aa r X i v : . [ m a t h . N T ] J a n Arithmetic statistics of Prym surfaces
Jef LagaJanuary 29, 2021
Abstract
We consider a family of abelian surfaces over Q arising as Prym varieties of double covers of genus- curves by genus- curves. These abelian surfaces carry a polarization of type (1 , and we show that theaverage size of the Selmer group of this polarization equals . Moreover we show that the average sizeof the -Selmer group of the abelian surfaces in the same family is bounded above by . This implies anupper bound on the average rank of these Prym varieties, and gives evidence for the heuristics of Poonenand Rains for a family of abelian varieties which are not principally polarized.The proof is a combination of an analysis of the Lie algebra embedding F ⊂ E , invariant theory, aclassical geometric construction due to Pantazis, a study of Néron component groups of Prym surfacesand Bhargava’s orbit-counting techniques. Contents V
487 Counting integral orbits in V ⋆
608 Proof of the main theorems 64 Introduction
Let λ : A → B be an isogeny of abelian varieties over Q . The λ - Selmer group of A is defined by Sel λ A := ker H ( Q , A [ λ ]) → Y v H ( Q v , A ) ! , where A [ λ ] denotes the kernel of λ , the cohomology groups are Galois cohomology and the product runs overall places v of Q . It is a finite group defined by local conditions and fits in an exact sequence → B ( Q ) /λ ( A ( Q )) → Sel λ A → X ( A/ Q )[ λ ] → . The determination of
Sel λ A , known as performing a λ -descent, is often the first step towards determiningthe finitely generated abelian groups A ( Q ) and B ( Q ) . One is therefore led to ask how Sel λ A behaveson average as λ varies in families. When A = B ranges over a family of Jacobian varieties and λ ismultiplication by an integer, the last ten years have seen spectacular progress in this direction; see forexample [BS15a, BS15b, BG13, SW18, RT] for works of particular relevance to this paper. There are someresults when A is not a Jacobian variety (see for example [BKLOS19, Mor19, MP20]) but they concern twistsof a single abelian variety over Q , therefore considering only an isotrivial family in the relevant moduli space.By contrast in this paper we study a non-isotrivial family of abelian varieties which are not Jacobians. Let E ⊂ Z be the subset of -tuples of integers b = ( p , p , p , p ) such that the projective closure of theequation y + p xy + p y = x + p x + p (1.2.1)defines a smooth genus- curve C b over Q . The quotient of C b by the involution τ ( x, y ) = ( x, − y ) is anelliptic curve E b given by the equation y + p xy + p y = x + p x + p . (1.2.2)The associated morphism f : C b → E b is a double cover ramified at four points, namely the ones with y = 0 and the point at infinity. The families of curves C b and E b parametrized by such b have a moduliinterpretation, see Remark 1.6.Let J b be the Jacobian variety of C b and let P b be the kernel of the norm map f ∗ : J b → E b . Then P b is anabelian surface carrying a polarization ρ : P b → P ∨ b of type (1 , . (This means that P b [ ρ ]( Q ) ≃ ( Z / Z ) .)It is called the Prym variety associated to the double cover C b → E b . The abelian threefold J b is isogenousto P b × E b .For b ∈ E we define the height of b as ht ( b ) = max | p i ( b ) | /i . Note that for every X ∈ R > , the set { b ∈ E | ht ( b ) < X } is finite. Theorem 1.1 (Theorem 8.7) . The average size of
Sel ρ P b for b ∈ E , when ordered by height, equals . Moreprecisely, we have lim X →∞ P b ∈ E , ht ( b ) Sel P b for b ∈ E , when ordered by height, is boundedabove by . More precisely, we have lim sup X →∞ P b ∈ E , ht ( b ) We expect that the limit in Theorem 1.2 exists and equals , see the end of §1.3. We mention a few standard consequences of the above theorems. The first one concerns the Mordell–Weilrank rk( P b ) of P b . Using the inequalities P b ) ≤ rk( P b ) ≤ P b , Theorem 1.2 immediately implies: Corollary 1.4. The average rank of P b for b ∈ E , when ordered by height, is bounded above by / . Because the rank of J b equals the sum of the ranks of its isogeny factors P b and E b , Corollary 1.4 also givesa bound on the average rank of the family of Jacobians J b for b ∈ E , once a bound for the average rank of E b is known. Since the statistical properties of Selmer groups of the family of elliptic curves E b reduce tothose of the family of elliptic curves in short Weierstrass form (see Remark 8.4), we may use the previouslyobtained estimates in the case of elliptic curves [BS13b, Theorem 3] to obtain: Corollary 1.5. The average rank of J b for b ∈ E , when ordered by height, is < / . 885 = 3 . . The basic proof strategy is the same as the one employed in previous works: for each of the isogenies ρ and [2] , we construct a representation of a reductive group over Q whose integral orbits parametrize Selmerelements and then count those orbits using the geometry-of-numbers techniques pioneered by Bhargava andhis collaborators. Given the robustness of these counting techniques, the crux of the matter is finding theright representation in the first place and showing that its rational orbits relate to the arithmetic of ourisogeny of interest.Previous cases suggest that relevant representations can very often be constructed using graded Lie algebras.In the special case of Z / Z -gradings on simply laced Lie algebras, Thorne [Tho13] has made this very explicitusing the connection with simple singularities [Slo80], paving the way for studying the -Selmer groups ofcertain families of curves using orbit-counting techniques. (See the introduction of [Lag20] for a more detailedexposition.) In the classical cases A n or D n the families of curves in question are hyperelliptic with markedpoints and most of these results were already obtained using different methods (where the papers [BG13,SW18, Sha19] handle the cases A n , A n +1 , D n +1 respectively), but in the exceptional cases E , E , E thecurves are not hyperelliptic and this framework has led to new results: see [Tho15, RT18, RT, Lag20].The present work is an attempt to incorporate non-simply laced Dynkin diagrams in the above picture, andmore specifically the Dynkin diagram of type F . The starting observation is the following. If h E is a simplecomplex Lie algebra of type E , then there exists an involution ζ : h E → h E whose fixed point subalgebra h ζ E is a simple complex Lie algebra of type F . This procedure is somewhat informally depicted as foldingthe Dynkin diagram of E : 3t suggests that studying the F case should correspond to studying the E case equivariantly with respectto the symmetry of the Dynkin diagram. This viewpoint is already present in the work of Slodowy [Slo80]where he identifies the restriction of the adjoint quotient of the F Lie algebra to a subregular transverseslice as the semi-universal deformation of the E surface singularity with ‘fixed symmetries’, and analogouslyfor other non-simply laced Lie algebras. We will approach Theorems 1.1 and 1.2 similarly.In more detail, we will define an involution ζ on the representation ( G E , V E ) constructed by Thorne in the E case, whose fixed points give rise to a representation V of a reductive group G . The family C of Equation(1.2.1) is then the subfamily of the semi-universal deformation of the E curve singularity (explicitly givenby Equation (3.1.1)) to which the involution τ ( x, y ) = ( x, − y ) lifts. In our previous work [Lag20] we haveconstructed an embedding of Sel J b in the set of G E ( Q ) -orbits of V E ( Q ) . The techniques of that papercombined with a detailed study of the action of τ allow us to embed Sel P b into the set of G ( Q ) -orbits of V ( Q ) admitting integral representatives.It then seems that Theorem 1.2 follows from geometry-of-numbers arguments to count integral orbits in V ,but there is a catch: such arguments will only allow us to count ‘strongly irreducible’ elements of Sel P b .To explain what this means, note that there exists a unique isogeny ˆ ρ : P ∨ b → P b such that [2] = ˆ ρ ◦ ρ , givingrise to the exact sequence Sel ρ P b → Sel P b → Sel ˆ ρ P ∨ b . We say an element of Sel P b is strongly irreducible if it has nontrivial image in Sel ˆ ρ P ∨ b . Estimating Sel P b then breaks up into two parts: estimating the strongly irreducible elements (which can be done using therepresentation V ), and Sel ρ P b . This is not unlike the situation of [BS13a], where the representation usedin that paper only counts elements of the -Selmer group of an elliptic curve of exact order , i.e. havingnontrivial image in the -Selmer group.Therefore to prove Theorem 1.2 it remains to prove Theorem 1.1, which we focus on now. Using a classicalgeometric construction going back to Pantazis, we may reduce to estimating the size of Sel ˆ ρ P ∨ b instead. Aconstruction in invariant theory which we call the ‘resolvent binary quartic’ allows us to embed Sel ˆ ρ P ∨ b in theset of PGL ( Q ) -orbits of binary quartic forms with rational coefficients. Counting orbits of integral binaryquartic forms using the techniques of [BS15a] and modifying the local conditions leads to the determinationof the average size of Sel ˆ ρ P ∨ b , proving Theorem 1.1 and consequently Theorem 1.2.We end this introduction by discussing some limitations, questions and remarks. We only obtain an upperbound in Theorem 1.2 because we are unable to prove a uniformity estimate similar to [BS15a, Theorem2.13] hence we cannot apply the so-called square-free sieve to obtain an equality in Theorem 7.7. We expectthat a similar such estimate holds and that the average size of Sel P b equals . For proving an equalityin Theorem 1.1, we bypassed proving such a uniformity estimate by reducing it to the one established byBhargava and Shankar [BS15a, Theorem 2.13]. The crucial ingredient for this reduction step is Corollary5.20 which is based on a detailed analysis of Néron component groups of certain Prym varieties in §5.3.The fact that the ˆ ρ -Selmer group of P ∨ b (and so consequently, by the ‘bigonal construction’ of Theorem 3.14,the ρ -Selmer group of P b ) has an interpretation in terms of binary quartic forms (Theorem 4.14) might beof independent interest. It seems conceivable that a further analysis would make the computation of Sel ρ P b possible using binary quartic forms, similar to the computation of the -Selmer group of an elliptic curve.4e compare our results with the heuristics of Poonen and Rains [PR12], which provide a framework forstatistics of Selmer groups using random matrix models. The self-dual isogeny ρ : P b → P ∨ b is defined bya symmetric line bundle, so [PR12, Theorem 4.13] shows that Sel ρ P b is the intersection of two maximalisotropic subspaces of an infinite-dimensional quadratic space over F . It is therefore natural to ask whetherthe distribution of ρ P b coincides with the one modelling -Selmer groups of elliptic curves (Conjecture1.1 of op. cit.); Theorem 1.1 provides evidence for this. On the other hand, the isogeny [2] : P b → P b is notself-dual and a different type of matrix model is needed. We hope to return to this in a future work. Remark 1.6. The families of curves considered here have a moduli interpretation. Loosely speaking, Equa-tion (1.2.2) defines the universal family of elliptic curves with a marked line in its Weierstrass embedding(here given by intersecting with the line { y = 0 } ) not meeting the origin ∞ , and Equation (1.2.1) describesthe double cover of this elliptic curve branched along the marked line and ∞ . Remark 1.7. Stable gradings on nonsimply laced Lie algebras have played a role before in arithmetic statis-tics. In [BES20] , the authors study the -isogeny Selmer group of the family of cubic twist elliptic curves y = x + k . There they implicitly use a Z / Z -grading on a Lie algebra of type G . This forms the startingpoint of the previously cited results of [BKLOS19] , so graded Lie algebras play a role there too. Remark 1.8. Bhargava and Ho have studied the representation V before in the context of invariant theoryof genus- curves (cf. Entry 10 of [BH16, Table 1] ). It would be interesting to relate their geometricconstructions to ours, and to see how the Prym variety fits in their description. In §2 we define the representation ( G , V ) , summarize its invariant theory and describe it explicitly. Moreoverwe describe the resolvent binary quartic of an element of V . In §3, we start by establishing a link betweenstable orbits in V and the family of curves C → B . Then we introduce the family of Prym varieties P → B and study its geometry. The construction of orbits associated with Selmer elements is the content of §4.We start by embedding the -Selmer group inside the space of rational orbits of the representation V . Wethen define a new representation V ⋆ of G ⋆ (very closely related to binary quartic forms) and embed the ˆ ρ -Selmer group inside the space of rational orbits of V ⋆ . In §5, we prove that orbits coming from Selmerelements admit integral representatives away from small primes. Then we count integral orbits of V and V ⋆ using geometry-of-numbers techniques in §6 and §7 respectively. Finally in §8 we combine all of the aboveingredients and prove Theorems 1.1 and 1.2. This