading of E 8
In the previous section we constructed a functor from Heisenberg groups to Z/3Z-graded Lie algebras. In order to count points, we need to have a 'reference' algebra in which one can do explicit calculations. In this section we introduce such an algebra using a principal grading as defined in [RLYG12] and give rigidifications of orbits and invariant polynomials using two special transverse slices to nilpotent orbits. The main results of this section, in §4.3, combine this work with the work done in §3 to define the map η f described in the introduction (in other words, to construct orbits from points of Jacobians).

Definition of the grading
Let H be a split reductive group of type E 8 over Z. Let T ⊂ H be a split maximal torus, and let Φ H ⊂ X * (T ) be the corresponding set of roots. Let S H ⊂ Φ H be a fixed choice of root basis. Let Φ + H be the corresponding set of positive roots. We suppose that H comes with a pinning {X α } α∈SH .
Letρ ∈ X * (T ) be the sum of the fundamental coweights with respect to S H , and let θ =ρ| µ3 : µ 3 → H. Let h = Lie (H). Then θ defines an action of µ 3 on h and thus determines a Z/3Z-grading (4.1) We let G = H θ , the centralizer of θ in H. We write V = h(1); it is a representation of G, free over Z of rank 84.
Proposition 4.1. The group G is a split reductive group isomorphic to SL 9 /µ 3 . The subgroup T ⊂ G is a split maximal torus. Over Z[1/3], θ is a stable Z/3Z-grading of H, in the sense of §2.
Proof. It follows from the discussion in [Con14, Remark 3.1.5] that G is smooth over Z, and moreover that the connected component G 0 (which agrees in each fibre G s with the connected component of G s ) is reductive. Moreover, T ⊂ G 0 is a split maximal torus. It remains therefore to check that G = G 0 and that its root datum is that of SL 9 /µ 3 . The quotient G/G 0 isétale over Z, so both of these last points can be checked at the generic point, in which case they follow from the general theory over C (see e.g. [Ree10]). The final statement can be checked in geometric fibres, in which case it is [RLYG12, Corollary 5.7].
We write H for the Q-fibre of H, and similarly for T , G, and V . In the coming sections we will describe the invariant theory of the pair (G, V ) and its relation to 3-descent on odd genus-2 curves. We will re-introduce integral structures into our discussion in §4.5 below.
We let B = V G = Q[V ] G , and write π : V → B for the quotient map. For a detailed study of the properties of the pair (G, V ), and its analogue over fields of sufficiently large positive characteristic, see [Lev09]. We invite the reader to become familiar at least with the results in the introduction to that paper before proceeding; in particular, we will make frequent use of the existence of Jordan decomposition of elements in V and of the fact that, if k is algebraically closed, then two semisimple elements of V (k) are G(k)-conjugate if and only if they have the same image in B(k). We

Kostant section
We define an affine linear subspace We write σ : B → V for the inverse σ = π| −1 κ , and call it the Kostant section. We define an action of G m on κ by the formula t · x = t Ad(ρ(t −1 ))(x). This is a contracting action with E as its unique fixed point, and the morphism π| κ is G m -equivariant (when G m acts on B = V G in the natural way, compatibly with its action on V by scalar multiplication). Hence σ is also G m -equivariant.
If k/Q is a field extension, and f ∈ B rs (k), then we can use the Kostant section to organise the set . Then µ f is a torsor for the group Z G (σ(f )) and determines a bijection The group scheme Z G (σ(f )) can be described explicitly as follows: More generally, the centralizer A := Z H (σ| B rs ) is a maximal torus in H B rs . We define Λ = X * (A) and Λ ∨ = Hom(Λ, Z). We define a pairing (·, ·) : Λ × Λ → Z by the formula (λ, µ) =λ(µ). Then Λ is anétale sheaf of E 8 root lattices on B rs . The grading θ determines a homomorphism µ 3 → Aut(Λ) that we also denote by θ, and which is an elliptic µ 3 -action on Λ. The stabilizer scheme Z G (σ| B rs ) is finiteétale over B rs , and can be identified with Λ θ (cf. Lemma 3.6). Moreover, Λ θ admits a symplectic, non-degenerate pairing ·, · : Λ θ × Λ θ → µ 3 (Lemma 2.2).
Proposition 4.3. We can choose polynomials c 12 , c 16 , c 24 , c 30 ∈ Q[V ] G with the following properties: 0 is (up to scalar) equal to the restriction to V of the usual Lie algebra discriminant of h.
2. Let C 0 → B be the family of affine curves given by the equation and let C → B be its completion inside weighted projective space P B (1, 1, 3), projective over B. Let J → B rs be the Jacobian of its smooth part. Then there is an isomorphism Λ θ ∼ = J [3] ofétale sheaves which sends the pairing ·, · on Λ θ to the Weil pairing on J [3].
The proof of this proposition will be given in §4.4.
Let P : B → C denote the section at infinity (which is a Weierstrass point in each smooth fibre of C ). The choice of P determines a symmetric line bundle M = O J (C − P) on J . We write L = M ⊗3 , and define G to be the 3-torsion subgroup of the Mumford theta group G (L ). Thus G is a central extension ofétale group schemes over B rs , and (Λ, θ, G ) is an object of the category Heis B rs defined in §3. We will soon show (Proposition 4.6) that the image of (Λ, θ, G ) under the functor defined in Theorem 3.1 is isomorphic in the groupoid GrLieT B rs to the triple (H B rs , θ B rs , A).

Twisting
We can now explain our construction of orbits. Let R be a Q-algebra. We recall that in §3 we have defined groupoids Heis R and GrLieT R , and a functor Heis R → GrLieT R . We now define some related groupoids.
We define GrLie R to be the groupoid of pairs (H ′ , θ ′ ), where H ′ is a reductive group over R of type E 8 and θ ′ : Lemma 4.4. The following sets are in canonical bijection: 1. The set of isomorphism classes of objects in GrLie R .
Proof. Note that GrLie R always contains the object (H R , θ R ). We have proved (Lemma 2.3) that any two objects of GrLie R are isomorphicétale locally on Spec R. Since Aut(H, θ) = G, the result follows by descent.
We define GrLieE R to be the groupoid of tuples (H ′ ′′ ) are given by isomorphisms H ′ → H ′′ intertwining θ ′ and θ ′′ and sending γ ′ to γ ′′ . If (H ′ , θ ′ , γ ′ ) ∈ GrLieE R , then we define an element π(γ ′ ) ∈ B(R) as follows: after passage to a faithfully flat extension R → R ′ , we can find an isomorphism α : , and π(α(γ ′ )) ∈ B(R ′ ) in fact lies in B(R) and is independent of the choice of α. Thus there is a functor π : GrLieE R → B(R), compatible with arbitrary base change on R. (We view the set B(R) as a discrete category, i.e. as a category with no non-identity morphisms.) There is an obvious functor V (R) → GrLieE R (where V (R) is viewed as a discrete category) given by If f ∈ B(R), then we define GrLieE R,f to be the full subcategory of GrLieE R consisting of tuples (H ′  2. The set ker(H 1 (R, Z G (σ(f ))) → H 1 (R, G)).