research has been carried while the author was a PhD student under the supervision of Jack Thorne.I want to thank him for suggesting the problem, providing many invaluable suggestions and his constantencouragement. I am also grateful to Beth Romano for useful discussions. This project has received fund-ing from the European Research Council (ERC) under the European Union’s Horizon 2020 research andinnovation programme (grant agreement No. 714405). For a field k we write ¯ k for a fixed algebraic closure of k . It is the group Sel ρ P b / X ( Q , P b [ ρ ]) that is the intersection of two maximal isotropic subspaces, but because P b [ ρ ] ≃ E b [2] , X ( Q , P b [ ρ ]) vanishes by [PR12, Proposition 3.3]. X is a scheme over S and T → S a morphism we write X T for the base change of X to T . If T = Spec A is an affine scheme we also write X A for X T . We write X ( S ) for the set of sections of the structure map X → S and X ( T ) = X T ( T ) .If λ : A → B is a morphism between group schemes we write A [ λ ] for the kernel of λ .If T is a torus over a field k and V a representation of T , we write Φ( V, T ) ⊂ X ∗ ( T ) for the set of weights of T on V . If H is a group scheme over k containing T , we write Φ( H, T ) for Φ(Ad H, T ) , where Ad H denotesthe adjoint representation of H .If G is a smooth group scheme over S we write H ( S, G ) for the set of isomorphism classes of étale sheaftorsors under G over S , which is a pointed set coming from non-abelian Čech cohomology. If S = Spec R wewrite H ( R, G ) for the same object. If G → S is affine then every sheaf torsor under G is representable by ascheme.If G → S is a group scheme acting on X → S and x ∈ X ( T ) is a T -valued point, we write Z G ( x ) → T forthe centralizer of x . If x is an element of a Lie algebra h , we write z h ( x ) for the centralizer of x , a subalgebraof h .A Z / Z - grading on a Lie algebra h over a field k is a direct sum decomposition h = M i ∈ Z / Z h ( i ) of linear subspaces of h such that [ h ( i ) , h ( j )] ⊂ h ( i + j ) for all i, j ∈ Z / Z . If is invertible in k then givinga Z / Z -grading is equivalent to giving an involution of h .If V is a finite free R -module over a ring R we write R [ V ] for the graded algebra Sym( R ∨ ) . Then V isnaturally identified with the R -points of the scheme Spec R [ V ] , and we call this latter scheme V as well. If G is a group scheme over R we write V // G := Spec R [ V ] G for the GIT quotient of V by G . V In this section we define the pair ( G , V ) using a Z / Z -grading on a Lie algebra of type F . We will define itby embedding it in a larger representation ( G E , V E ) defined in [Lag20, §2.1] using a Z / Z -grading on a Liealgebra of type E , which we recall first. Objects related to V E will usually denoted by a subscript ( − ) E .Let H E be a split adjoint semisimple group of type E over Q with Lie algebra h E . We suppose that H E comes with a pinning ( T E , P E , { Y α } ) . So T E ⊂ H E is a split maximal torus (which determines a rootsystem Φ( H E , T E ) ⊂ X ∗ ( T E ) ), P E ⊂ H E is a Borel subgroup containing T E (which determines a root basis S H E ⊂ Φ( H E , T E ) ) and Y α is a generator for each root space ( h E ) α for α ∈ S H E . The group H E is of dimension .Let ˇ ρ E ∈ X ∗ ( T E ) be the sum of the fundamental coweights with respect to S H E , defined by the propertythat h ˇ ρ E , α i = 1 for all α ∈ S H E . Write ζ : H E → H E for the unique nontrivial automorphism preserving thepinning: it is an involution inducing the order- symmetry of the Dynkin diagram of E . Let θ E := ζ ◦ Ad(ˇ ρ E ( − ρ E ( − ◦ ζ. H Split adjoint group of type F §2.1 θ Stable involution of H §2.1 G Fixed points of θ on H §2.1 V ( − -part of action of θ on h §2.1 B GIT quotient V // G §2.1 ∆ ∈ Q [ B ] Discriminant polynomial §2.1 π : V → B Invariant map §2.1 σ : B → V Kostant section §2.2 H E Split adjoint group of type E §2.1 ζ : H E → H E Pinned automorphism of H E §2.1 θ E , G E , V E Analogous objects of H E §2.1 B E , π E , σ E Analogous objects of H E §2.1,2.2 Q v Resolvent binary quartic of v ∈ V §2.5 p , p , p , p G -invariant polynomials of V §3.1 C → B Family of projective curves §3.1, Eq.(3.1.2) τ : C → C Involution ( x, y ) ( x, − y ) §3.1 J → B rs Jacobian variety of C rs → B rs §3.1 Λ , W E E root lattice and its Weyl group §3.2 E → B Quotient of C by τ §3.3, Eq.(3.3.2) P → B rs Prym variety of the cover C rs → E §3.3 ρ : P → P ∨ Polarization of type (1 , §3.3 χ : B → B Automorphism arising from bigonal construction §3.4 ˆ X Pullback of a B -scheme X along χ §3.4 P → B Compactified Prym variety §3.5 G ⋆ PGL over Q §4.2 V ⋆ G ⋆ -representation Q ⊕ Q ⊕ Sym (2) §4.2 B ⋆ GIT quotient V ⋆ // G ⋆ §4.2 Q : V → V ⋆ Map v ( p ( v ) , p ( v ) , Q v ) §4.2 S Z [1 /N ] , where N sufficiently large integer §5.1 H , G , V , B , G ⋆ , . . . Extensions of above objects over Z §5.1 C → B , E → B Extension of C and E over Z §5.1 J → B rs S Jacobian of C rs S → B rs S §5.1 P → B rs S Prym variety of C rs S → E §5.1Table 1: Notation used throughout the paper7roup Type Isomorphism class Dimension H E E Adjoint G E C PSp H F Adjoint G C × A (Sp × SL ) /µ Table 2: Properties of the semisimple groups.Then θ E defines an involution of h E and thus by considering ( ± -eigenspaces it determines a Z / Z -grading h E = h E (0) ⊕ h E (1) . Let G E := H θ E E be the centralizer of θ E in H E ⊂ Aut( h E ) and write V E := h E (1) ; the space V E defines arepresentation of G E and its Lie algebra g E by restricting the adjoint representation. The pair ( G E , V E ) hasbeen studied extensively in [Lag20].We now consider the ζ -fixed points of the above objects. Let H := H ζ E and h := h ζ E . Then H is a splitadjoint semisimple group of type F with Lie algebra h , and the pinning of H E induces a pinning of H , cf.[Ree10, §3.1]. Indeed, T := T ζ E is a split maximal torus and P := P ζ E is a Borel subgroup containing T .They determine a root system Φ( H , T ) ⊂ X ∗ ( T ) and a root basis S H ⊂ Φ( H , T ) respectively. The naturalmap X ∗ ( T E ) → X ∗ ( T ) restricts to a surjection S H E → S H where two different elements β, β ′ ∈ S H E definethe same element of S H if and only if β ′ = ζ ( β ) . (The map S H E → S H can be seen as ‘folding’ the E Dynkin diagram alluded to in the introduction.) If α ∈ S H we write [ α ] for its inverse image in S H E underthe projection map. We define X α := P β ∈ [ α ] Y β ∈ h α . Then the triple ( T , P , { X α } ) is a pinning of H .Since θ E commutes with ζ , the restriction θ := θ E | H defines an involution H → H . We have θ = Ad ˇ ρ ( − ,where ˇ ρ ∈ X ∗ ( T ) is the sum of the fundamental coweights with respect to S H . As before this determines a Z / Z -grading h = h (0) ⊕ h (1) . Let G = H θ be the centralizer of θ in H and write V := h (1) . Again V defines a representation of G and itsLie algebra g . The pair ( G , V ) is the central object of study in this paper. We summarize some of its basicproperties here. Proposition 2.1. The groups G E , H , G are split connected semisimple groups over Q with maximal torus T .Their properties are listed in Table 2. The vector spaces V E and V have dimension and respectively.Proof. The properties of H follow from [Ree10, Lemma 3.1]. The isomorphism class of G E and G over Q canbe deduced from the analysis of the Kač diagrams of the automorphisms θ E , θ given in [RLYG12, §7.1, Tables2 and 6], using the results of [Ree10]. (The notation (Sp × SL ) /µ means the quotient by the diagonallyembedded µ ֒ → Sp × SL .) These groups are split since T is a split torus of maximal rank.The next proposition shows that regular semisimple orbits of V over algebraically closed fields are wellunderstood. For a field k/ Q , we say v ∈ V ( k ) is regular, nilpotent, semisimple respectively if it is so whenconsidered as an element of h ( k ) . Proposition 2.2. Let k/ Q be a field. The following properties are satisfied:1. V k satisfies the Chevalley restriction theorem: if a ⊂ V k is a Cartan subalgebra, then the map N G ( a ) → W a = N H ( a ) /Z H ( a ) is surjective, and the inclusions a ⊂ V k ⊂ h k induce isomorphisms a // W a ≃ V k // G ≃ h k // H . In particular, the quotient is isomorphic to affine space. . Suppose that k is algebraically closed and let x, y ∈ V ( k ) be regular semisimple elements. Then x is G ( k ) -conjugate to y if and only if x, y have the same image in V // G .Proof. These are classical results in the invariant theory of graded Lie algebras due to Vinberg and Kostant–Rallis; we refer to [Tho13, §2] for precise references.We now give some alternative characterizations of regular semisimple elements in V , after introducingsome more notation. First recall that the discriminant of h is the image under the Chevalley isomorphism Q [ t ] W ( H , T ) → Q [ h ] H of the product of all roots α ∈ Φ( H , T ) , where t := Lie T . Write ∆ ∈ Q [ V ] G for its restric-tion to V ⊂ h . Next we introduce weights of one-parameter subgroups. If k/ Q is a field and λ : G m → G k ahomomorphism, we may decompose V ( k ) as ⊕ i ∈ Z V i where V i = { v ∈ V ( k ) | λ ( t ) · v = t i v } . Every v ∈ V ( k ) can be written as v = P v i and we call integers i with v i = 0 the weights of v with respect to λ . Proposition 2.3. Let k/ Q be field and v ∈ V ( k ) . Then the following are equivalent:1. v is regular semisimple.2. ∆( v ) = 0 .3. The G -orbit of v is closed in V and Z G ( v ) is finite (i.e. v is stable in the sense of geometric invarianttheory).4. For every nontrivial homomorphism λ : G m → G ¯ k , v has a negative weight with respect to λ .Proof. The equivalence between the first two properties is a well known property of the discriminant. Thefirst property implies the third by [Tho13, Proposition 2.8], and the converse follows from [RLYG12, Lemma5.6]. Finally, the equivalence between the last two properties is the content of the Hilbert-Mumford stabilitycriterion [Mum77].We write B := V // G = Spec Q [ V ] G , B E := V E // G E = Spec Q [ V E ] G E and π : V → B , π E : V E → B E for thenatural quotient maps. Scaling defines G m -actions on V and V E , and there are unique G m -actions on B and B E such that the morphisms π and π E are G m -equivariant. In §3.1 we will describe the weights of B and B E . We describe a section of the quotient map π : V → B whose construction is originally due to Kostant. Let E := P α ∈ S H E Y α ∈ h E . Then E is a regular nilpotent element of h E which lies in h (1) . Using [Tho13, Proposition2.7], there exists a unique normal sl -triple ( E, X, F ) containing E . By definition, this means that ( E, X, F ) satisfies the identities [ X, E ] = 2 E, [ X, F ] = − F, [ E, F ] = H, with the additional property that X ∈ h E (0) and F ∈ h E (1) . Since ( E, ζ ( X ) , ζ ( F )) is also a normal sl -triplecontaining E , we see that ( E, ζ ( X ) , ζ ( F )) = ( E, X, F ) hence X and F lie in h .We define affine linear subspaces κ E := E + z h E ( F ) ⊂ V E and κ := κ ζ E = E + z h ( F ) ⊂ V . Proposition 2.4. 1. The composite maps κ ֒ → V → B and κ E ֒ → V E → B E are isomorphisms.2. κ and κ E are contained in the open subscheme of regular elements of V and V E respectively.3. The morphisms G × κ → V , ( g, v ) g · v and G E × κ E → V E , ( g, v ) g · v are étale. roof. Parts 1 and 2 are [Tho13, Lemma 3.5]; the last part is [Tho13, Proposition 3.4]. (These facts arestated only for simply laced groups in [Tho13] but they remain valid in the F case by the same proof.)Write σ : B → V for the inverse of π | κ and σ E : B E → V E for the inverse of π E | κ E . We call σ the Kostantsection for the pair ( G , V ) . It determines, for every field k/ Q and b ∈ B ( k ) , a distinguished orbit in V ( k ) with invariants b , playing an analogous role to reducible binary quartic forms as studied in [BS15a]. It willbe used to organize the set of G ( k ) -orbits of V ( k ) . Definition 2.5. Let k/ Q be a field and v ∈ V ( k ) . We say v is k -reducible if v is not regular semisimple or v is G ( k ) -conjugate to σ ( b ) with b = π ( v ) . If k/ Q is algebraically closed, every element of V ( k ) is k -reducible by Proposition 2.2. ζ on B E The involution ζ : V E → V E induces an involution B E → B E , still denoted by ζ . Proposition 2.6. 