The set of isomorphism classes of objects
Proof. The group scheme Z G (σ(f )) is a finiteétale R-scheme, and the action map The existence of the bijection between the first and second sets is therefore a consequence of e.g. [Con14, Exercise 2.4.11].
The category GrLieE R,f contains the triple (H R , θ R , σ(f )). Its automorphisms may be identified with the sections over R of Z G (σ(f )). Moreover, any two objects of GrLieE R,f are isomorphicétale locally on Spec R. This implies the existence of the bijection between the second and third sets. Let f τ ∈ B rs (B rs ) be the tautological section. According to Proposition 4.3, (Λ, θ, G ) defines an element of the groupoid Heis B rs ,fτ , hence a tuple (H τ , θ τ , A τ ) ∈ GrLieT B rs (and in fact A τ = A). The Lie algebra a τ has a tautological section γ τ (which is in fact none other than σ). Thus (H τ , θ τ , γ τ ) ∈ GrLieE B rs ,fτ .
Proof. The proof relies upon the fact that for each triple, the associated Z/3Z-grading can be naturally extended to a Z/6Z-grading. A Z/6Z-grading of h B rs extending the grading of (H B rs , θ B rs , σ(f τ )) is given byρ| µ6 . Let V ′ ⊂ h B rs denote the 1-part of this Z/6Z-grading, and note that σ(f τ ) ∈ V ′ (B rs ).
To define a Z/6Z-grading of h τ extending the grading of (H τ , θ τ , γ τ ), observe that M is a symmetric line bundle. A choice of isomorphism M ∼ = [−1] * M determines an automorphism (ω, α) → (−ω, [−1] * α) of G which acts as the identity on the central µ 3 and as −1 on the quotient J [3]. Since this automorphism is compatible with the action of −1 on Λ, Theorem 3.1 implies the existence of an involution θ ′ : H τ → H τ that commutes with θ τ , that normalizes the torus Z Hτ (γ τ ), and that acts −1 on the character group of this torus. Therefore θ ′ · θ τ defines a Z/6Z-grading of h τ such that, if V ′ τ ⊂ h τ is the 1-part of the grading, then To complete the proof, consider the B rs -scheme T of isomorphisms H → H τ intertwiningρ| µ6 and θ ′ · θ τ and sending σ(f τ ) to γ τ . Then T is anétale B rs -scheme, which is in fact a torsor for Z H (σ(f τ ))ρ |µ 6 (the argument is the same as in the proof of Lemma 2.3). Since Z H (σ(f τ ))ρ |µ 6 → B rs is the trivial group scheme, we have T = B rs and it follows that there is a unique isomorphism (H τ , θ τ , γ τ ) ∼ = (H B rs , θ B rs , σ(f τ )) in GrLieE B rs ,fτ that is compatible with the given Z/6Z-gradings.
Theorem 4.7. Let f ∈ B rs (R). Then there is an equivalence of categories Heis R,f → GrLieE R,f , compatible with base change on R.
Proof. Let (f * Λ, θ, H ) ∈ Heis R,f , and let (H ′ , θ ′ , A ′ ) ∈ GrLieT R be its image under the functor of Theorem 3.1. Since It is fully faithful, by Lemma 3.6. The category Heis R,f contains the object f * (Λ, θ, G ), which corresponds to the object (H R , θ R , f * (γ τ )), by Proposition 4.6. The objects of both categories are therefore classified by the group ). This shows that our functor is essentially surjective, and completes the proof of the lemma.
Corollary 4.8. Let R be a Q-algebra over which every locally free module of finite rank is free. Let f ∈ B rs (R). Then there is a canonical injection η f : Proof. The group G acts on H 0 (J f , L ), which is a locally free R-module of rank 9. By Proposition 4.6 and Theorem 3.7, there is a diagram of R-groups with exact rows: Let P ∈ J f (R). Let L P = t * P M ⊗ M ⊗ M , and let G P denote the 3-torsion subgroup of the Mumford theta group G (L P ). Then G P ∈ Heis R,f . Let (H P , θ P , γ P ) ∈ GrLieE R,f denote the tuple corresponding to G P under the equivalence of Theorem 4.7. Then the class ϕ ∈ H 1 (R, J f [3]) corresponding to (H P , θ P , γ P ) under the bijection of Lemma 4.5 is the Kummer class of the point P (as follows from Lemma 3.6).
To prove the corollary, we will show that this class lifts to H 1 (R, G ). This will imply that the image of ϕ in H 1 (R, G) lies in the image of the map H 1 (R, G sc ) → H 1 (R, G), which is trivial (by our assumption on R). To show that the class lifts, it will even suffice to show that it lifts to H 1 (R, G (L )), where G (L ) is the Mumford theta group of L , sitting in the short exact sequence of R-groups is injective, again by our assumption on R. We define T P to be the scheme of pairs (ω, α), where ω ∈ J f and α : L P → t * ω L is an isomorphism. Note that forgetting ω leads to a surjective map T P → [3] −1 (P ). Thus T P is a torsor for G (L ), defining a class in H 1 (R, G (L )) that lifts the class ϕ ∈ H 1 (R, J f [3]). This completes the proof of the corollary.
Remark 4.9. We could prove a stronger version of Corollary 4.8 where we replace our assumption that every locally free module is free with the assumption that H 1 (R, G sc ) is trivial, by refining the torsor for G (L ) constructed in the proof to a torsor for G using the canonical isomorphism M ⊗9 ∼ = [3] * M afforded by the theorem of the cube. However, since we don't need this extra generality we have chosen not to include the details here.
We restate the corollary in the case that R = k is a field extension of Q.
Corollary 4.10. Let k/Q be a field, and let f ∈ B rs (k). Then there is a canonical injection η f, : We also record for future use the fact that when R = Q, we can extend the above construction of orbits from rational points to 3-Selmer elements: Proposition 4.11. Let f ∈ B rs (Q). Then the map η f : Proof. Taking into account Lemma 4.5, we just need to show that the map H 1 (Q, G) → v H 1 (Q v , G) is trivial kernel, where the product runs over the set of all places v of Q. This is an exercise using class field theory.