1. The inclusion V ⊂ V E induces a closed embedding B ֒ → B E whose image is thesubset of ζ -fixed points of B E .2. The involution ζ : B E → B E coincides with the involution ( − 1) : B E → B E induced by the G m -actionon B E .3. Let k/ Q be a field and v ∈ V ( k ) . Then v is regular semisimple as an element of h ( k ) if and only if v is regular semisimple as an element of h E ( k ) .Proof. Because the inclusion V ⊂ V E restricts to the inclusion κ ⊂ κ E , the first claim follows from Part 1 ofProposition 2.4.To prove the second claim, recall that T E ⊂ H E denotes a split maximal torus. Write t E ⊂ h E for its Liealgebra and W E for its Weyl group. By the classical Chevalley restriction theorem and Proposition 2.2respectively, the inclusions t E ֒ → h E , V E ֒ → h E induce isomorphisms t E // W E ≃ h E // H E , B E ≃ h E // H E ,equivariant with respect to the actions of G m and ζ . So it suffices to prove that the action of ζ on t E // W E is given by − . Since ζ and − are not contained in W E and this group has index in N G E ( t E ) , the product − ζ lies in W E . Therefore − ζ acts trivially on t E // W E , as desired.To prove the third claim, we may assume that k is algebraically closed and after conjugating by H ( k ) that v ∈ t ( k ) := t ζ E ( k ) . Then v is regular semisimple as an element of h ( k ) if and only if dα ( v ) = 0 for all α ∈ Φ( H , T ) , and v is regular semisimple as an element of h E ( k ) if and only if dα ( v ) = 0 for all α ∈ Φ( H E , T E ) .These two statements are equivalent because the restriction map Φ( H E , T E ) → Φ( H , T ) is surjective. V In this section we give an explicit description of V which will be convenient for doing explicit computationsin §2.5, §2.6 and §6.10. Recall from §2.1 that h is a Lie algebra of type F and that there is a direct sumdecomposition h = g + V where g ≃ sp ⊕ sl and V is a -dimensional representation of g . The splitmaximal torus T ⊂ H gives rise to three subsets of X ∗ ( T ) : Φ( H , T ) , Φ( G , T ) and Φ( V , T ) . They will bedenoted by Φ H , Φ G and Φ V and satisfy Φ H = Φ G ⊔ Φ V . Using the root basis S H fixed in §2.1, Φ V (resp. Φ G )consists of those roots in Φ H which have odd root height (resp. even root height).10ollowing Bourbaki [Bou68, Planche VIII], we label the elements of S H = { α , α , α , α } according to thefollowing labelling of the nodes of the Dynkin diagram: α α α α Define β , β , β , β to be α + α , α + α , α + α , α + α + 2 α respectively. Then S G := { β , β , β , β } is a root basis of Φ G , according to the following labelling of the Dynkin diagram of type C × A : β β β β With respect to this root basis the positive roots of Φ G , denoted Φ + G , are given by { β , β , β , β + β , β + β , β + β , β + β + β , β + 2 β + β , β + 2 β + β } ∪ { β } . Another basis of X ∗ ( T ) ⊗ Q will be convenient for describing Φ G and Φ V . We define L = (2 β + 2 β + β ) / ,L = (2 β + β ) / ,L = β / ,L = β / . Then S G = { L − L , L − L , L , L } and the elements of Φ G are given by {± L i ± L j | ≤ i, j ≤ } ∪ {± L } . Using the above explicit description or a general recipe applied to the Kač diagram of θ (given in [RLYG12,§7.1,Table 6]), we see that V is isomorphic to W ⊠ (2) where W is a -dimensional irreducible representationof sp with highest weight L + L + L (we choose an explicit realization of this representation in a moment)and (2) denotes the standard representation of sl . The elements of Φ V are of the form x ± L , where x isany element of the set Φ W := {± L i | i = 1 , , } ∪ {± L ± L ± L } . Every element α ∈ X ∗ ( T ) ⊗ Q has a unique expression of the form P i n i ( α ) β i with n i ( α ) ∈ Q . We define apartial ordering on X ∗ ( T ) ⊗ Q by declaring for x, y ∈ X ∗ ( T ) ⊗ Q that x ≥ y if n i ( x − y ) ≥ for all i = 1 , . . . , . (2.4.1)This induces a partial ordering on Φ V .We have tabulated the elements of Φ V in Table 3; the second column displays the coordinates of a weight in thebasis { β / , β / , β / , β / } . For example, the first entry is α = L + L + L + L = 2 α +3 α +4 α +2 α =(2 β + 4 β + 3 β + β ) / ∈ Φ V ; it is the highest root of Φ H and the unique maximal element of Φ V withrespect to the partial ordering.We now describe the Sp -representation W explicitly following [IR05, §2.2]. Fix a vector space Q withstandard basis e , . . . , e . We define Sp as the symplectic group stabilizing the -form ω on Q given by thematrix (cid:18) I − I (cid:19) . ω defines a Sp -equivariant contraction map cont ω : V ( Q ) → Q , x ∧ x ∧ x ω ( x , x ) − ω ( x , x ) + ω ( x , x ) . Define W := ker cont ω ⊂ ^ ( Q ) . We may organize an element P c ijk e i ∧ e j ∧ e k ∈ V Q in the matrices: ( u, X, Y, z ) = c , c c c c c c c c c , c c c c c c c c c , c . Then elements of W correspond to -tuples ( u, X, Y, z ) such that X and Y are symmetric matrices. Anelement of W will be usually thought of as such a -tuple.The ring of invariant polynomials Q [ W ] Sp is freely generated by one degree-4 polynomial F explicitly givenby F ( u, X, Y, z ) := ( uz − tr XY ) + 4 u det Y + 4 z det X − X ij det( ˆ X ij ) det( ˆ Y ij ) , (2.4.2)where for a matrix A we denote by ˆ A ij the matrix obtained by crossing out the i th row and j th column. Proposition 2.7. Let k/ Q be an algebraically closed field. Then W ( k ) has finitely many Sp ( k ) × k × -orbits.Moreover: • { w ∈ W ( k ) | F ( w ) = 0 } is the unique open dense orbit. • If w ∈ W ( k ) is nonzero with F ( w ) = 0 , then w is Sp ( k ) × k × -conjugate to an element of the form , ∗ ∗ 00 0 ∗ , , . Proof. It is well-known that W ( k ) has finitely many Sp ( k ) × k × -orbits with { F = 0 } the unique open denseone; see [IR05, §2.3] for precise references. The description of the remaining orbits and Proposition 2.3.3 ofloc. cit. implies the existence of the representatives above.We now fix the identifications of this subsection to remove any ambiguities. There exists an isomorphism G ≃ (Sp × SL ) /µ such that:• the weights L , L , L correspond to the weights of e , e , e in the defining representation of Sp (and SL acts trivially),• the weight L corresponds to the weight (cid:18) t t − (cid:19) t of SL .Then there exists a unique isomorphism V ≃ W ⊠ (2) of G -representations which sends X α ∈ V α (part ofthe pinning of H fixed in 2.1) to the element ( e ∧ e ∧ e , . This choice is somewhat arbitrary but whatis important for us is that it preserves the ‘obvious’ integral structures on both sides; this will be relevantin §5.1. We fix these isomorphisms for the remainder of the paper. It is therefore permitted, for every field k/ Q , to view an element v ∈ V ( k ) as a pair ( w , w ) of elements of W ( k ) , where A ∈ SL ( k ) acts on ( w , w ) via ( w , w ) · A t . 12 Weights Basis S H Basis { L i } α + 3 α + 4 α + 2 α L + L + L + L α + 2 α + 4 α + 2 α L + L − L + L α + 2 α + 3 α + α L + L − α + 2 α + 2 α + 2 α L + L + L − L α + α + 2 α + α L + L α + 2 α + 2 α L − L + L + L − α + 2 α + 2 α L + L − L − L − α + α + α L − L α + α + α L + L 10 2 0 − α + 2 α L − L − L + L 11 0 2 1 − α L − L 12 0 0 − α − L + L 13 2 0 1 − α L − L + L − L − α − L + L + L + L 15 2 0 − − − α L − L − L − L − − − α − L + L − L + L 17 0 0 1 − − α L − L 18 0 − − − α − L + L − − − α − α − L + L + L − L 20 0 0 − − − α − α − α − L − L − − − − α − α − α − L + L − − − − α − α − α − L − L + L + L − − − − α − α − α − L + L − L − L 24 0 − − − − α − α − α − α − L − L − − − − α − α − α − α − L − L − L + L − − − − − α − α − α − α − L − L − − − − − α − α − α − α − L − L + L − L − − − − − α − α − α − α − L − L − L − L Table 3: The elements of Φ V .13 .5 The resolvent binary quartic In this section we define for every v ∈ V ( k ) a binary quartic form Q v . At the end of §2.4 we fixed anisomorphism G ≃ (Sp × SL ) /µ ; let p : G → PGL be the corresponding projection map. Moreover wehave fixed an isomorphism V ≃ W ⊠ (2) , where W is the -dimensional Sp -representation described in §2.4. Definition 2.8. Let k/ Q be a field and v ∈ V ( k ) , giving rise to a pair of elements ( w , w ) in W ( k ) . Wedefine the resolvent binary quartic form Q v by the formula Q v := F ( xw + yw ) ∈ k [ x, y ] deg=4 . Note that Q λv = λ Q v and Q g · v = p ( g ) · Q v , where an element [ A ] ∈ PGL ( k ) acts on a binary quartic form Q ( x, y ) by [ A ] · Q ( x, y ) := Q (( x, y ) · A ) / (det A ) . Definition 2.9. Let k/ Q be a field and v ∈ V ( k ) . We say v is almost regular semisimple if Q v has distinctroots in P (¯ k ) . Lemma 2.10. Let k/ Q be a field and v ∈ V ( k ) . If v is regular semisimple, then v is almost regularsemisimple.Proof. We may assume that k is algebraically closed. Assume for contradiction that Q v does not have distinctroots. Then there exists an element γ ∈ PGL ( k ) so that the coefficients of γ · Q v at x and x y vanish.Choosing a lift g ∈ G ( k ) of γ and replacing v by g · v , we may assume that this holds for Q v . Therefore if v = ( w , w ) ∈ V ( k ) and g ( t ) := F ( w + tw ) then g (0) = g ′ (0) = 0 . Since F ( w ) = 0 , Proposition 2.7 showsthat we may assume after conjugation by Sp ( k ) that w is of the form , ∗ ∗ 00 0 ∗ , , . To derive a contradiction we will use the equivalence between Parts 1 and 4 of Proposition 2.3 repeatedly.We first claim that all the elements ∗ on the diagonal are nonzero. If not, then we may assume that the onein the bottom right corner is zero. But then the one-parameter subgroup (in the explicit realizations of Sp described in §2.4) t diag (1 , , t, , , t − ) × diag ( t − , t ) does not have a negative weight with respect to v , contradicting the assumption that v is regular semisim-ple. Secondly, the condition g ′ (0) = 0 translates into the condition that the coordinate of w at z in thedecomposition ( u, X, Y, z ) vanishes, by an explicit computation using Formula (2.4.2). But then the one-parameter subgroup t diag ( t, t, t, t − , t − , t − ) × diag ( t − , t ) again has no negative weight with respect to v , a contradiction. Definition 2.11. For a field k/ Q and an element v ∈ V ( k ) , we say v is almost k -reducible if it is not regularsemisimple or the resolvent binary quartic form Q v has a k -rational linear factor. Lemma 2.12. Let k/ Q be a field and v ∈ V ( k ) . If v is k -reducible (Definition 2.5), then v is almost k -reducible.Proof. We may assume that v is regular semisimple and of the form σ ( b ) for some b ∈ B ( k ) . A well-knownresult of Kostant determines the adjoint action of the sl -subalgebra generated by ( E, X, F ) on h in termsof the exponents , , , of F [Kos59, Corollary 8.7]. It implies that σ ( b ) ∈ κ ( k ) is supported on vectors14hose weights, considered as elements of Φ H , have root height , − , − , − , − with respect to S H . UsingTable 3 it follows that if σ ( b ) = ( w , w ) with w i ∈ W ( k ) then w is of the form , ∗ , ∗ ∗∗ ∗ ∗ ∗ , ∗ . Formula (2.4.2) shows that the polynomial F vanishes on elements of such form, so Q v is divisible by y . Remark 2.13. Not every element v ∈ V ( R ) is almost R -reducible. For example, let v = ( w , w ) ∈ V ( R ) begiven by: w = , , − − 12 3 1 , ,w = − , − , − , . Then one computes that Q v = 376 x + 507 x y + 1697 x y + 846 xy + 119 y , which has no real roots norrepeated roots. If v is regular semisimple, we have obtained a valid example; if not, then we may replace v bya small perturbation which is regular semisimple whose resolvent binary quartic form has no real roots either.This observation will be used in the proof of Lemma 4.9. Let Φ + H denote the positive roots of Φ H with respect to S H and write Φ + V := Φ V ∩ Φ + H . If v ∈ V we candecompose v as P α ∈ Φ V v α with v α in the weight space corresponding to α . For any subset M of Φ V wedefine the linear subspace V ( M ) = { v ∈ V | v α = 0 for all α ∈ M } ⊂ V . We state a lemma which describes sufficient conditions for an element v ∈ V to be almost Q -reducible. Thiswill (only) be useful when estimating the number of irreducible orbits in the cuspidal region in §6.10. Recallthat we write α = P n i ( α ) β i . Lemma 2.14. Let M be a subset of Φ V , and suppose that one of the following three conditions is satisfied:1. There exist integers b , . . . , b not all equal to zero such that ( α ∈ Φ V | X i =1 b i n i ( α ) > ) ⊂ M. 2. For every v = ( w , w ) ∈ V ( M )( Q ) , we have F ( w ) = 0 .Then every element of V ( M )( Q ) is almost Q -reducible.Proof. In the first case, the integers b , . . . , b determine a cocharacter of T with respect to which everyelement of V ( M )( Q ) has only nonnegative weights. By the Hilbert-Mumford stability criterion (Proposition2.3), V ( M )( Q ) then contains no regular semisimple elements so consists solely of almost Q -reducible elements.If the second condition is satisfied, then for every v ∈ V ( Q ) the resolvent binary quartic form Q v has a Q -rational linear factor, so v is almost Q -reducible too.15 emma 2.15. Let M be a subset of Φ V , and suppose M contains one of the following subsets, in the notationof Table 3: { , , , , , , , } , { , , , , , , , , , } , { , , , , , , } , { , , , , , , } . Then every element of V ( M )( Q ) is almost Q -reducible.Proof. We show that M satisfies one of the conditions of Lemma 2.14. If M contains the first displayedsubset, we may use Condition 1 with ( b , b , b , b ) = (0 , , , . If M contains the second subset, we usethe same condition with ( b , b , b , b ) = (1 , , , . The last two cases follow from Condition 2: indeed for v ∈ V ( M )( Q ) the vector w is either of the form , , ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ , ∗ or , ∗ , ∗ ∗∗ ∗ ∗∗ ∗ ∗ , ∗ . In both cases we see using Formula (2.4.2) that F ( w ) = 0 . In this section we relate the representation ( G , V ) to our family of curves of interest, see Proposition 3.3.The proof involves a similar result for the representation ( G E , V E ) and a study of the involution ζ . We firstrecall this result for ( G E , V E ) , after introducing some notation.Let V rsE denote the open subscheme of regular semisimple elements of V E ⊂ h E , and let B rsE be its image under π E : V E → B E . Define the B rsE -scheme A E := Z H E ( σ E | B rsE ) , the centralizer of the B rsE -point σ E | B rsE of V E . It isa family of maximal tori in H E parametrized by B rsE . We also define Λ E := Hom( A E , G m ) as the