Proof of Proposition 4.3
We now prove Proposition 4.3. We will use a special transverse slice to the orbit of a subregular nilpotent element in V in order to form the bridge between the group H and the family of abelian surfaces J .
We begin by fixing a subregular nilpotent element e ∈ V (the existence of such an element can be read off from the tables in [VÈ78], which also show that there is a unique G-orbit of subregular nilpotents in V ). We can complete e to a normal sl 2 -triple (e, h, f ) in h. We define X 0 = e + z h (f ), an affine linear subspace of h. We define a µ 3 -action on X 0 by the formula ζ · x = ζ −1 Ad(θ(ζ))(x). We define a G m -action on X 0 by the formula t · x = t 2 Ad(λ(t −1 ))(x), where λ : G m → G is the cocharacter with dλ(1) = h. These two actions commute, giving a µ 3 × G m -action on X 0 . Let B 0 = h H, and let p 0 : X 0 → B 0 denote the restriction of the adjoint quotient π 0 : h → h H to X 0 . This is equivariant for the action of µ 3 × G m on source and target if G m acts on h H by the square of its usual action. We identify X * (µ 3 × G m ) = Z/3Z × Z. If v is an eigenvector for an action of µ 3 × G m , then we define its weight to be the image in Z/3Z × Z of the character by which µ 3 × G m acts on v. 2. The restriction of the 7 elements c 2 , c 8 , c 12 , c 14 , c 18 , c 20 , c 24 to X 0 , together with x, y, z, are algebraically independent and generate Q[X 0 ]. Moreover, the morphism p 0 : X 0 → B 0 is given by the formula Proof. View X 0 as a vector space with origin e; then the action of µ 3 × G m on X 0 is linear. By direct calculation, the weights of µ 3 × G m in X 0 are as follows: ( By comparison with the results of [Slo80, §8.7], we see that the differential dp 0,e has rank 7, mapping the subspace where G m acts with weights 4, 16, 24, 28, 36, 40, 48 isomorphically into the Zariski tangent space T 0 (h H) and annihilating the subspace where G m acts with weights 12, 20, and 30.
Following through the argument of [Slo80, §8.7, Theorem] with this µ 3 × G m -action now shows that there is a is the semi-universal µ 3 × G m -deformation of the singularity y 2 = z 3 + x 5 given by the formula where x, y, z have weights (0, 12), (0, 30) and (−1, 20), respectively, and each c i has weight (−i, 2i). We fix our invariant polynomials c 2 , . . . , c 30 ∈ Q[h] H to be the images under this isomorphism of the elements with the same names in the affine ring of B ′ 0 . This completes the proof of the proposition.
We fix elements c 2 , c 8 , c 12 , c 14 , c 18 , c 20 , c 24 , c 30 and x, y, z as in Proposition 4.12. Thus we have identified X 0 explicitly as given by the equation We view (4.3) as an affine Weierstrass equation over A 1 B0 . This allows us to compactify X 0 to obtain a projective Weierstrass fibration (in the sense of [Mir81]) Y 0 → P 1 B0 which contains X 0 as an open subscheme. More precisely, X 0 is the complement in Y 0 of the zero section O and the fibre F above the point x = ∞ of P 1 B0 .
Proof. To show that this morphism is an isomorphism, we use the existence of the Springer resolution. Recall that T ⊂ H is a maximal torus with Lie algebra t and root basis S H ⊂ Φ(H, T ). We write P ⊂ H for the Borel subgroup corresponding to this choice of root basis. Let X 0,t denote the pullback of X 0 → B 0 along the finite map t → B 0 = t W (H, T ), and define Then X 0,t → t is the Springer resolution of the transverse slice X 0 : it is smooth, and the morphism X 0,t → X 0,t is a proper morphism which is an isomorphism away from the singular points in each fibre of X 0,t → t (cf. [Slo80, §5.3]). In particular, we can glue X 0,t with Y 0,t to obtain a smooth proper surface Y 0,t → t which is itself a resolution of Y 0,t → t.
We have Pic(H/P ) = X * (T ). The projection X 0,t → H/P therefore induces a map X * (T ) → Pic( X 0,t ). Let ξ denote the generic point of t, and let ξ denote a lift of η 0 to a geometric point above ξ. We claim that the composite X * (T ) → Pic( X 0,t ) → Pic( X 0,t,ξ ) is an isomorphism. In fact, it suffices to prove the analogous claim above the central point 0 of t, as we now explain. In order to avoid confusing notation, let us write s for this point, and s for a geometric point above it. Then there is a commutative diagram where the right vertical map is given by specialization and is injective with torsion-free cokernel (cf. [MP12, Proposition 3.6], which gives a specialization morphism for Néron-Severi groups of the fibres of the proper morphism Y 0,t → t; we are using that the fibres of Y 0,t → t are rational elliptic surfaces, and that in each geometric fibre the free rank 2 subgroup O, F of the Picard group splits off as a direct summand in a way that is compatible with specialization). Since the source and target of the right vertical arrow are both free Z-modules of rank 8, this arrow is an isomorphism. It follows that if the bottom arrow in the above diagram is an isomorphism, then so is the top arrow.
Here, however, we can make everything explicit, using the description of the exceptional fibre of the morphism X 0,t,s → X 0,t,s given in [Hin91]. The upshot is that this exceptional divisor is a union of projective lines C α indexed by simple roots α ∈ S H , the classes of which freely generate Pic( X 0,t,0 ); and if α ∈ S H is a simple root, then the image of α ∈ X * (T ) in the Picard group is the class of the curve C α (this is [Hin91, 5.3, Lemma]).
The same argument shows that the morphism X * (T ) → Pic( X 0,t,ξ ) ∼ = W ⊥ 0 intertwines the root lattice on X * (T ) with the negative of the intersection pairing on W ⊥ 0 . Indeed, this can be checked in the central fibre of the Springer resolution, where it follows from the formula (C α , C β ) = −β(α), itself a consequence of [Hin91, Proposition 5.2].
It remains to see why the above results on the map X * (T ) → Pic( X 0,t,ξ ) imply what we need for the map X * (A η 0 ) → Pic(X 0,η 0 ). However, the points ξ and σ 0 (η 0 ) of h(k(η 0 )) are conjugate under the action of H(k(η 0 )), so what we really need to check is that the two maps X * (T ) → Pic(X 0,t,ξ ), one arising by pullback from Pic(H/P ), and the other arising from the existence of the T k(η 0 ) -torsor are the same. This follows from the definitions.
The morphism X 0 → B 0 is µ 3 -equivariant, and µ 3 acts on B 0 = Spec Q[c 2 , c 8 , c 12 , c 14 , c 18 , c 20 , c 24 , c 30 ] by the formula ζ · c i = ζ −i c i . If X denotes the restriction of X 0 to B = Spec Q[c 12 , c 18 , c 24 , c 30 ] = B µ3 0 , then θ acts on the fibres of X → B by the formula θ(ζ)(x, y, z) = (x, y, ζ −1 z). If we write Y → B for the pullback of Y 0 to B, then µ 3 acts on the fibres of Y → B by the same formula, and we can identify the fixed locus Y µ3 with the projective completion C of the family of affine curves C 0 : y 2 = x 5 + c 12 x 3 + c 18 x 2 + c 24 x + c 30 (4.4) that is the object of our study in this paper.
We now come back to Proposition 4.3. We recall that we have defined Λ = X * (A), where A = Z H (σ| B rs ) is a torus over B rs . Thus Λ is anétale sheaf of E 8 root lattices, which may be identified with the pullback of Λ 0 along the closed immersion B → B 0 . The image of the stable Z/3Z-grading θ : µ 3 → H normalizes A, and determines an elliptic µ 3 -action θ : µ 3 → Aut(Λ). We must show that there is an isomorphism Λ θ → J [3] intertwining the pairing ·, · on Λ θ , described in §2, with the Weil pairing on J [3].
We define the morphism Λ θ → J [3] in stages. Since both Λ and J [3] are locally constantétale sheaves over B rs , it is enough to define this morphism at the generic point η of B rs . Let η now denote the generic point of B rs , and let η be a geometric point above it. We have the following corollary of Lemma 4.13: Corollary 4.14. There is a π 1 (η, η)-equivariant isomorphism X * (A η ) → Pic(X η ) which intertwines the Weyl-invariant pairing on the source with the negative of the intersection pairing of W ⊥ on the target.
The morphism X * (A η ) → Pic 0 (C η ) is defined to be the composite where the first arrow is the one given by Corollary 4.14, the second is pullback along C 0 → X , and the third one is the natural isomorphism Pic(C 0 η ) ∼ = Pic(C η )/ P η ∼ = Pic 0 (C η ). We could equivalently define it as the composite is the orthogonal complement, and where the last arrow is now pullback along C → Y .
Proof. It suffices to check this statement at the geometric generic point of B rs . Let us write ·, · W for the Weil pairing on J [3] η = Pic(C η ) [3]. We will consider the factorization Let us write (·, ·) W ⊥ for the (negative definite) intersection pairing on Pic(X η ), and (·, ·) for its negative. The pairing (·, ·) corresponds under the isomorphism X * (A η ) ∼ = Pic(X η ) of Corollary 4.14 to the pairing on X * (A η ) which is also denoted by (·, ·). To prove the proposition, it is enough to show that the map Pic(X η ) → Pic(C 0 η ) factors through an isomorphism ψ : Pic(X η ) θ → Pic 0 (C η ) [3] which satisfies the identity for all α, β ∈ Pic(X η ). In order to do this, we will make everything explicit. Recall (cf. §2) that Pic(X η ) θ is isomorphic as an abelian group to (Z/3Z) 4 . Its 80 non-trivial elements are in bijective correspondence with the θ-orbits of root vectors in Pic(X η ). To prove the result, it will suffice to show that these 80 non-trivial elements are in bijection with the non-trivial elements of Pic(C η ) [3], and that if α, β ∈ Pic(X η ) are two root vectors, then they satisfy the identity (4.5).
The first case is where the values ω(γ), γ ∈ Σ(α, β), are pairwise distinct. In this case we see that both the Weil pairing ψ(α), ψ(β) W and the value given by (4.6) are equal to 1. Indeed, the Weil pairing can be computed using [BFT14, Lemma 5], while for (4.6) this is obvious.
The second case is where some value ω(γ) occurs exactly twice. Since our pairing does not depend on the choice of ζ, we can suppose without loss of generality that it is ζ that appears twice. Then we see that The only part of Proposition 4.3 that remains to be proved is that ∆ 0 = disc(x 5 + c 12 x 3 + c 16 x 2 + c 24 x + c 30 ) has the property that ∆ 2 0 is (up to scalar) the restriction to V of the usual Lie algebra discriminant ∆. Note that ∆ 2 0 and ∆ both have degree 240, that ∆ 0 is irreducible, and that ∆ 2 0 and ∆ vanish along the same points (as follows from e.g. [Slo80,§6.6]). This implies that they are equal up to scalar, as desired.

Spreading out
So far we have described the structure of the representation (G, V ) over Q. We recall (see §4.1) that it has a natural extension (G, V ) over Z. We now observe that all of the above works equally well over Z[1/N ] for some integer N ≥ 1.
Indeed, we can choose the invariant polynomials c 12 , c 18 , c 24 , c 30 ∈ Q[V ] G to lie in Z[V ] G (by using the G m -equivariant structure of X 0 described at the beginning of §4.4 to clear denominators). We set B = Spec Z[c 12 , . . . , c 30 ] and write π : V → B for the corresponding morphism (which extends the morphism between V → B on Q-fibres already denoted by π). Note that this implies that ∆ 0 = disc(x 5 + c 12 x 3 + c 18 x 2 + c 24 x + c 30 ) ∈ Z[V ] G too. We define B rs = Spec Z[c 12 , c 18 , c 24 , c 30 ][∆ −1 0 ]. We extend C to a family of projective curves C → B given by the same equation as before.
Then the analogue of Proposition 4.6 holds in GrLieE B rs ,fτ .
With these data in hand, we can extend our constructions of orbits from sections of Jacobians. We can therefore apply the results of §4.3 for S-algebras R (and not just Q-algebras). We mention in particular: 1. Let R be an S-algebra and let f ∈ B rs (R). Suppose given a tuple (H ′ , θ ′ , γ ′ ) ∈ GrLieE R,f . If (H ′ , θ ′ ) ∼ = (H R , θ R ), then (H ′ , θ ′ , γ ′ ) determines an element of G(R)\V f (R), a set which is in turn in canonical bijection with the set ker(H 1 (R, Z G (σ(f ))) → H 1 (R, G)).

Let
R be an S-algebra, and let f ∈ B rs (R). Suppose that every locally free R-module is free. Then there is an injective map η f :

Measures
The results of this section are used in the calculations of §6 -7. Let ω G be a generator for the (free rank 1 Z-module of) left-invariant top forms on G. It is uniquely determined up to sign, and determines Haar measures dg on G(R) and on G(Q p ) for each prime p.
Proof. Note that G has class number 1 (i.e. G(Q)\G(A ∞ )/G( Z) has 1 element). Therefore the product expresses the Tamagawa number of the simple group G = SL 9 /µ 3 , which equals 3 (apply the results of [Lan66] and [Ono65]).
Let ω V be a generator for the for the left-invariant volume forms on V , which is again determined up to sign, and determines Haar measures dv on V (R) and on V (Q p ) for every prime p. We write ω B for the volume form dc 12 ∧ dc 18 ∧ dc 24 ∧ dc 30 on B. It determines measures df on B(R) and on B(Q p ) for every prime p.
Proposition 4.17. There exists a constant W 0 ∈ Q × with the following properties: , and define a function m p : V (Z p ) rs → R ≥0 by the formula Then m p (v) is locally constant.
3. Let U 0 ⊂ G(R) and U 1 ⊂ B(R) rs be open subsets such that the product morphism µ : Here we write | · | p for the usual p-adic absolute value on Q p (with |p| p = p −1 ).
Proof. All of these identities can be proved in the same way as in [RTa,Proposition 3.3] and [Tho15, The key input in the proof is the equality dim Q V = i deg c i , which holds here since 84 = 12 + 18 + 24 + 30.