charactergroup of A E . Then Λ E is an étale sheaf of E root lattices on B rsE . By definition, this means that Λ E is alocally constant étale sheaf of finite free Z -modules, equipped with a pairing ( · , · ) : Λ E × Λ E → Z such thatfor every geometric point ¯ x of B rsE , the stalk of Λ E at ¯ x is a root lattice of type E . This induces a pairing Λ E / E × Λ E / E → {± } : ( λ, µ ) ( − ( λ,µ ) . Proposition 3.1. We can choose polynomials p , p , p , p , p , p ∈ Q [ V E ] G E with the following properties:1. Each polynomial p i is homogeneous of degree i and Q [ V E ] G E ≃ Q [ p , p , p , p , p , p ] . Consequently,there is an isomorphism B E ≃ A Q .2. Let C E → B E be the family of projective curves inside P B E with affine model y + x ( p y + p y ) + ( p y + p y ) = x + p x + p . (3.1.1) If k/ Q is a field and b ∈ B E ( k ) , then ( C E ) b is smooth if and only if b ∈ B rsE ( k ) . . Let J E → B rsE be the Jacobian of its smooth part [BLR90, §9.3; Theorem 1] . Then there is a uniqueisomorphism Λ E / E ≃ J E [2] of finite étale group schemes over B rsE that sends the pairing on Λ E / E to the Weil pairing J E [2] × J E [2] → {± } .4. There exists an isomorphism Z G E ( σ E | B rsE ) ≃ J E [2] of finite étale group schemes over B rsE .Proof. This is a combination of classical results and Thorne’s thesis [Tho13]. We refer to [Lag20, Proposition2.5] for precise references, with the caveat that the role of the coordinates x and y is interchanged here. Theonly part that remains to be proven is the uniqueness of the isomorphism Λ E / E ≃ J E [2] that preservesthe pairings on both sides. This follows from [Lag20, Proposition 2.6(4)].We now incorporate the involution ζ in the picture, and compare it to an involution defined on the level ofcurves. Recall that ζ : G E → G E is an involution with fixed points G . Since ζ commutes with σ E , it defines aninvolution of the scheme Z H E ( σ E | B rsE ) lifting the involution ζ : B rsE → B rsE . It induces an involution of Λ E / E ,still denoted by ζ .On the other hand, if C E → B E denotes the family of Equation (3.1.1), then the map ( x, y ) ( x, − y ) definesan involution τ : C E → C E lifting the involution ( − 1) : B E → B E . It induces an involution of J E [2] , denotedby τ ∗ . Lemma 3.2. Under the isomorphism φ : Λ E / E ∼ −→ J E [2] from Proposition 3.1, the involutions ζ and τ ∗ are identified.Proof. Write φ ′ = τ ∗ ◦ φ ◦ ζ . We need to prove that φ ′ = φ . By the second part of Proposition 2.6, φ ′ : Λ E / E → J E [2] is an isomorphism of B rsE -schemes. Moreover, ζ and τ ∗ respect the pairings on Λ E / E and J E [2] respectively. The result follows from the uniqueness statement in Part 3 of Proposition 3.1.Proposition 3.1 and Lemma 3.2 have the following important consequence, which connects the representation ( G , V ) with the subfamily C → B of C E → B E . Again let V rs denote the open subscheme of regular semisimpleelements of V and let B rs be its image under π : V → B . Proposition 3.3. We can choose polynomials p , p , p , p ∈ Q [ V ] G with the following properties:1. Each polynomial p i is homogeneous of degree i and Q [ V ] G ≃ Q [ p , p , p , p ] . Consequently, there isan isomorphism B ≃ A Q .2. Let C → B be the family of projective curves inside P B with affine model y + p xy + p y = x + p x + p . (3.1.2) If k/ Q is a field and b ∈ B ( k ) , then C b is smooth if and only if b ∈ B rs ( k ) .3. Let J → B rs be the Jacobian of the morphism C rs → B rs . Let τ : C rs → C rs be the involution of B rs -schemes sending ( x, y ) to ( x, − y ) and let τ ∗ : J → J be the induced morphism on J . Then theisomorphism Z G E ( σ | B rs ) ≃ J [2] obtained from Proposition 3.1 intertwines the involutions ζ and τ ∗ andrestricts to an isomorphism Z G ( σ | B rs ) ≃ J [2] τ ∗ Proof. Let p ′ , p ′ , p ′ , p ′ , p ′ , p ′ ∈ Q [ V E ] G E be a choice of polynomials satisfying the conclusion of Proposition3.1. Write p i ∈ Q [ V ] G for the restriction of p ′ i to V . The first two parts of Proposition 2.6 imply that p = p = 0 and Q [ V ] G = Q [ p , p , p , p ] . The family C → B is the pullback of the family C E → B E along B → B E . Moreover, Part 3 of Proposition 2.6 shows that B rs = B ∩ B rsE . The proposition now follows fromProposition 3.1 and Lemma 3.2. 17e henceforth fix p , p , p , p ∈ Q [ V ] G satisfying the conclusions of Proposition 3.3. Recall that we havedefined a G m -action on B which satisfies λ · p i = λ i p i . The assignment λ · ( x, y ) = ( λ x, λ y ) defines a G m -action on C such that the morphism C → B is G m -equivariant. J [2] We give some additional properties of the finite étale group scheme J [2] → B rs . Recall that T is a splitmaximal torus of H ; let t be its Lie algebra and W its Weyl group W := N G ( T ) / T . We define a map f : t rs → B rs as follows. The inclusions t ⊂ h and V ⊂ h induce isomorphisms t //W ≃ h // H and B ≃ h // H bythe classical Chevalley isomorphism and Proposition 2.2 respectively. Composing the first with the inverseof the second determines an isomorphism t //W ∼ −→ B . Precomposing this isomorphism with the naturalprojection t → t //W and restricting to regular semisimple elements defines a morphism f : t rs → B rs . Since t rs → t rs //W is a torsor under W , f is a W -torsor too.Let W E be the Weyl group of the split maximal torus T E of H E . It it known that the inclusion T ⊂ T E induces an isomorphism of W onto W ζ E , the centralizer of ζ in W E [Car72, §13.3.3]. We therefore obtain anaction of W on Λ := X ∗ ( T E ) , a root lattice of type E . Proposition 3.4. The finite étale group scheme J [2] → B rs becomes trivial after the base change f : t rs → B rs , where it is isomorphic to the constant group scheme Λ / . The monodromy action is given by thenatural action of W ≃ W ζ E .Proof. By [Lag20, Part 1 of Proposition 2.6], the group scheme J E [2] → B rsE becomes trivial after the basechange f E : t rsE → B rsE where f E is defined analogously as before. Moreover the monodromy action is given bythe natural action of W E on Λ / . The proposition is thus implied by the following commutative diagram: t rs t rsE B rs B rsE f f E Corollary 3.5. The finite étale B rs -subgroup schemes of J [2] are ⊂ (1 + τ ∗ ) J [2] ⊂ J [2] τ ∗ ⊂ J [2] (3.2.1) of order , , , respectively. Moreover the B rs -group schemes (1 + τ ∗ ) J [2] and J [2] /J [2] τ ∗ are notisomorphic, even after base change to k for any field extension k/ Q .Proof. In light of Proposition 3.4, the above claims are reduced to analysing the action of W ζ E on Λ / . Forexample for the first part it suffices to determine the W ζ E -invariant subgroups of Λ / and for the second part,it suffices to find an element of W ζ E which acts trivially on (1 + ζ ) (Λ / but not so on (Λ / / (Λ / ζ .Both are direct computations in the E root lattice, which we omit.18 .3 A family of Prym varieties In this section we introduce the family of Prym surfaces P → B rs and discuss some of its properties. Wefirst discuss it in a more general set-up.Let k be a field of characteristic different from and X/k a smooth projective genus- curve. Let τ : X → X be an involution with four fixed points. Suppose we are given a k -point ∞ ∈ X ( k ) fixed by τ . Let E := X/τ be the quotient of X by τ and f : X → E be the associated double cover which is branched at four points.By the Riemann–Hurwitz formula, E is an elliptic curve with origin f ( ∞ ) . This defines an isomorphism E ≃ J E between E and its Jacobian which sends f ( ∞ ) to the identity of J E .The Jacobian variety J X of X is not simple. Indeed the map f induces a surjective norm homomorphism f ∗ : J X → J E ≃ E which sends the equivalence class [ D ] of a divisor to [ f ( D )] , so E is an isogeny factor of J X . To describe the remaining part of J X we use the following classical definition. Definition 3.6. We define the Prym variety P X,τ of the pair ( X, τ ) as the kernel of the norm map: P X,τ := ker ( f ∗ : J X → E ) . Prym varieties have been studied by Mumford in a much more general set-up [Mum74]. We warn the readerthat many authors only consider fixed-point free involutions when defining Prym varieties or equivalently,unramified double covers. In our case the algebraic group P X,τ satisfies the following properties:1. Let f ∗ : E → J X be the pullback map on divisors. Then f ∗ is injective, f ∗ ◦ f ∗ = [2] and f ∗ ◦ f ∗ = 1+ τ ∗ .Hence P X,τ = ker (1 + τ ∗ : J X → J X ) . (3.3.1)2. P X,τ is connected, hence an abelian surface.3. The restriction of f ∗ : E → J X to E [2] induces an isomorphism E [2] ∼ −→ image( f ∗ ) ∩ P X,τ . Consequentlythere is an injective morphism ψ : E [2] ֒ → P X,τ [2] .4. The map E × P X,τ → J X , determined by f ∗ and the inclusion P X,τ ֒ → J X , is surjective with kernelequal to the graph of ψ , given by { ( x, ψ ( x )) | x ∈ E [2] } . Consequently there is an isomorphism J X ≃ ( E × P X,τ ) / { ( x, ψ ( x )) | x ∈ E [2] } . So J X is isogenous to E × P X,τ .5. The cokernel of f ∗ is naturally identified with the dual abelian variety of P X,τ , written P ∨ X,τ , and thecomposite P X,τ ֒ → J X ։ P ∨ X,τ , denoted ρ , is a polarization of type (1 , . (This means that P X,τ [ ρ ] isisomorphic to ( Z / over ¯ k .) The map f ∗ : E [2] → P X,τ [ ρ ] is an isomorphism.Indeed, to verify the above properties we may assume that k is algebraically closed. Then Property 1 followsfrom [Mum74, §3; Lemma 1] and the fact that X → E is ramified at four points. The other properties followfrom going through the correspondence described in [Mum74, §2]: in the notation of that paper, we startwith Data I of the form ( X, Y, φ ) = ( E, X, f ∗ ) whose invariants are ( a, b, c ) = (1 , , . Equation (2.1) of loc.cit. holds by the discussion in §1 of op. cit. We additionally record the following important fact. Lemma 3.7. The isogeny ρ : P X,τ → P ∨ X,τ is self-dual. roof. This follows from the fact that f ∗ and f ∗ are dual to each other when transported along the principalpolarizations of J X and E , see [Mum74, End of §1].We now specialize to our situation of interest. Recall that C rs → B rs consists of the smooth members of theprojective closure of the family of curves y + p xy + p y = x + p x + p and that τ : C → C, ( x, y ) ( x, − y ) is the involution which defines, for every field k/ Q and b ∈ B rs ( k ) , aninvolution τ b : C b → C b with four fixed points fixing the point at infinity ∞ ∈ C b ( k ) .Define E → B to be the projective completion of the family of plane curves given by y + p xy + p y = x + p x + p . (3.3.2)Define E → B rs to be its restriction to B rs . Then there is a unique morphism of B -schemes f : C → E sending a point ( x, y ) to ( x, y ) . This identifies, for each field k/ Q and b ∈ B rs ( k ) , E b with the quotient of C b by τ b .The morphism τ defines via pullback a morphism of abelian schemes τ ∗ : J → J . Define P := ker(1 + τ ∗ : J → J ) . (3.3.3)The morphism P → B rs is proper and by Equation (3.3.1) its fibres are abelian surfaces enjoying theproperties described above. The next useful lemma [Edi92, Proposition 3.5] applied to the Z / Z -action − τ ∗ on J → B rs shows that P → B rs is smooth, hence an abelian scheme. Lemma 3.8. Let G be a finite group, acting equivariantly on a smooth morphism of schemes X → S . Ifthe order of G is invertible on S , then the induced morphism on fixed points X G → S G is smooth. Lemma 3.9. The filtration ⊂ (1 + τ ∗ ) J [2] ⊂ J [2] τ ∗ ⊂ J [2] of Corollary 3.5 is identified with the filtration ⊂ E [2] ⊂ P [2] ⊂ J [2] . Here we see E [2] as a subgroup of P [2] using the pullback map f ∗ : E [2] ֒ → J [2] .Proof. This follows from Corollary 3.5 and the fact that E [2] and P [2] have order and respectively.The map τ ∗ : J [2] → J [2] has image E [2] and kernel P [2] , so J [2] /P [2] ≃ E [2] . The remaining gradedpiece P [2] /E [2] of the filtration of Lemma 3.9 will be determined in Corollary 3.15.Since P [2] = J [2] τ ∗ , Proposition 3.3 immediately implies the following. Proposition 3.10. The isomorphism Z G E ( σ | B rs ) ≃ J [2] of Proposition 3.3 restricts to an isomorphism Z G ( σ | B rs ) ≃ P [2] of finite étale group schemes over B rs . The following diagram of smooth group schemes over B rs summarizes the situation.20 E P J E P ∨ f ∗ × ρ f ∗ Let k/ Q be a field and let ( X, τ ) be a pair where X/k a smooth projective genus-3 curve and τ : X → X aninvolution with four fixed points. Let ∞ ∈ X ( k ) be a k -point fixed by τ . In §3.3 we have associated to thisdata a Prym variety P X,τ . In this section we will, under the presence of additional assumptions, realize thedual P ∨ X,τ as the Prym variety of another such pair ( ˆ X, ˆ τ ) . This is a special case of the bigonal constructiongoing back to Pantazis [Pan86] but we present it in a way closer in spirit to Barth [Bar87] who analysed theabove situation in great detail. Only Theorem 3.14 will be used later.Recall that we have defined a polarization ρ : P X,τ → P ∨ X,τ of type (1 , ; let ˆ ρ : P ∨ X,τ → P X,τ be the uniqueisogeny such that ˆ ρ ◦ ρ = [2] . (Warning: ˆ ρ is not the dual of ρ !) Define i : X → P ∨ X,τ as the composite of theAbel–Jacobi map X → J X with respect to ∞ with the projection J X → P ∨ X,τ . Proposition 3.11 (Barth) . 