Constructing integral orbit representatives
We continue with the notation of §4. Let E denote the set of polynomials f (x) = x 5 + c 12 x 3 + c 18 x 4 + c 24 x + c 30 ∈ Z[x] of non-zero discriminant. If p is a prime, let E p denote the set of polynomials f (x) = x 5 + c 12 x 3 + c 18 x 4 + c 24 x + c 30 ∈ Z p [x] of non-zero discriminant. Thus we can identify E p = B(Z p ) rs := B(Z p ) ∩ B rs (Q p ).
This section is devoted to the proof of the following theorem concerning the map η f of Corollary 4.10.
Theorem 5.1. Let N be the integer of §4.5. Then for each prime p > N , for each polynomial f (x) ∈ E p , and for each Most of §5 is devoted to the proof of this theorem. We first prove the theorem for polynomials of square-free discriminant in §5.1. This is then used as an ingredient in the proof of the theorem in the general case in §5.2.

The case of square-free discriminant
In this section we establish Theorem 5.1 for polynomials f (x) ∈ E p of square-free discriminant. We first prove two useful lemmas.
acts nilpotently in h 0,k and invertibly in h 1,k . In fact, if x k = y s + y n is the Jordan decomposition of x k as a sum of its semisimple and nilpotent parts, then h 0,k = z h (y s ). We must show that y n is a regular nilpotent element of z h (y s ).
To see this, we first observe that there exists a unique closed subgroup L ⊂ H R such that Lie L = h 0,R and such that L is smooth over R with connected fibres. Moreover, we have L k = Z H (y s ). The uniqueness follows from [SGA70, Exp. XIV, Proposition 3.12]. To show existence, choose a regular semisimple element r ∈ z h (y s ) and an arbitrary lift r ∈ h R,0 . The centralizer Z H (r) is a maximal torus of H R with Lie algebra contained in h R,0 , and we can construct a Levi subgroup of H R with Lie algebra h R,0 after passage to ań etale extension R → R ′ where Z H (r) is split.
Using the results of [Slo80, §6.6], we see that the derived group of L has type A 2 and that the centre Z L has rank 6. Moreover, the action of µ 3 determined by θ restricts to an action on L, and the induced morphism θ L : µ 3 → Aut(L) is a stable Z/3Z-grading. (We have defined this notion in §2 for a reductive group of type E 8 , but the definition is the same here: in each geometric fibre, there is a maximal torus of L on which θ L defines an elliptic µ 3 -action.) To show that x k is regular in h k , we must show that y n is a regular nilpotent element in h 0,k . After passage to anétale extension R → R ′ of discrete valuation rings, we can find an isomorphism h der 0,R ∼ = sl 3,R under which θ L corresponds to the homomorphism ζ → Ad(diag(1, ζ, ζ 2 )). (The proof is the same as the proof of Lemma 2.3, using that the automorphism group of the A 2 root lattice contains a unique conjugacy class of elements of order 3.) Let ∆ ′ denote the Lie algebra discriminant of h 0,R . Then ord K ∆(x) = ord K ∆ ′ (x).
Let x ′ denote the projection of x to h der 0,R (this projection exists because of our assumption on the residue characteristic of R).
The image of x ′ in sl 3,R is given by a matrix and the discriminant ∆ ′ (x) equals (abc) 2 . If ord K (abc) 2 = 2 then exactly one of a, b or c is divisible by ̟, and in this case we see that the reduction modulo ̟ of x ′ (which coincides with y n ) is a regular nilpotent element. This proves our claim that V reg We next claim that Z G (σ(f )) satisfies the "Néron mapping property" Z G (σ(f ))(R ′ ) = Z G (σ(f ))(Frac(R ′ )) for anyétale extension R → R ′ of discrete valuation rings. In view of the identification of Z G (σ(f )) K with J f [3], we just need to show that the isomorphism Z G (σ(f )) K ∼ = J f [3] extends uniquely to an isomorphism Z G (σ(f )) ∼ = J f [3]. This will follow if we can show that the special fibre of Z G (σ(f )) has order 3 3 . This is the case. Writing now y s + y n for the Jordan decomposition of σ(f ) k and carrying through the above computation, we see that Z G (σ(f )) k can be identified with the θ L -fixed points in the centre of the group L k . Since the centre is a rank-6 torus on which θ L defines an elliptic µ 3 -action, this group indeed has order 3 3 .
(See [Tho13, Proposition 2.8] for a similar calculation.) The map G → V reg f , g → g · σ(f ) is surjective andétale, and in fact a torsor for theétale group scheme Z G (σ(f )). The only part of this claim that we have not already established is the fact that this map is surjective in the special fibre G k → V reg f,k . This is equivalent to showing that if y s ∈ V k is a semisimple element such that Z H (y s ) has derived group of type A 2 , then Z G (y s ) = Z H (y s ) θ acts transitively on the regular nilpotent elements of z h (y s )(1). This is true. (Note that in the (Z/3Z)-grading of sl 3,k given by ξ : ζ → Ad(1, ζ, ζ 2 ), SL ξ 3 does not act transitively on the regular nilpotent elements, but PGL ξ 3 does. Luckily in our situation the group Z H (y s ) fits into a θ-equivariant short exact sequence where C is a θ-elliptic torus. This implies that the map Z H (y s ) θ → PGL θ 3 is surjective.) It follows that the set G(R)\V f (R) is in bijection with ker(H 1 (R, Z G (σ(f ))) → H 1 (R, G)). By Lemma 5.3, the map H 1 (R, Z G (σ(f ))) → H 1 (K, Z G (σ(f ))) is injective, implying that the map α : We therefore just need to show that each class in the image of . This follows from the fact that the map Finally, suppose once more that R has finite residue field. Lang's theorem once again implies that H 1 (R, G) = {1} and H 1 (R, J f ) = {1}. This completes the proof.
Corollary 5.5. Let R be a PID in which N is a unit, and let f ∈ B(R) be a polynomial such that disc(f ) is square-free (as an element of R). Let K = Frac(R). Let P ∈ J f (K), and let γ P ∈ V f (K) be a representative of the orbit η f (P ). Then there exists g ∈ G(K) such that g · γ P ∈ V f (R).
After localizing, we can assume that R is a DVR. In this case, Proposition 5.4 implies that we can find g ∈ G(K) such that g · γ P ∈ V f (R). In other words, g defines an isomorphism between (H K , θ K , γ K ) and (H K , θ K , g · γ P ), and the latter triple extends naturally to (H R , θ R , g · γ P ) ∈ GrLieE R,f .

The general case
We now use the results just established in §5.1 to complete the proof of Theorem 5.1. Let us therefore take a prime p > N , a polynomial f (x) ∈ E p , and a point P ∈ J f (Q p ). We must show that the orbit η f (P ) ⊂ V f (Q p ) contains an element of V f (Z p ).
We first give an explicit representation of the point P . Arguing as in the proof of [BG13, Proposition 19], we can assume (after possibly changing P without changing its image in are monic of degrees ν and 5 − ν, respectively, and r 0 (x) has degree at most ν − 1. (This is the Mumford representation of P : thus P corresponds to the linear equivalence class of the divisor D − ν∞, where D ⊂ C 0 f,Qp is the effective divisor of degree ν determined by the equations y = r 0 (x), u 0 (x) = 0.) Let D ν denote the scheme (over Z p ) of tuples of polynomials (u(x), v(x), r(x)), where u(x), v(x) are monic of degrees ν and 5 − ν, respectively, and r(x) has degree at most ν − 1, and u(x)v(x) + r(x) 2 = x 5 + a 1 x 4 + a 2 x 3 + a 3 x 2 + a 4 x + a 5 satisfies a 1 = 0. Thus the tuple (u 0 (x), v 0 (x), r 0 (x)) determines a point of D ν (Z p ). Let δ ∈ H 0 (D ν , O Dν ) denote the discriminant of the (monic, degree 5) polynomial u(x)v(x) + r(x) 2 , and let D δ ν ⊂ D ν denote the closed subscheme defined by the vanishing of δ. Then D δ ν has codimension 1 in each fibre of D ν over Z p . (In fact, D δ ν is flat over B Zp ).
. Then disc f 1 = δ(u 1 , v 1 , r 1 ) is square-free, when viewed as an element of the ring Q p [λ], and its image in F p [λ] is non-zero.
(We can accomplish this by choosing e.g. . We first choose them so that the discriminant of f 1 (x) is not zero in F p [λ]. If the discriminant is not already square-free in Q p [λ] then by Bertini's theorem we can choose a small p-adic perturbation to make it so.) . We have constructed a smooth projective curve C f1 → U 1 , together with a section P 1 ∈ J f1 (U 1 ). Applying the construction described in §4.5, we obtain a tuple (H 1 , θ 1 , γ 1 ) ∈ GrLieE U1,f1 . The pullback of this tuple to GrLieE Qp,f along the point {λ = 0} ∈ U 1 (Q p ) corresponds to the orbit η f (P ) under the bijection of Lemma 4.5.
Using that disc(f 1 ) is square-free when viewed as an element of Q p [λ], we can apply Corollary 5.5 to find that there is an extension of the triple (H 1 [1/p], θ 1 [1/p], γ 1 [1/p]) to a similar triple (H 2 , θ 2 , γ 2 ) over U 2 . We can glue these triples to obtain a similar triple (H 0 Observe that by construction, θ 0 is a stable Z/3Z-grading of H 0 .
Note that the complement of U 0 in Spec Z p [λ] is a union of finitely many closed points in the special fibre. We now apply the following lemma.
Lemma 5.6. Let T be an integral regular scheme of dimension 2, and let Z ⊂ T be a closed subset of dimension 0. Let U = T − Z. Then restriction M → M U defines an equivalence between the following two categories: 1. The category of reductive groups over T , with morphisms given by isomorphisms of group schemes. Applying Lemma 5.6, we see that H 0 extends uniquely to a reductive group H 2 over Spec Z p [λ], and that θ 0 extends uniquely to a grading θ 2 : µ 3 → H 2 , and γ 0 comes from a unique section γ 2 ∈ V 2 = h 2 (1). Note that θ 2 is a stable Z/3Z-grading of H 2 . It follows that (H 2 , θ 2 , γ 2 ) is an object of the category GrLieE Zp[λ],f1 considered in §4.3. By construction, its pullback to GrLieE Qp,f along the map λ = 0 corresponds, under the bijection of Lemma 4.5, to the orbit η f (P ) ∈ G(Q p )\V f (Q p ).
. This completes the proof of Theorem 5.1.

Complements
We conclude §5 with a weak result that holds for every prime (not just primes p > N ). The G m -action on B here is the standard one (where t · c i = t i c i ).
Proposition 5.7. Let p be a prime, and let f 0 (x) ∈ E p . Then there exists an integer n ≥ 1 and an open neighbourhood W p of f 0 in E p such that for all f ∈ W p and for all y ∈ J p n ·f (Q p ), the orbit η p n ·f (y) ∈ G(Q p )\V p n ·f (Q p ) contains an element of V p n ·f (Z p ).
Proof. Choose n ≥ 1 such that each orbit in the image of η p n ·f0 intersects V p n ·f0 (Z p ). Let σ 1 , . . . , σ r ∈ V p n ·f0 (Z p ) be representatives for the distinct G(Q p )-orbits in the image of η p n ·f0 . For each i = 1, . . . , r, we can find an open neighbourhood U ′ p,i ⊂ V (Z p ) of σ i with the following properties: 3. For each g ∈ U p , the elements s i (g) represent the distinct G(Q p )-orbits in the image of η g . This essentially follows from the fact that the action map G → V f attached to any f ∈ B rs (Q p ), x ∈ V f (Q p ), isétale. After possibly shrinking U p , we can assume that it has the form p n · W p for some open compact subset W p ⊂ B(Z p ) rs which contains f 0 . This completes the proof. Proof. We have G(A ∞ ) = G(Q)G( Z). It follows that for a given element v ∈ V (Q), finding g ∈ G(Q) such that g · v ∈ V (Z) is equivalent to finding for each prime p an element g p ∈ G(Q p ) such that g · v ∈ V (Z p ). The result therefore follows on combining Theorem 5.1 and Proposition 5.7.

Counting points
We retain the notation of §4. In particular, we have a reductive group G over Z acting on a free Z-module V , and a G-equivariant morphism π : V → B = Spec Z[c 12 , c 18 , c 24 , c 30 ] (where G acts trivially on B). For f ∈ B(Z), we define ht(f ) = sup i |c i (f )| 120/i . If v ∈ V (Z), then we define ht(v) = ht(π(v)).

Counting points with finitely many congruence conditions
Suppose given an integer M ≥ 1 and a G(Z/M Z)-invariant function w : V (Z/M Z) → R ≥0 . We define We define µ w to be the average value of w (where V (Z/M Z) gets its uniform probability measure).
For a field k/Q, we say that v ∈ V (k) is k-reducible if v has zero discriminant or if v is G(k)-conjugate to the Kostant section σ(π(v)) ∈ V (k). Otherwise we say that v is k-irreducible. If X ⊂ V (Q) is any subset, then we write X irr for its intersection with the set of Q-irreducible elements.
The first main result of this section concerns the number of G(Z)-orbits of Q-irreducible elements of V (Z) of bounded height: Theorem 6.1. We have where W 0 denotes the constant of Proposition 4.17.
The proof of Theorem 6.1 is very similar to the proofs of earlier results like [BG13, Theorem 36] (see also [Tho15,§3]). Rather than repeat details word for word here, we instead give the key propositions, which can be inserted into the arguments at the appropriate points. In comparing what we prove here with the results of [BG13] it's useful to note that because σ(B rs (R)) contains exactly one representative for each orbit of G(R) on V (R) rs , it may be used to construct a fundamental set for the action of R >0 × G(R) on V (R) rs (cf. [BG13, Section 9.1]), and also that the stabilizer in G(R) of every element in V (R) rs has order 9 (because for any f ∈ B rs (R), J f (R) [3] has order 9).
Following, e.g., [BG13, Section 10], the only arguments that do not carry over easily are those that show we can bound the contribution from the cusp region in a fundamental domain for the action of G(Z) on V (R). To do this, by the same logic as in the proof of [Tho15, Theorem 3.6], it suffices to check that certain combinatorial properties hold in the set of weights for the action of G on V .
We start by defining some notation. We write S G = {β 1 , ..., β 8 } for the root basis of Φ G fixed in §4.1. Then any γ ∈ X * (T ) may be written uniquely as γ = 8 i=1 n i (γ)β i for some n i (γ) ∈ Q. Note that the Cartan decomposition h = t ⊕ α∈ΦH h α of h is preserved by the action of µ 3 via θ. We define Φ V = {α ∈ Φ H | h α ⊂ V }; then V = ⊕ α∈ΦV h α . Given a vector v ∈ V , we write v = α∈ΦV v α for its decomposition as a sum T -eigenvectors. We write Φ + V for Φ + H ∩ Φ V . Given a subset M ⊂ Φ V , we define V (M ) = {v ∈ V | v α = 0 for all α in M }. The following lemma, which is a variant of [RTa, Proposition 2.15], gives criteria for the vectors in V (M ) to be reducible.
Lemma 6.2. Let k/Q be a field. Given a subset M ⊂ Φ V , suppose one of the following three conditions is satisfied: 2. There exist integers a 1 , . . . , a 8 not all equal to zero such that if α ∈ Φ V and 8 i=1 a i n i (α) > 0, then α ∈ M .
3. There exist β ∈ Φ G , α ∈ Φ V − M , and integers a 1 , . . . , a 8 not all equal to zero such that the following conditions hold: integers a 1 , . . . , a 8 not all equal to zero, and integers b 1 , ..., b 8 not all equal to zero such that the following conditions hold: Then every element of V (M )(k) is k-reducible.
Proof. If one of the first three conditions is satisfied, the fact that the elements of V (M )(k) are k-reducible is given by a proof identical to that of [RTa,Proposition 2.15]. To prove that the fourth criterion implies reducibility, and so is reducible by condition 2 of the lemma. Thus we may assume v α = 0 and v α+β = 0. Let U −β ⊂ G be the root subgroup corresponding to −β ∈ Φ G . Note that there exists u ∈ U −β such that u · (v α + v α+β ) = cv α+β for some constant c. By condition (a), we have u · v ∈ V (M )(k), and so by our choice of u we have u · v ∈ V (M ∪ {α}). Thus v is k-reducible as desired. Proposition 6.3. There exists a unique root λ 0 ∈ Φ V that is maximal with respect to the partial ordering induced by the root basis S G , or in other words such that n i (λ 0 ) ≥ n i (λ) for all i ∈ {1, . . . , 8} and all λ ∈ Φ V . There exists a collection C of cusp data such that Proof. Each weight λ ∈ Φ V admits a unique expression λ = 8 i=1 n i (λ)β i for some n i (λ) ∈ Q. One checks that the height h G (λ) := 8 i=1 n i (λ) achieves its maximal value 9 exactly once, at the maximal weight λ 0 (which we note is not in this case the highest root of Φ H with respect to the root basis S H ).
We use a computer to generate the collection C using a procedure very similar to the proof of [RTa,Proposition 4.5]. Yet here the criteria corresponding to [RTa, Proposition 2.15] is not enough to complete the proof: we must use part 4 of Lemma 6.2 to eliminate additional cusp data (M 0 , M 1 ) such that V (M 0 , M 1 )(Q) irr = ∅. The details of this computation can be found in the Mathematica notebook https://www.dpmms.cam.ac.uk/~jat58/E8(3)CuspData.nb.
Let N be the integer of §4.5, and let p > N be a prime. We define V red p ⊂ V (Z p ) to be the set of vectors v ∈ V (Z p ) such that either p|∆ 0 (v), or p ∤ ∆ 0 (v) and the image v of v in V (F p ) is G(F p )-conjugate to σ(π(v)). Similarly, we define V bigstab p ⊂ V (Z p ) to be the set of vectors v ∈ V (Z p ) such that either p|∆ 0 (v), or p ∤ ∆ 0 (v) and the image v of v in V (F p ) has non-trivial stabilizer in G(F p ). Proof. This can be proved using [Ser12, Proposition 9.15]. We illustrate the method for V bigstab p . The number of points of V (F p ) of zero discriminant is O(p 83 ). The number of points of V (F p ) of non-zero discriminant equals |B rs (F p )||G(F p )|. For a prime p ≡ 1 mod 3, let C ⊂ Sp 4 (F 3 ) be the set of elements γ which have 1 as an eigenvalue. Then [Ser12, Proposition 9.15] gives v∈V bigstab