1. The morphism i : X ֒ → P ∨ X,τ is a closed embedding.2. The divisor i ( X ) is ample and the induced polarization of P ∨ X,τ coincides with ˆ ρ .3. If A/k is an abelian surface and j : X ֒ → A is a closed embedding mapping ∞ to such that [ − restricts to τ on X , then there exists a unique isomorphism of abelian varieties P ∨ X,τ → A sending i to j .Proof. We may suppose that k is algebraically closed. Part 1 follows from the proof of [Bar87, Proposition1.8]. For Part 2, note that by the adjunction formula we have i ( X ) · i ( X ) = 2 p a ( X ) − and if a curve Y on A is not numerically equivalent to i ( X ) we can translate Y using A so that it intersects i ( X ) in afinite non-empty subscheme, implying that Y · i ( X ) > . Therefore i ( X ) is ample by the Nakai-Moishezoncriterion. Let λ : P ∨ X,τ → P X,τ be the corresponding polarization. The fact that λ = ˆ ρ follows from theequality ker λ = ker ˆ ρ [Bar87, Lemma 1.11]. Part 3 is [Bar87, Proposition 1.10].We now describe the bigonal construction. Suppose in addition to the above that X is not hyperellipticand we are given an effective divisor κ on X fixed by τ such that κ is canonical. Let Θ κ ⊂ J X be thecorresponding theta divisor, namely the pullback of the image of the natural summing map X × X → Pic ( X ) along the translation-by- κ map J X → Pic ( X ) , D D + κ . The divisor Θ κ is symmetric and induces theprincipal polarization on J X ; we refer to [BL04, Chapter 11, §2] for these classical facts. Set ˆ X := Θ κ ∩ P X,τ . Then ˆ X induces the polarization ρ : P X,τ → P ∨ X,τ , by construction of ρ . Let ˆ τ be the restriction of [ − to ˆ X , which coincides with the restriction of τ ∗ to ˆ X . 21 emma 3.12. The curve ˆ X is smooth, geometrically connected and of genus . The involution ˆ τ : ˆ X → ˆ X has fixed points over ¯ k .Proof. We may suppose that k is algebraically closed. Since ˆ X is an ample divisor on the smooth projectivesurface P X,τ , it is connected by the Kodaira vanishing theorem. Moreover because ˆ X defines a polarizationof degree , it has self-intersection so arithmetic genus by the adjunction formula. Because we assumedthat X is not hyperelliptic and of genus , Θ κ is smooth by Riemann’s singularity theorem [BL04, Chapter11, §2.5]. Therefore ˆ X is smooth by Lemma 3.8, being the fixed points of the involution [ − ◦ τ : Θ κ → Θ κ .It remains to calculate the number of fixed points of ˆ τ . Let f : X → E be the quotient of X by τ andlet g : E → P be the morphism induced by the degree- divisor f ∗ ( κ ) . Since Θ κ is smooth and X is nothyperelliptic, the summing map Sym X → Θ κ , D D − κ is an isomorphism; let e X be the inverse imageof ˆ X under this isomorphism. An effective degree- divisor D lies on e X if and only if D + τ ( D ) ∼ κ . Since f ∗ : Pic( E ) → Pic( X ) is injective and f ∗ ◦ f ∗ = 1 + τ ∗ (Property 1 of §3.3), the latter holds if and only if f ∗ ( D ) ∼ f ∗ ( κ ) .It suffices to prove that the involution D τ ( D ) on e X has fixed points. If e , . . . , e are the ramificationpoints of g then f ∗ ( e ) , . . . , f ∗ ( e ) are fixed points; we claim that these are the only ones. Arguing bycontradiction, suppose that D = P + P ∈ e X is fixed by τ and not of this form. Then τ ( P i ) = P i for i = 1 , and P = P ; write P , P for the remaining fixed points of τ on X . We have equivalences of divisors P + 2 P = D + τ ( D ) ∼ κ ∼ P + P + P + P where last equivalence follows from the Riemann-Hurwitzformula applied to f . This implies that P + P ∼ P + P . Since X is not hyperelliptic and P , . . . , P aredistinct, we obtain a contradiction.The effective degree-2 divisor κ defines a point ˆ ∞ ∈ ˆ X ( k ) fixed by ˆ τ . We thus obtain a Prym variety P ˆ X, ˆ τ and an embedding ˆ i : ˆ X ֒ → P ∨ ˆ X, ˆ τ as defined above. The inclusion ˆ X ֒ → P X,τ maps ˆ ∞ to and extends to ahomomorphism J ˆ X → P X,τ from the Jacobian of ˆ X . Proposition 3.13. The homomorphism J ˆ X → P X,τ factors through an isomorphism of abelian varieties P ∨ ˆ X, ˆ τ → P X,τ which identifies the polarizations ˆ ρ ˆ X and ρ X .Proof. The first claim follows from Part 3 of Proposition 3.11 applied to the closed embedding ˆ X ֒ → P X,τ .Since the polarizations of P X,τ and P ∨ ˆ X, ˆ τ are defined by the embedded curve ˆ X , the isomorphism identifiesthe polarizations.We apply the above generalities to the family of curves that concern us. If b = ( p , p , p , p ) ∈ B rs ( k ) then C b and E b are of the form C b : y + p xy + p y = x + p x + p ,E b : y + p xy + p y = x + p x + p . The point ∞ ∈ C b ( k ) is the unique point at infinity, τ b : C b → C b is the involution sending ( x, y ) to ( x, − y ) and f b : C b → E b the quotient of C b by τ b . The divisor κ = 2 ∞ is a theta characteristic fixed by τ b .The proof of Lemma 3.12 shows that ˆ C b is isomorphic to the closed subscheme of Sym C b consisting ofdegree- divisors D with the property that f b ( D ) ∼ f b ( ∞ ) . It follows that ˆ C b has an affine open given by22he closed subscheme of A defined by the equations y + p xy + p y = x + p x + p ,y ′ + p x ′ y ′ + p y ′ = x ′ + p x ′ + p ,x = x ′ ,y + y ′ + p x + p = 0 , quotiented by the involution ( x, y, x ′ , y ′ ) ( x ′ , y ′ , x, y ) . This quotient can be realized by introducing thevariables y + y ′ and yy ′ ; a computation then shows that ˆ C b and its quotient by ˆ τ are given by (the projectiveclosure of) the equations ˆ C b : ( y + p x + p ) = − x + p x + p ) , (3.4.1) ˆ E b : ( y + p x + p ) = − x + p x + p ) . (3.4.2)This construction motivates us to define a G m -equivariant morphism χ : B → B by sending ( p , p , p , p ) to · (cid:18) − p , p − p , p − p p p , − p − p 27 + 8 p p − p + 16 p p (cid:19) . (3.4.3)(We include the factor in front so that χ has integer coefficients.) We have defined χ so that for all b ∈ B rs ( k ) , C χ ( b ) is isomorphic to ˆ C b . We also write ˆ b for χ ( b ) , thinking of it as the ‘bigonal dual’ of b . Theorem 3.14 (The bigonal construction) . The morphism χ : B → B satisfies the following properties: forany field k/ Q and b = ( p , p , p , p ) ∈ B rs ( k ) , we have • The projective curve C χ ( b ) is isomorphic to the projective closure of the curve (cid:0) y + p x + p (cid:1) = − x + p x + p ) , (3.4.4) and τ : C χ ( b ) → C χ ( b ) maps ( x, y ) to ( x, − y ) . • There exists an isomorphism P χ ( b ) ≃ P ∨ b of (1 , -polarized abelian varieties. • We have χ ( χ ( b )) = 18 · b for all b ∈ B .Proof. Only the last part is not yet established, which follows from an explicit computation.For any B -scheme U we define the B -scheme ˆ U as the pullback of U → B along χ : B → B . In particular, weobtain the B -schemes ˆ C, ˆ E, ˆ J, ˆ P . In this notation, one can prove that there exists an isomorphism P ∨ ≃ ˆ P of polarized abelian schemes over B rs . However, we will not need this fact in what follows. Corollary 3.15. There exists an exact sequence of finite étale group schemes over B rs : → E [2] → P [2] → ˆ E [2] → isomorphic to the exact sequence → P [ ρ ] → P [2] ρ −→ P ∨ [ˆ ρ ] → . Moreover, the B rs -groups E [2] and ˆ E [2] are not isomorphic, even after base change to k for any field extension k/ Q . roof. Since B rs is normal, it suffices to prove the corollary over the generic point. The second exact sequenceof the corollary follows from the identity ˆ ρ ◦ ρ = [2] . We have seen in §3.3 (Property 5) that the kernel P [ ρ ] of ρ is identified with E [2] . Since P ∨ [ˆ ρ ] ≃ ˆ P [ ρ ] by Theorem 3.14, we see that P ∨ [ˆ ρ ] ≃ ˆ E [2] . The last claimfollows from the last claim of Corollary 3.5. In Section 3.3 we have constructed a family of abelian varieties P → B rs . In this section we construct aprojective scheme P → B containing P as a dense open subscheme. We establish some basic properties of P which will be used in §5.4 to construct integral orbit representatives; they are summarized in Proposition3.24.Recall that C E → B E is the family of projective curves given by Equation (3.1.1). Let J E → B rsE be therelative Jacobian of its smooth part, a smooth and proper morphism. In [Lag20, §4.3], a proper morphism ¯ J E → B E is constructed which parametrizes torsion-free rank-1 sheaves on the fibres of C E → B E . (In fact,in that paper the compactified Jacobian ¯ J E was constructed over Spec Z [1 /N ] for some N ≥ ; we define ¯ J E as the Q -fibre of ¯ J E .) We state some of its properties here, referring to [Lag20, Corollary 4.13] for proofsand references. Proposition 3.16. 1. For any B E -scheme T , the T -points of ¯ J E are in natural bijection with the setof isomorphism classes of locally finitely presented O C E × T -modules F , flat over T , with the propertythat F t is torsion-free rank of degree zero for every geometric point t of T , and that there exists anisomorphism of O T -modules ∞ ∗ T F ≃ O T , where ∞ : B E → C E denotes the section at infinity.2. The morphism ¯ J E → B E is flat, projective and its restriction to B rsE ⊂ B E is isomorphic to J E → B rsE .3. The variety ¯ J E → Spec Q is smooth. Recall from §3.1 that the involution τ : ( x, y ) ( x, − y ) of C E lifts the involution ( − 1) : B E → B E . It inducesan involution τ ∗ of ¯ J E , sending a rank torsion-free sheaf F to its pullback τ ∗ ( F ) .On the other hand, we may construct a different involution of ¯ J E extending [ − 1] : J E → J E , as follows. If F is a coherent sheaf on a scheme X , we define F ∨ := H om ( F , O X ) . Lemma 3.17. Let T be a B E -scheme and F an O C E × T -module, corresponding to a T -point of ¯ J E . Then the O C E × T -module F ∨ corresponds to a T -point of ¯ J E , and the corresponding morphism ¯ J E → ¯ J E , F F ∨ isan involution.Proof. Since the fibres of C E → B E are all Gorenstein curves (being complete intersections), [Har86, Lemma1.1(a)] shows that E xt O C E ,t ( F t , O C E ,t ) = 0 for all geometric points t of T . Therefore [AK80, Theorem1.10(ii)] implies that F ∨ is locally finitely presented and flat over T , and that ( F ∨ ) S ≃ ( F S ) ∨ for everymorphism S → T . It follows that ( F ∨ ) t = F ∨ t is torsion-free rank of degree zero since the same is truefor F t . Moreover ∞ ∗ T ( F ∨ ) ≃ ( ∞ ∗ T F ) ∨ ≃ O ∨ T ≃ O T . Therefore by Proposition 3.16, F ∨ corresponds to a T -point of ¯ J E .It remains to prove that the natural map F → F ∨∨ is an isomorphism. Since the formation of F ∨∨ commutes with base change, we may assume that T is the spectrum of an algebraically closed field. In thiscase the claim follows from [Har86, Lemma 1.1(b)]. 24rite µ for the composite of the commuting involutions τ ∗ and F F ∨ . Write ¯ J → B for the restrictionof ¯ J E to B ֒ → B E . Definition 3.18. We define the compactified Prym variety P → B as the µ -fixed points of the morphism ¯ J E → B E . The scheme P is a closed subscheme of ¯ J E so the morphism P → B is projective. By definition of P (cf.Equation (3.3.3)) the restriction of P to B rs ⊂ B is isomorphic to P . Lemma 3.19. The scheme P is smooth over Q .Proof. Apply Lemma 3.8 to the Z / Z -action of µ on the smooth morphism ¯ J E → Spec Q .We now analyse the irreducible components of P . The following lemma contains the key calculation of thecentral fibre of P → B . Proposition 3.20. The fibre of the morphism P → B above is geometrically irreducible of dimension .Proof. In the course of the proof we may and will assume that all schemes are base changed to C and byabuse of notation will identify them with their set of complex points. Write J for the identity componentof the Picard scheme of the projective curve C given by the equation ( y = x ) , an open subscheme of ¯ J stable under µ . To prove the proposition it suffices to prove that P := J µ is irreducible of dimension and P ⊂ P is a dense open subscheme. Because the normalization π : ˜ C → C is rational and C is Gorenstein,we may appeal to the results of [Bea99] (originally due to Rego [Reg80]) to describe ¯ J explicitly.Define the local rings ˜ O = C [[ t ]] , O = C [[ t , t ]] ⊂ ˜ O and the truncated versions ˜ A = ˜ O /t and A =image( O → ˜ O /t ) ⊂ ˜ A . Then O is the completed local ring of C at the origin and ˜ O its normalization.Let Gr (3 , ˜ A ) be the Grassmannian parametrizing -dimensional subspaces of ˜ A . Let M ⊂ Gr (3 , ˜ A ) be thereduced closed subscheme parametrizing those subspaces which are stable under the action of A . The map M M ⊗ O A establishes a bijection between the O -submodules M of ˜ O with dim C ˜ O /M = 3 and M (by[GP93, Lemma 1.1(iv)] and the fact that dim C ˜ O / O = 3 ), whose inverse we denote by M M O . We have anatural morphism of O C -modules π ∗ O ˜ C → ˜ A , where ˜ A is considered as the structure sheaf of the degree- divisor supported at the preimage under π of the singular point. The assignment M F M := ker( π ∗ O ˜ C → ˜ A/M ) defines a morphism e : M → ¯ J which is bijective and proper ([Bea99, Proposition 3.7]), hence a homeomor-phism in the Zariski topology. The sheaf F M is invertible if and only if M O is a cyclic O -module; the locusof such M define an open subscheme M ◦ ⊂ M . Let τ : ˜ A → ˜ A be the C -algebra homomorphism sending t to − t . For M ∈ M ( C ) define M ∨ = { x ∈ ˜ A | x · M ⊂ A } and τ ∗ M = { τ ( m ) | m ∈ M } ; they define involutions ( − ) ∨ and τ ∗ of M with composite µ . Since F µ ( M ) ≃ µ ( F M ) , it suffices to prove that M ◦ P := ( M ◦ ) µ isirreducible, two-dimensional and dense in M P := M µ .The map a A · a defines a bijection ˜ A × /A × → M ◦ . We have a group isomorphism G a → ˜ A × /A × givenby sending ( a , a , a ) to the coset ofexp ( a t + a t + a t ) = 1 + ( a t + a t + a t ) + ( a t + a t + a t ) / · · · + ( a t + a t + a t ) / . The composite G a → M ◦ is an isomorphism of varieties and gives M ◦ the structure of an algebraic groupwhich acts on M . The restriction of µ to M ◦ corresponds to the involution ( a , a , a ) ( a , − a , a ) under the above isomorphism. Therefore M ◦ P is isomorphic to G a , hence irreducible and two-dimensional; itremains to prove that it is dense in M P . The orbits of the action of M ◦ stratifies M into affine cells which25tratum Module Type Dimension Image under µX O {0,3,4} X X t O + t O {1,4,5} X X t O + t O {2,3,5} X X t O + t O {2,4,5} X X t O + t O + t O {3,4,5} X Table 4: Stratification of M are described in [Coo98, §4]. They correspond to isomorphism classes of torsion-free rank O -modules andtheir properties are described in Table 4. The second column gives an O -module representative M O for some M ∈ X i ; the third column depicts the powers of t generating M .Since µ preserves M ◦ it permutes the strata. By dimension reasons it can only permute X and X . Sincethe dual of tA is t A + t A and τ fixes tA we see that µ ( X ) = X . Therefore M P = M ◦ P ⊔ X µ ⊔ X µ . So itwill be enough to show that the closure of M ◦ P contains X µ ⊔ X µ .Using the description of M ◦ given above and the exponential map, every element of M ◦ P is an A -modulegenerated by (cid:18) at + a t + a t + a t + (cid:18) a 120 + b (cid:19) t (cid:19) for some a, b ∈ C . Using the Plucker coordinates { t i ∧ t j ∧ t k } in Gr (3 , ˜ A ) , one can compute that the closureof M ◦ P contains λ ( t ∧ t ∧ t ) + µ ( t ∧ t ∧ t ) for all λ, µ ∈ C . Since every element of X or X is of thisform, this proves the proposition.Recall that we have defined a G m -action on C → B in §3.1 after Proposition 3.3. By functoriality thisinduces a G m -action on P such that the morphism P → B is G m -equivariant. The following fact will be usedin the next three lemmas: if Z ⊂ B is a closed, nonempty and G m -invariant subscheme, then it contains thecentral point . Lemma 3.21. The scheme P is geometrically irreducible.Proof. Since P is smooth (Lemma 3.19), the irreducible components of P Q coincide with its connectedcomponents so in particular are disjoint. The image of each connected component under P → B is closed(using properness) and G m -invariant, hence contains the central point. But P , Q is irreducible by Proposition3.20 so there exists at most one such connected component, as required. Lemma 3.22. The morphism P → B is flat.Proof. We first claim that all the fibres of P → B are -dimensional. Since P → B is proper, the fibredimension of this morphism is upper semicontinuous on B [Gro66, Corollaire 13.1.5]. The general fibre is -dimensional; let Z ⊂ B be the closed subset where the fibre has larger dimension. The G m -action on P → B shows that this locus is invariant under G m hence it must contain the central point , if it is non-empty. ButProposition 3.20 shows that Z , proving the claim.The lemma now follows from the smoothness and irreducibility of P (Lemmas 3.19 and 3.21) and B andMiracle Flatness [Mat86, Theorem 23.1]. Lemma 3.23. The fibres of the morphism P → B are geometrically integral. roof. We first claim that P is geometrically reduced. Since it contains the smooth dense open subscheme P by Proposition 3.20, it suffices to prove that it is (geometrically) Cohen-Macaulay. But since P → B isflat (Lemma 3.22) and ֒ → B is a complete intersection, the pullback P ֒ → P is a complete intersection.Since P is smooth (Lemma 3.19), P , Q is a local complete intersection hence Cohen-Macaulay. We concludethat P is geometrically reduced hence by Proposition 3.20 geometrically integral.The proposition now follows from the contracting G m -action. Indeed, the locus Z of elements of B abovewhich the fibre fails to be geometrically integral is closed [Gro66, Théorème 12.2.1(x)] and G m -invariant.Since we have just shown that Z does not contain the central point, it must be empty.We summarize the properties of P for later reference in the following proposition. Proposition 3.24. The morphism P → B constructed above is flat, projective and its restriction to B rs isisomorphic to P . Moreover P is smooth and geometrically integral. The locus of P where the morphism P → B is smooth is an open subset whose complement has codimension at least two.Proof. The only thing that remains to be proven is the statement about the smooth locus of P → B ; denotethis morphism by φ . Let Z ⊂ P be the (reduced) closed subscheme where φ fails to be smooth. Thesmoothness of P → B rs shows that Z is supported above the discriminant hypersurface of B . Moreover sincethe fibres of φ are geometrically integral by Lemma 3.23, they intersect Z in a proper closed subset of smallerdimension. Combining the last two sentences proves the statement.The discussion of this section has another geometric consequence, which will be useful in §5.3. Proposition 3.25. Let Pic C/ B → B be the identity component of the relative Picard scheme of C → B [BLR90, §9.3, Theorem 1] . Then the fibres of Pic [1 + τ ∗ ] → B are geometrically integral.Proof. By construction of P , there exists a morphism of B -schemes Pic [1 + τ ∗ ] → P which is an openimmersion. Therefore for every b ∈ B , Pic [1 + τ ∗ ] b is a non-empty open subset of P b . Since the fibres of P → B are geometrically integral (Lemma 3.23), the proposition follows. We give an explicit description of the discriminant polynomial ∆ ∈ Q [ V ] G = Q [ B ] introduced in §2.1 beforeProposition 2.3. Recall that we have fixed an isomorphism Q [ V ] G ≃ Q [ p , p , p , p ] from Proposition 3.3,so we consider ∆ as a polynomial in p , p , p , p . Since the F root system has roots, ∆ is homogeneousof degree with respect to the G m -action on B .Set ∆ ˆ E := 4 p + 27 p and ∆ E := ∆ ˆ E ◦ χ , where χ is defined by Formula (3.4.3), both elements of Q [ B ] .Then ∆ E and ∆ ˆ E are up to elements of Q × the discriminants of the curves E → B and ˆ E → B . Lemma 3.26. The polynomial ∆ ∈ Q [ p , p , p , p ] equals, up to an element of Q × , the polynomial ∆ E · ∆ ˆ E .In other words, there exists a constant A ∈ Q × such that ∆( b ) = A (cid:16) p (ˆ b ) + 27 p (ˆ b ) (cid:17) (cid:0) p ( b ) + 27 p ( b ) (cid:1) . Proof. It suffices to prove the claim when base changed to an algebraically closed field k/ Q . The polynomials ∆ and ∆ E · ∆ ˆ E both have degree . Moreover ∆ E and ∆ ˆ E are irreducible coprime polynomials in k [ B ] . So27o prove the claim it suffices to prove that the vanishing loci of ∆ and ∆ E · ∆ ˆ E agree. By Proposition 3.3,if b ∈ B ( k ) then ∆( b ) = 0 if and only if the curve ( y + p xy + p y = x + p x + p ) is smooth. By theJacobian criterion for smoothness, this happens if and only if the curve y + p xy + p y = x + p x + p issmooth and the polynomial x + p x + p has no multiple roots. The lemma then follows from the explicitdescriptions of E → B and ˆ E → B given by Equations (3.3.2) and (3.4.2) respectively. Remark 3.27. The factorization of ∆ into a product of two degree- polynomials of Lemma 3.26 canbe interpreted Lie-theoretically. It corresponds to the fact that the Weyl group W ( H , T ) has two orbits on Φ( H , T ) , namely an orbit consisting of the short roots and one consisting of the long roots. It is true(although we do not prove this) that ∆ ˆ E corresponds the short root orbit and ∆ E to the long root orbit. In this section we construct, for each b ∈ B rs ( Q ) , an embedding of Sel P b inside the set of G ( Q ) -orbits of V ( Q ) with invariants b . Moreover we introduce a different representation ( G ⋆ , V ⋆ ) and similarly prove that Sel ˆ ρ P ∨ b embeds in its rational orbits.We first recall a well-known lemma which gives a cohomological description of orbits. Lemma 4.1. Let G → S be a smooth affine group scheme. Suppose that G acts on the S -scheme X and let e ∈ X ( S ) . Suppose that the action map m : G → X, g g · e is smooth and surjective. Then the assignment x m − ( x ) induces a bijection between the set of G ( S ) -orbits on X ( S ) and the kernel of the map of pointedsets H ( S, Z G ( e )) → H ( S, G ) .Proof. This is [Con14, Exercise 2.4.11]: the conditions imply that X ≃ G/Z G ( e ) and since G and Z G ( e ) (thefibre above e of a smooth map) are S -smooth we can replace fppf cohomology by étale cohomology.Lemma 4.1 has the following concrete consequence. Let k be a field and G/k a smooth algebraic group whichacts on a k -scheme X . Suppose that e ∈ X ( k ) has smooth stabilizer Z G ( e ) and the action of G ( k s ) on X ( k s ) is transitive, where k s denotes a separable closure of k . Then the G ( k ) -orbits of X ( k ) are in bijection with ker(H ( k, Z G ( e )) → H ( k, G )) . This fact allows us to make the connection with Galois cohomology and liesat the basis of our orbit parametrizations in §4.1 and §4.3. -Selmer group The purpose of this section is to prove Theorem 4.5 and its consequence, Corollary 4.6. For any scheme-theoretic point b of B we write V b for the fibre of π : V → B under b and similarly for V E . Lemma 4.2. Let R be a Q -algebra and b ∈ B rs ( R ) . Then there are canonical bijections of sets1. G ( R ) \ V b ( R ) ≃ ker (cid:0) H ( R, P b [2]) → H ( R, G ) (cid:1) . G E ( R ) \ V E ,b ( R ) ≃ ker (cid:0) H ( R, J b [2]) → H ( R, G E ) (cid:1) . The reducible orbits G ( R ) · σ ( b ) and G E ( R ) · σ E ( b ) correspond to the trivial element in H ( R, P b [2]) and H ( R, J b [2]) respectively. Moreover the following diagram is commutative: ( R ) \ V b ( R ) G E ( R ) \ V E ,b ( R )H ( R, P b [2]) H ( R, J b [2]) Here the horizontal maps are induced by the natural inclusions and the vertical maps are the injectionsinduced by the above bijections.Proof. We consider the case of ( G , V ) , the case of ( G E , V E ) being analogous. The bijection then follows fromLemma 4.1 applied to the action of G B rs on V rs . Indeed, the action map G × B rs → V rs , ( g, b ) g · σ ( b ) isétale (Proposition 2.4) and it is surjective by Part 2 of Proposition 2.2. Moreover we have an isomorphism Z G ( σ | B rs ) ≃ P [2] by Proposition 3.10. Pulling back along b : Spec R → B rs gives the desired bijection.The claim about G ( R ) · σ ( b ) follows from the explicit description of the bijection of Lemma 4.1. Thecommutative diagram follows from the definition of the pushout of torsors and the compatibility betweenthe isomorphisms Z G E ( σ | B rs ) ≃ J [2] and Z G ( σ | B rs ) ≃ P [2] . Lemma 4.3. Let G sc → G be the simply connected cover of G . Let R be a Q -algebra such that every locallyfree R -module of constant rank is free. Then the pointed set H ( R, G sc ) is trivial.Proof. We have G sc ≃ SL × Sp (Proposition 2.1). The result now follows from the triviality of H ( R, SL ) (by Hilbert’s theorem 90) and H ( R, Sp ) [Lag20, Lemma 3.12]. Lemma 4.4. Let R be a Q -algebra such that every locally free R -module of constant rank is free. Then thenatural map of pointed sets H ( R, G ) → H ( R, G E ) has trivial kernel.Proof. Let G sc E → G E be the simply connected cover of G E . We have a commutative diagram with exactrows over R : µ G sc E G E µ G sc G = Here the maps are the natural ones and we omit the subscript R from the notation. Considering the longexact sequence in cohomology we obtain a commutative diagram with exact rows of pointed sets: H ( R, G sc E ) H ( R, G E ) H ( R, µ )H ( R, G sc ) H ( R, G ) H ( R, µ ) = Lemma 6.2 implies that H ( R, G sc ) is trivial. The exactness of the rows and the commutativity of thediagram imply that the kernel of the map H ( R, G ) → H ( R, G E ) is trivial, as desired.29 heorem 4.5. Let R be a Q -algebra such that every locally free R -module is free and b ∈ B rs ( R ) . Thenthere is a canonical injection η b : P b ( R ) / P b ( R ) ֒ → G ( R ) \ V b ( R ) compatible with base change. Moreover themap η b sends the identity element to the orbit of σ ( b ) .Proof. If A ∈ P b ( R ) , define η b ( A ) ∈ H ( R, P b [2]) as the image of A under the -descent map, namely theisomorphism class of the P b [2] -torsor [2] − ( A ) . It suffices to prove, under the identification of Lemma 4.2,that the class η b ( A ) is killed under the map H ( R, P b [2]) → H ( R, G ) . By Lemma 4.4 it suffices to prove thatthis class is trivial in H ( R, G E ) . By the parametrization of orbits of the representation V E [Lag20, Theorem3.13], the composite J b ( R ) / J b ( R ) ֒ → H ( R, J b [2]) → H ( R, G E ) is trivial. The commutative diagram P b ( R ) / P b ( R ) J b ( R ) / J b ( R )H ( R, P b [2]) H ( R, J b [2])H ( R, G ) H ( R, G E ) then implies the theorem.We obtain the following concrete corollary of the parametrization of -Selmer elements. Corollary 4.6. Let b ∈ B rs ( Q ) and write Sel P b for the -Selmer group of P b . Then the injection η b : P b ( Q ) / P b ( Q ) ֒ → G ( Q ) \ V b ( Q ) of Theorem 4.5 extends to an injection Sel P b ֒ → G ( Q ) \ V b ( Q ) . Proof. To prove the corollary it suffices to prove that -Selmer elements in H ( Q , P b [2]) are killed under thenatural map H ( Q , P b [2]) → H ( Q , G ) . By definition, an element of Sel P b consists of a class in H ( Q , P b [2]) whose restriction to H ( Q v , P b [2]) lies in the image of the -descent map for every place v . By Theorem4.5 the image of such an element in H ( Q v , G ) is trivial for every v . Since the restriction map H ( Q , µ ) → Q v H ( Q v , µ ) has trivial kernel by the Hasse principle for the Brauer group, the kernel of H ( Q , G ) → Q v H ( Q v , G ) is trivial too. ( G ⋆ , V ⋆ ) We define a representation ( G ⋆ , V ⋆ ) and study its relation to ( G , V ) using the binary quartic resolvent mapfrom §2.5. Definition 4.7. Define the Q -group G ⋆ := PGL . Define the G ⋆ -representation V ⋆ := Q ⊕ Q ⊕ Sym (2) ,where Q denotes a copy of the trivial representation and Sym (2) denotes the space of binary quartic forms { q | q ( x, y ) = ax + bx y + cx y + dxy + ey } . An element [ A ] ∈ PGL ( Q ) acts on q via [ A ] · q ( x, y ) = q (( x, y ) · A ) / (det A ) . Define B ⋆ := V ⋆ // G ⋆ . We will typically write an element of V ⋆ as a triple ( b , b , q ) . We define a G m -action on V ⋆ by λ · ( b , b , q ) =( λ b , λ b , λ q ) . Write Q : V → V ⋆ for the morphism v ( p ( v ) , p ( v ) , Q v ) , where Q v denotes the resolvent30inary quartic from §2.5 and p , p denote the invariant polynomials fixed in Proposition 3.3. The odd choiceof G m -action on V ⋆ is explained by the fact it makes Q equivariant with respect to the G m -actions on V and V ⋆ . Similarly to §2.5 write p : G → G ⋆ for the projection associated to the identification G ≃ (Sp × SL ) /µ chosen in §2.4. There exists a unique G m -action on B ⋆ such that the quotient morphism π ⋆ : V ⋆ → B ⋆ is G m -equivariant.If q ( x, y ) = ax + bx y + cx y + dxy + ey , we define I ( q ) := − ae − bd + c ) , (4.2.1) J ( q ) := (72 ace + 9 bcd − ad − eb − c ) . (4.2.2)Then I, J generate the ring of invariants of a binary quartic form. (Our I ( q ) is − times the degree- invariantdefined in [BS15a, §2, Eq. (4)].) We obtain an isomorphism of graded Q -algebras Q [ B ⋆ ] ≃ Q [ b , b , I, J ] where b , b , I, J have degree , , , respectively. Moreover a binary quartic form q with coefficients in afield extension k/ Q has distinct roots in P (¯ k ) if and only if I ( q ) + 27 J ( q ) = 0 .We describe centralizers of elements of V ⋆ in two ways. First we recall their classical relation to -torsion ofelliptic curves. If k is a field and I, J ∈ k write E I,J for the elliptic curve over k given by the Weierstrassequation y = x + Ix + J . Lemma 4.8. Let k/ Q be a field and v ∈ V ⋆ ( k ) have invariants ( I, J ) := ( I ( v ) , J ( v )) ∈ k such that I + 27 J = 0 . Then there is an isomorphism of finite étale group schemes over k : Z G ⋆ ( v ) ≃ E I,J [2] . Proof. Up to scaling the invariants and changing an elliptic curve by a quadratic twist which doesn’t affectthe -torsion group scheme, this is contained in [BS15a, Theorem 3.2].Next we give an alternative interpretation of centralizers in V ⋆ using the results of §3. Recall from Corollary3.15 that we have an exact sequence of finite étale group schemes over B rs : → E [2] → P [2] → ˆ E [2] → . Lemma 4.9. The following two morphisms are canonically identified: • The morphism p : Z G ( σ | B rs ) → Z G ⋆ ( Q ◦ σ | B rs ) . • The morphism P [2] → ˆ E [2] .In particular for every field k/ Q and b ∈ B rs ( k ) , we have an isomorphism of k -group schemes Z G ⋆ ( Q ( σ ( b ))) ≃ E p ( b ) ,p ( b ) [2] . Proof. The last sentence follows from the first claim and the fact that ˆ E b and E p ( b ) ,p ( b ) are quadratictwists so have isomorphic -torsion group scheme. To prove the first claim it suffices to prove that the map Z G ( σ | B rs ) → Z G ⋆ ( Q ◦ σ | B rs ) is a nonconstant morphism of finite étale group schemes and Z G ⋆ ( Q ◦ σ | B rs ) has order ; its kernel must then correspond, under the isomorphism Z G ( σ | B rs ) ≃ P [2] of Proposition 3.10,to the unique finite étale subgroup scheme of P [2] of order by Corollary 3.5. Lemma 2.10 implies that Z G ⋆ ( Q ◦ σ | B rs ) is finite étale and Lemma 4.8 implies that it is of order . Assume for contradiction that p isconstant. Then by Lemma 4.1 we obtain a commutative diagram for every field k/ Q and b ∈ B rs ( k ) :31 ( k ) \ V b ( k ) G ⋆ ( k ) \ V ⋆ Q ( b ) ( k )H ( k, Z G ( σ ( b ))) H ( k, Z G ⋆ ( Q ( σ ( b )))) Q where the bottom map is constant. This implies that for every field k/ Q and for every two v , v ∈ V b ( k ) ,the binary quartic forms Q v and Q v are PGL ( k ) -equivalent. In particular by taking v = σ ( b ) , Lemma2.12 shows that Q v has a k -rational linear factor for every v ∈ V rs ( k ) . It is now simple to exhibit an explicit v ∈ V rs ( k ) for which this fails; an example with k = R is given in Remark 2.13.The morphism Q : V → V ⋆ induces a morphism B → B ⋆ , still denoted by Q . We write Q , Q , Q I , Q J ∈ Q [ B ] for the components of Q using the coordinates b , b , I, J . Evidently, we have Q = p and Q = p . Thenext lemma determines Q I and Q J up to a constant. Proposition 4.10. There exists λ ∈ Q × such that Q I = λ p , Q J = λ p . Proof. Since Q is G m -equivariant, the elements Q I and Q J of Q [ B ] are homogeneous of degree , respec-tively. Lemma 2.10 implies that Q maps B rs in the locus of B ⋆ where Q I + 27 Q J does not vanish. In otherwords, we have a divisibility of polynomials in Q [ B ] : Q I ( b ) + 27 Q J ( b ) | (4 p (ˆ b ) + 27 p (ˆ b ) )(4 p ( b ) + 27 p ( b ) ) . (4.2.3)Here we have replaced ∆ ∈ Q [ B ] by its explicit description afforded by Lemma 3.26. By degree considerationsand the fact that the right hand side of (4.2.3) is a product of two irreducible polynomials, we know that upto a nonzero constant Q I ( b ) + 27 Q J ( b ) equals either p ( b ) + 27 p ( b ) or p (ˆ b ) + 27 p (ˆ b ) . In the firstcase, an explicit computation (using that Q I is a Q -linear combination of elements of the form p , p p , p and analogously for Q J ) one see that we must have ( Q I ( b ) , Q J ( b )) = ( λ p ( b ) , λ p ( b )) for some λ ∈ Q × .In the second case, we must have ( Q I ( b ) , Q J ( b )) = ( λ p (ˆ b ) , λ p (ˆ b )) for some λ ∈ Q × since b ˆ b is anisomorphism (Theorem 3.14).We argue by contradiction to exclude the second case, so suppose that it holds. Let k/ Q be an algebraicallyclosed field extension and µ ∈ k × a fourth root of λ . Then ( Q I ( b ) , Q J ( b )) = ( p ( µ · ˆ b ) , p ( µ · ˆ b )) . Let η : Spec k ( η ) → B k be the generic point of B k and for ease of notation write v ⋆ = Q ( σ ( η )) . By Lemma 4.8we have an isomorphism Z G ⋆ ( v ⋆ ) ≃ E Q I ( η ) , Q J ( η ) [2] . On the other hand by Lemma 4.9 we have Z G ⋆ ( v ⋆ ) ≃ E p ( η ) ,p ( η ) [2] . Using the assumption ( Q I ( b ) , Q J ( b )) = ( p ( µ · ˆ b ) , p ( µ · ˆ b )) and the fact that ˆ E b [2] ≃ E p ( b ) ,p ( b ) [2] , we obtain a chain of isomorphisms ˆ E η [2] ≃ E p ( η ) ,p ( η ) [2] ≃ Z G ⋆ ( v ⋆ ) ≃ E Q I ( η ) , Q J ( η ) [2] = E p ( µ · ˆ η ) ,p ( µ · ˆ η ) [2] ≃ E η [2] . But by Corollary 3.15, E [2] and ˆ E [2] are not isomorphic as finite étale group schemes over B rs . By [Sta18, Tag0BQM] and the fact that B is normal, the k ( η ) -groups E η [2] and ˆ E η [2] are not isomorphic either. This is acontradiction, proving the proposition. Corollary 4.11. The map Q : B → B ⋆ is a G m -equivariant isomorphism.Proof. In the coordinates B ≃ A p ,p ,p ,p ) and B ⋆ ≃ A b ,b ,I,J ) , Q takes the form ( p , p , p , p ) ( p , p , λ p , λ p for some λ ∈ Q × by Proposition 4.10.32 .3 Embedding the ˆ ρ -Selmer group The following proposition follows from Lemmas 4.1 and 4.9 by the same proof as Lemma 4.2. If b ⋆ is ascheme-theoretic point of B ⋆ we write V ⋆b ⋆ for the fibre of π ⋆ : V ⋆ → B ⋆ under b ⋆ . Proposition 4.12. Let R be a Q -algebra. Let b ∈ B rs ( R ) with b ⋆ := Q ( b ) . Then there are canonicalbijections of sets:1. G ( R ) \ V b ( R ) ≃ ker (cid:0) H ( R, P b [2]) → H ( R, G ) (cid:1) . G ⋆ ( R ) \ V ⋆b ⋆ ( R ) ≃ ker (cid:16) H ( R, ˆ E b [2]) → H ( R, G ⋆ ) (cid:17) . The reducible orbits G ( R ) · σ ( b ) and G ⋆ ( R ) · Q ( σ ( b )) correspond to the trivial element in H ( R, P b [2]) and H ( R, ˆ E b [2]) respectively. Moreover the following diagram is commutative. G ( R ) \ V b ( R ) G ⋆ ( R ) \ V ⋆b ⋆ ( R )H ( R, P b [2]) H ( R, ˆ E b [2]) Here the horizontal maps are induced by Q : V → V ⋆ and the projection P b [2] → ˆ E b [2] respectively and thevertical maps are the injections induced by the above identifications. The following corollary will be useful later and connects the notion of almost reducibility to a more arithmeticone. It follows from the commutative diagram of Proposition 4.12. Corollary 4.13. Let k/ Q be a field and b ∈ B rs ( k ) . Then the following are equivalent for v ∈ V b ( k ) : • v is almost k -reducible (Definition 2.11). • The class of G ( k ) · v in H ( k, P b [2]) under the bijection of Proposition 4.12 lies in the kernel of the map H ( k, P b [2]) → H ( k, ˆ E b [2]) . Theorem 4.14. Let R be a Q -algebra such that every locally free R -module of constant rank is free and let b ∈ B rs ( R ) with b ⋆ := Q ( b ) . Then there exists a natural embedding η ⋆b : P b ( R ) / ˆ ρ ( P ∨ b ( R )) ֒ → G ⋆ ( R ) \ V ⋆b ⋆ ( R ) compatible with base change on R . Moreover it sends the identity element to the orbit G ⋆ ( R ) · Q ( σ ( b )) .Proof. Recall from Corollary 3.15 that we have an isomorphism P ∨ b [ˆ ρ ] ≃ ˆ E b [2] . For A ∈ P b ( R ) , write η ⋆b ( A ) ∈ H ( R, ˆ E b [2]) for the image of A under the ˆ ρ -descent map transported along the isomorphism H ( R, P ∨ b [ˆ ρ ]) ≃ H ( R, ˆ E b [2]) . Using the identification of Proposition 4.12 it suffices to prove that η ⋆b ( A ) iskilled under the map H ( R, ˆ E b [2]) → H ( R, G ⋆ ) . The commutative diagram P b ( R ) / P b ( R ) P b ( R ) / ˆ ρ ( P ∨ b ( R ))H ( R, P b [2]) H ( R, P b [ˆ ρ ]) η ⋆b ( A ) lifts to a class in H ( R, P b [2]) lying in the image of the -descent map. By the proof ofTheorem 4.5, the image of this class in H ( R, G ) is trivial. Therefore the image of η ⋆b ( A ) in H ( R, G ⋆ ) istrivial too.We obtain the following consequence for the ˆ ρ -Selmer group, whose proof is identical to the proof of Corollary4.6 and uses the fact that H ( Q , G ⋆ ) → Q v H ( Q v , G ⋆ ) has trivial kernel. Corollary 4.15. Let b ∈ B rs ( Q ) with b ⋆ := Q ( b ) and write Sel ˆ ρ P ∨ b for the ˆ ρ -Selmer group of P ∨ b . Then theinjection η ⋆b : P b ( Q ) / ˆ ρ ( P ∨ b ( Q )) ֒ → G ⋆ ( Q ) \ V ⋆b ⋆ ( Q ) of Theorem 4.14 extends to an injection Sel ˆ ρ P ∨ b ֒ → G ⋆ ( Q ) \ V ⋆b ⋆ ( Q ) . So far we have considered properties of the pair ( G , V ) and ( G ⋆ , V ⋆ ) over Q . In this subsection we definethese objects over Z and observe that the above results and constructions are still valid over Z [1 /N ] for anappropriate choice of integer N ≥ .Indeed, our choice of pinning of H in §2.1 determines a Chevalley basis of h , hence a Z -form h of h (in thesense of [Bor70, §1]) with adjoint group H , a split reductive group of type F over Z . The Z -lattice V = V ∩ h is admissible [Bor70, Definition 2.2]; define G as the Zariski closure of G in GL( V ) . The Z -group scheme G has generic fibre G and acts faithfully on the free Z -module V of rank . The automorphism θ : H → H extends by the same formula to an automorphism H → H , still denoted by θ . We may similarly define H E , G E and V E and extend θ E , ζ to involutions H E → H E . Lemma 5.1. G is a split reductive group over Z of type C × A .2. The equality H θ = G extends to an isomorphism H θ Z [1 / ≃ G Z [1 / .3. The equality H ζ E = H extends to an isomorphism H ζ E , Z [1 / ≃ H Z [1 / .Proof. For the first claim, it suffices to prove that G → Spec Z is smooth and affine and that its geometricfibres are connected reductive groups. But G is Z -flat and affine by construction, and its fibres are reductiveby [Bor70, §4.3]. The second claim follows from the fact that H θ Z [1 / is a reductive group scheme of thesame type as G Z [1 / , which follows from [Con14, Remark 3.1.5]. The third claim follows from the fact that H ζ E , Z [1 / is Z [1 / -smooth by Lemma 3.8, and that its geometric fibres are adjoint semisimple of type F (bythe same reasoning as [Ree10, §3.1]).We define the smooth Z -group G ⋆ := PGL and G ⋆ -representation V ⋆ := Z ⊕ Z ⊕ Sym (2) , where Sym (2) denotes the space of binary quartic forms ax + bx y + cx y + dxy + ey with a, . . . , e ∈ Z . The Z -module V ⋆ is free of rank .Recall that in §3.1 we have fixed polynomials p , p , p , p ∈ Q [ V ] G satisfying the conclusions of Proposition3.3. Note that those conclusions are invariant under the G m -action on B . By rescaling the polynomials p , . . . , p using this G m -action, we can assume they lie in Z [ V ] G . We may additionally assume that thediscriminant ∆ from §3.6 lies in Z [ V ] G . Define B := Spec Z [ p , p , p , p ] . We have an invariant map π : V → B . Define B rs := B [∆ − ] . 