Counting points with infinitely many congruence conditions
We now observe that using the results of [Bha] (see also [BS15]), we can get a strengthened version of Theorem 6.1 where we impose infinitely many congruence conditions. This is the analogue of [BG13, Theorem 42]. We state this following [BG13]. Suppose given for each prime p a G(Z p )-invariant function w p : V (Z p ) → [0, 1] satisfying the following conditions: • w p is locally constant outside the closed subset V (Z p ) − V (Z p ) rs ⊂ V (Z p ).
• For all sufficiently large primes p, we have w p (v) = 1 for all v ∈ V (Z p ) such that p 2 ∤ ∆ 0 (v).
Then we can define a function w : V (Z) → [0, 1] by the formula w(v) = p w p (v) if ∆ 0 (v) = 0, and w(v) = 0 otherwise. If X ⊂ V (Z) is an G(Z)-invariant subset, then we define Our strengthened theorem is then as follows.
Following the proof of [BS15, Proposition 25], for primes p > N we define Let W 1 p ⊂ W p denote the set of points v such that either π(v) mod p has either more than 1 repeated root or a triple root, or such that v mod p is not regular. (The proof of Proposition 5.4 shows that if v is such an element, then ∆ 0 (v) is necessarily divisible by p 2 .) Let W 2 p ⊂ W p denote the set of points v such that π(v) mod p has 1 double root and no other repeated roots, and such that v mod p is regular. Then W p = W 1 p ∪ W 2 p . In order to prove Theorem 6.5 using the method of [Bha] (or [BS15, Theorem 24]), it will suffice to define a map ψ : G(Z)\(V (Z) ∩ W 2 p ) → G(Z)\(V (Z) ∩ W 1 p ) with the following properties: There is an isomorphism h der 0,Zp ∼ = sl 3,Zp which intertwines θ| h der 0,Zp with ζ → Ad(diag(1, ζ, ζ 2 )), and which sends y n to the element   0 1 0 0 0 1 0 0 0   .
of sl 3,Fp . (Indeed, there is a unique such isomorphism modulo p, which then lifts by Hensel's lemma to an isomorphism over Z p .) Similarly, there is a map ϕ v : SL 3,Zp → L which intertwines Ad(diag(1, ζ, ζ 2 )) with θ L = θ| L and which is compatible with the above isomorphism on Lie algebras. The map ϕ v is uniquely determined up to conjugation by diagonal matrices in PGL 3 (Z p ); the element g v,p = ϕ v (diag(p, 1, p −1 )) ∈ L(Q p ) is therefore independent of any choices.
To see that this g v,p has the desired properties, let v ′ denote the projection of v to h der 0,Zp , and note that the image of v ′ in sl 3,Zp has the form where a ≡ b ≡ 1 mod p and p 2 |c (because of our assumption that p 2 divides ∆ 0 (v)). Thus we have The reduction modulo p of g v,p · v is no longer regular, showing that g v,p · v ∈ W 1 p . This defines the map ψ p .
We now need to describe the set Π p (w) for w ∈ W 1 p . Let w Fp = z s + z n be the Jordan decomposition. As before, we get a decomposition h Zp = h 0,Zp ⊕ h 1,Zp where ad(w) acts topologically nilpotently in h 0,Zp and invertibly in h 1,Zp , and h 0,Zp is the Lie algebra of a Levi subgroup L ⊂ H Zp .
Observe that if w = g v,p · v for some v ∈ W 2 p , then the derived subalgebra of z h (z s ) is isomorphic to sl 3,Fp (i.e. is split) and its grading is conjugate to the Z/3Z-grading given by the formula ζ → Ad(diag(1, ζ, ζ 2 )) (in fact, it coincides with the derived subalgebra of z h (y s ) in the above discussion). We can therefore assume without loss of generality that z h (z s ) is split and has a grading of this form (otherwise Π p (w) is empty).
If we fix an isomorphism between z h (z s ) der and sl 3,Fp which identifies θ| z h (zs) der with the Z/3Z-grading ζ → Ad(diag(1, ζ, ζ 2 )), then there is a compatible morphism SL 3,Zp → L, uniquely determined up to conjugation by diagonal elements of PGL 3 (Z p ), and we get an element h p ∈ H(Q p ), image of diag(p −1 , 1, p) ∈ SL 3 (Q p ).
There are three possible choices of isomorphism between z h (z s ) der and sl 3,Fp , up to SL θ 3,Fp -conjugacy, so we get three elements h p ∈ L(Q p ). The set Π p (w) is contained in the set of elements h p constructed this way, showing that Π p (w) has cardinality at most 3. The other claimed properties of the set Π p (w) follow from the definition. We have therefore completed the proof of Theorem 6.5.

The main theorem
We can now prove the theorems stated in the introduction. We begin by re-establishing notation. Thus E denotes the set of polynomials f (x) = x 5 + c 12 x 3 + c 18 x 2 + c 24 x + c 30 ∈ Z[x] of non-zero discriminant, and E min ⊂ E denotes the set of polynomials f (x) not of the form n · g = n 10 g(x/n 2 ) ∈ E for any g ∈ E and integer n ≥ 2. If f ∈ E , then we define the height of f by the formula ht(f ) = sup i |c i | 120/i . Thus for any a > 0, the set {f ∈ E | ht(f ) < a} is finite. We recall that the set E min is in bijection with the set of isomorphism classes of pairs (C , P) where C is a (smooth, projective, connected) genus-2 curve over Q and P ∈ C (Q) is a marked Weierstrass point, via the map which takes f ∈ E to the projective completion of the affine curve C 0 f : y 2 = f (x).
We first prove a 'local' result. Let G, V be the group and representation defined in §4, and let N ≥ 1 be the integer of §4.5; thus our main constructions make sense over Z[1/N ]. If p is a prime, then we write E p for the set of polynomials f (x) = x 5 + c 12 x 3 + c 18 x 2 + c 24 x + c 30 ∈ Z p [x] of non-zero discriminant, and E p,min ⊂ E p for the set of polynomials not of the form p 10 g(x/p 2 ) for any polynomial g(x) ∈ E p .
Proposition 7.2. Let f 0 (x) ∈ E min . Then we can find for each prime p ≤ N an open compact neighbourhood W p of f 0 (x) in E p such that the following condition holds. Let E W = E ∩ ( p≤N W p ), and let E W,min = E W ∩ E min . Then we have lim a→∞ f ∈EW,min,ht(f )<a | Sel 3 (J f )| |{f ∈ E W,min | ht(f ) < a}| = 4.
(The intersection E ∩ ( p≤N W p )) is taken in p≤N E p , where we view E as a subset via the diagonal embedding.) Proof. We choose the sets W p for p ≤ N , together with integers n p ≥ 0, so that the conclusion of Corollary 5.8 holds. If p > N , let W p = E p,min and n p = 0. Let M = p p np . After possibly shrinking the W p with p ≤ N , we can assume that the W p with p ≤ N satisfy W p ⊂ E p,min .
For v ∈ V (Z) with π(v) = f , define w(v) ∈ Q ≥0 by the following formula: For v ∈ V (Z p ) with π(v) = f , define w p (v) ∈ Q ≥0 by the following formula: otherwise.
Then for any v ∈ V (Z), we have w(v) = p w p (v), and the function w satisfies the conditions described before the statement of Theorem 6.5.
Let W 0 ∈ Q × be the constant of Proposition 4.17. That proposition implies that for any prime p, we have the formula v∈V (Zp) Running through the same argument as in the proof of Proposition 7.2, we get lim sup a→∞ f ∈E k ,ht(f )<a | Sel 3 (J f ) r | − 1 a 7/10 |J f [3](Q)| ≤ 3 p≤N vol(⊔ i≥k W p,i ), which becomes arbitrarily small as k → ∞. This completes the proof of Theorem 7.1.
Remark 7.3. Using Theorem 6.5 and [BG13, Theorem 44], one can prove the analogue of Theorem 7.1 for any 'large' subset of E min , where 'large' has the same meaning as in [BG13,§11]; this includes in particular any subset defined by finitely many congruence conditions on the cofficients of f (x) = x 5 +c 12 x 3 +c 18 x 2 +c 24 x+c 30 .
Proof. According to [PS14,Remark 10.5], this follows if one can establish property Eq 2 (3) of op. cit., which asserts that after fixing a 'trivializing congruence class' U 3 ⊂ E 3,min in which the groups J f (Q 3 )/3J f (Q 3 ) = F are independent of f ∈ U 3 , the images x| 3 of 3-Selmer elements x ∈ Sel 3 (J f ) in the local groups J f (Q 3 )/3J f (Q 3 ) = F are equidistributed for f ∈ E min ∩ U 3 . This can be proved by a small modification of the proof of Theorem 7.1, analogous to the proof of [BG13, Theorem 47]. We omit the details.