34ecall from §4.2 that we have defined a morphism Q : V → V ⋆ using the binary quartic resolvent from §2.5,which extends by the same formula to a morphism Q : V → V ⋆ (This follows from Formula (2.4.2) and ourchoice of isomorphism V ≃ W ⊠ (2) made at the end of §2.4). Define B ⋆ = Spec Z [ b , b , I, J ] and write π ⋆ : V ⋆ → B ⋆ for the invariant map.Extend the morphism χ from §3.4 to the morphism χ : B → B given by the same Formula (3.4.3). Following§3.6 we define ∆ ˆ E := 4 p + 27 p and ∆ E := ∆ E ◦ χ , both elements of Z [ B ] .We extend the family of curves given by Equation (3.1.2) to the family C → B given by that same equation.Similarly we define E → B by the family of curves given by Equation (3.3.2). They are defined by theprojective closures of the equations C : y + p xy + p y = x + p x + p , (5.1.1) E : y + p xy + p y = x + p x + p . (5.1.2)As before if X is a B -scheme we write ˆ X for the pullback of X along χ : B → B .We can find an integer N with the following properties (set S = Z [1 /N ] ):1. The integer N is good in the sense of [Lag20, Proposition 4.1]. In particular, , and are invertiblein S and C S → B S is flat and proper with geometrically integral fibres and smooth exactly above B rs S .2. The morphism Q : B → B ⋆ of §4.2 extends to an isomorphism Q : B S → B ⋆S , and there exists λ ∈ S × such that ( Q I , Q J ) = ( λ p , λ p ) (Proposition 4.10).3. The discriminant locus { ∆ = 0 } S → B S has geometrically reduced fibres. Moreover ∆ agrees with ∆ E ∆ ˆ E up to a unit in Z [1 /N ] . (Proposition 3.26.)4. There exists open subschemes V rs ⊂ V reg ⊂ V S such that if S → k is a map to a field and v ∈ V ( k ) then v is regular if and only if v ∈ V reg ( k ) and v is regular semisimple if and only if v ∈ V rs ( k ) . Moreover, V rs is the open subscheme defined by the nonvanishing of the discriminant polynomial ∆ ∈ S [ V ] .5. S [ V ] G = S [ p , p , p , p ] . The Kostant section of §2.2 extends to a section σ : B S → V reg of π satisfyingthe following property: for any b ∈ B ( Z ) ⊂ B S ( S ) we have σ ( N · b ) ∈ V ( Z ) .6. Define J → B rs S to be the Jacobian of the family of smooth curves C rs S → B rs S [BLR90, §9.3, Theorem1]. Let E → B rs S denote the restriction of E S to B rs S . Let P → B S be the Prym variety of the cover C rs S → E as defined in §3.3. Then the isomorphism from Proposition 3.10 extends to an isomorphism P [2] ≃ Z G S ( σ | B rs S )) of finite étale group schemes over B rs S .7. The action map G S × B S → V reg , ( g, b ) g · σ ( b ) is étale and its image contains V rs . (Proposition 2.4.)8. The B -scheme P constructed in §3.5 extends to a B S -scheme P → B S which is flat, projective, withgeometrically integral fibres and whose restriction to B rs S is isomorphic to P . Moreover, P → S issmooth with geometrically integral fibres, and the smooth locus of the morphism P → B S is an opensubscheme of P whose complement is S -fibrewise of codimension at least two. (Proposition 3.24.)9. Let Pic C S / B S denote the identity component of the relative Picard scheme of C S → B S . Then the fibresof Pic C S / B S [1 + τ ∗ ] → B S are geometrically integral. (Proposition 3.25.)10. For every field k of characteristic not dividing N and b ∈ B rs ( k ) , there exists an isomorphism ˆ P b ≃ P ∨ b of (1 , -polarized abelian varieties. (Theorem 3.14)35he existence of such an N follows from the principle of spreading out. (See [Lag20, Proposition 4.1] formore details.) We fix such an integer for the remainder of the paper.Using these properties, we can extend our previous results to S -algebras rather than Q -algebras. We mentionin particular: Proposition 5.2 (Analogue of Lemma 4.2 and Proposition 4.12) . Let R be an S -algebra and b ∈ B rs ( R ) with b ⋆ := Q ( b ) . Then we have natural bijections of pointed sets:1. G ( R ) \ V b ( R ) ≃ ker (cid:0) H ( R, P b [2]) → H ( R, G ) (cid:1) . G E ( R ) \ V E ,b ( R ) ≃ ker (cid:0) H ( R, J b [2]) → H ( R, G E ) (cid:1) . G ⋆ ( R ) \ V ⋆b ⋆ ( R ) ≃ ker (cid:16) H ( R, ˆ E b [2]) → H ( R, G ⋆ ) (cid:17) . Proposition 5.3 (Analogue of Theorems 4.5 and 4.14) . Let R be an S -algebra and b ∈ B rs ( R ) with b ⋆ := Q ( b ) . Suppose that every locally free R -module of constant rank is free. Then there is a commutative diagram P b ( R ) / P b ( R ) G ( R ) \ V b ( R ) P b ( R ) / ˆ ρ ( P ∨ b ( R )) G ⋆ ( R ) \ V ⋆b ⋆ ( R ) η b Q η ⋆b Here the horizontal arrows η b and η ⋆b are injections and send the identity to the orbit of σ ( b ) and Q ( σ ( b )) respectively. The main aim of §5 is to prove the following two theorems concerning integral orbit representatives. Bothhave consequences for orbits over Z , see Corollaries 5.27 and 5.28. Theorem 5.4. Let p be a prime not dividing N . Let b ∈ B ( Z p ) with ∆( b ) = 0 . Then every orbit in theimage of the map η b : P b ( Q p ) / P b ( Q p ) → G ( Q p ) \ V b ( Q p ) of Theorem 4.5 has a representative in V ( Z p ) . Theorem 5.5. Let p be a prime not dividing N . Let b ∈ B ( Z p ) with ∆( b ) = 0 and write b ⋆ := Q ( b ) . Thenevery orbit in the image of the map η ⋆b : P b ( Q p ) / ˆ ρ ( P ∨ b ( Q p )) → G ⋆ ( Q p ) \ V ⋆b ⋆ ( Q p ) of Theorem 4.14 has a representative in V ⋆ ( Z p ) . Theorem 5.5 will follow easily from Theorem 5.4, so we spend most of §5 proving Theorem 5.4. In this section we define some groupoids which will be a convenient way to think about orbits in our repre-sentations and a crucial ingredient for the proof of Theorem 5.4. It is closely modelled on the correspondingsection [RT, §4.3]; the reader may also consult [Lag20, §4.2]. Throughout this section we fix a scheme X over S = Z [1 /N ] .We define the groupoid GrLie X whose objects are pairs ( H ′ , θ ′ ) where36 H ′ is a reductive group scheme over X whose geometric fibres are simple of Dynkin type F . (See[Con14, Definition 3.1.1] for the definition of a reductive group scheme over a general base.)• θ ′ : H ′ → H ′ is an involution of reductive X -group schemes such that for each geometric point ¯ x of X there exists a maximal torus A ¯ x of H ′ ¯ x such that θ ′ acts as − on X ∗ ( A ¯ x ) .A morphism ( H ′ , θ ′ ) → ( H ′′ , θ ′′ ) in GrLie X is given by an isomorphism φ : H ′ → H ′′ such that φ ◦ θ ′ = θ ′′ ◦ φ .There is a natural notion of base change and the groupoids GrLie X form a stack over the category of schemesover S in the étale topology. Recall that in §5.1 we have defined a pair ( H S , θ S ) which by [RLYG12, Corollary14] defines an object of GrLie S . Proposition 5.6. Let X be an S -scheme. The assignment ( H ′ , θ ′ ) Isom(( H X , θ X ) , ( H ′ , θ ′ )) defines abijection between: • The isomorphism classes of objects in GrLie X . • The set H ( X, G ) .Proof. Since GrLie is a stack in the étale topology of S -schemes and Aut(( H X , θ X )) = G X , it suffices toprove that any two objects ( H, θ ) , ( H ′ , θ ′ ) of GrLie X are étale locally isomorphic. The proof of this factgiven below is very similar to the proof of [RT, Lemma 2.3]; we reproduce it here for convenience.Since H ′ is étale locally isomorphic to H we may assume that H ′ = H . Let T denote the X -scheme of elements h ∈ H such that Ad( h ) ◦ θ = θ ′ ; it is a closed subscheme of H that is X -smooth by [Con14, Proposition 2.1.2].Since smooth surjective morphisms have sections étale locally, it suffices to prove that T → X is surjective.Since the construction of T is compatible with base change we may assume that X = Spec k where k is analgebraically closed field.By assumption, there exist maximal tori A, A ′ ⊂ H on which θ, θ ′ ∈ H ( k ) act through − . Using theconjugacy of maximal tori, we may assume that A = A ′ so θ ′ = a · θ for some a ∈ A ( k ) . Writing a = b forsome b ∈ A ( k ) , we see that θ ′ = b · b · θ = b · θ · b − . Therefore θ ′ is H ( k ) -conjugate (even A ( k ) -conjugate)to θ , as desired.We define the groupoid GrLieE X whose objects are triples ( H ′ , θ ′ , γ ′ ) where ( H ′ , θ ′ ) is an object of GrLie X and γ ′ ∈ h ′ (the Lie algebra of H ′ ) satisfying θ ′ ( γ ′ ) = − γ ′ . A morphism ( H ′ , θ ′ , γ ′ ) → ( H ′′ , θ ′′ , γ ′′ ) in GrLieE X is given by a morphism φ : H ′ → H ′′ in GrLie X mapping γ ′ to γ ′′ .We define a functor GrLieE X → B ( X ) (where B ( X ) is seen as a discrete category) as follows. For an object ( H ′ , θ ′ , γ ′ ) in GrLieE X , choose a faithfully flat extension X ′ → X such that there exists an isomorphism φ :( H ′ , θ ′ ) X ′ → ( H S , θ ) X ′ in GrLie X . We define the image of the object ( H ′ , θ ′ , γ ′ ) under the map GrLieE X → B ( X ) by π ( φ ( γ ′ )) . This procedure is independent of the choice of φ and X ′ and by descent defines an elementof B ( X ) . For b ∈ B ( X ) we write GrLieE X,b for the full subcategory of elements of GrLieE X,b mapping to b under this map. In §5.1 we have defined an object ( H B S , θ B S , σ ) of GrLieE B S .Recall that for b ∈ B ( X ) , V b denotes the fibre of b of the map π : V → B . Proposition 5.7. Let X be an S -scheme and let b ∈ B rs ( X ) . The assignment ( H ′ , θ ′ , γ ′ ) Isom(( H X , θ X , σ ( b )) , ( H ′ , θ ′ , γ ′ )) defines a bijection between Isomorphism classes of objects in GrLieE X,b . • The set H ( X, Z G ( σ ( b ))) .Proof. The object ( H X , θ X , σ ( b )) of GrLieE X,b has automorphism group Z G ( σ ( b )) . By descent, it suffices toprove that every object ( H ′ , θ ′ , γ ′ ) in GrLieE X,b is étale locally isomorphic to ( H X , θ X , σ ( b )) . By Proposition5.6, we may assume that ( H ′ , θ ′ ) = ( H X , θ X ) . By Property 7 of §5.1 (which is a spreading out of Proposition2.4 over S ), the action map G S × B rs S → V rs is étale and surjective. Therefore it has sections étale locally,hence γ ′ is étale locally G -conjugate to σ ( b ) .The following important proposition gives an interpretation of the (not necessarily regular semisimple) G ( X ) -orbits of V ( X ) in terms of the groupoids GrLie X and GrLieE X . Proposition 5.8. Let X be an S -scheme and let b ∈ B ( X ) . The following sets are in canonical bijection: • The set of G ( X ) -orbits on V b ( X ) . • Isomorphism classes of objects ( H ′ , θ ′ , γ ′ ) in GrLieE X,b such that ( H ′ , θ ′ ) ≃ ( H S , θ ) X in GrLie X .Proof. If v ∈ V b ( X ) we define the object A v = ( H X , θ X , v ) of GrLieE X,b . The assignment v 7→ A v establishesa well-defined bijection between the two sets of the proposition; we omit the formal verification.The following lemma is analogous to [RT, Lemma 5.6] and will be useful in §5.4 to extend orbits over a baseof dimension . Lemma 5.9. Let X be an integral regular scheme of dimension . Let U ⊂ X be an open subschemewhose complement has dimension . If b ∈ B S ( X ) , then restriction induces an equivalence of categories GrLieE X,b → GrLieE U,b | U .Proof. We will use the following fact [CTS79, Lemme 2.1(iii)] repeatedly: if Y is an affine X -scheme of finitetype, then restriction of sections Y ( X ) → Y ( U ) is bijective. To prove essential surjectivity, let ( H ′ , θ ′ , γ ′ ) be an object of GrLieE U,b | U . By [CTS79, Théoreme 6.13] and Proposition 5.6, ( H ′ , θ ′ ) extends to an object ( H ′′ , θ ′′ ) of GrLie X . If Y is the closed subscheme of h ′′ of elements γ satisfying θ ′′ ( γ ) = − γ and γ mapsto b in B ( X ) , then Y is affine and of finite type over X . It follows by the fact above that γ ′ lifts to anelement γ ′′ ∈ h ′′ ( X ) and that ( H ′′ , θ ′′ , γ ′′ ) defines an object of GrLieE X,b . Since the scheme of isomorphisms Isom GrLieE ( A , A ′ ) is X -affine, fully faithfulness follows again from the above fact. In this subsection we perform some calculations with component groups of Néron models of Prym varieties.For the purposes of constructing integral representatives (Theorems 5.4 and 5.5), only Proposition 5.12 willbe needed. However, the finer analysis succeeding it is necessary to obtain a lower bound in Theorem 1.1;of this analysis only Corollary 5.20 will be used later. Notation. For the remainder of §5.3, let R be a discrete valuation ring with residue field k and fractionfield K . We suppose that N is invertible in R .Recall that we have defined in §5.1 abelian schemes J , P and E over B rs S . We will use a minor abuse ofnotation and for any b ∈ B ( R ) with ∆( b ) = 0 , we write J b (which is a priori only defined when ∆( b ) ∈ R × )38or the K -scheme J b K , and similarly for P b and E b . For such b we write J b , P b , P ∨ b , E b for the Néronmodels of J b , P b , P ∨ b , E b respectively. The involution τ ∗ b of J b uniquely extends to an involution of J b , againdenoted by τ ∗ b . Lemma 5.10. Let b ∈ B ( R ) with ∆( b ) = 0 . Then the equality P b = ker(1 + τ ∗ b : J b → J b ) from (3.3.1)extends to an isomorphism P b ≃ ker(1 + τ ∗ b : J b → J b ) .Proof. It suffices to prove that ker(1 + τ ∗ b : J b → J b ) is smooth over R and satisfies the Néron mappingproperty for P b . The smoothness follows from Lemma 3.8 applied to − τ ∗ b . The Néron mapping propertyfollows from that of J b . Lemma 5.11. Let b ∈ B ( R ) with ∆( b ) = 0 . Suppose that the curve C b /R is regular. Then J b and P b haveconnected fibres.Proof. By [BLR90, §9.5, Theorem 1], J b is isomorphic to Pic C b /R , the identity component of the Picardscheme of C b → Spec R . Since Pic C b /R has connected fibres by definition, the same holds for J b .It remains to consider P b . Lemma 5.10 and the previous paragraph shows that P b is isomorphic to Pic C b /R [1 + τ ∗ b ] . It therefore suffices to prove that Pic C b /R [1 + τ ∗ b ] has connected fibres. By Proposition3.25 (and its analogue over B S : Property 9 of §5.1), Pic C S / B S [1 + τ ∗ ] → B S has connected fibres. Since Pic C b /R [1 + τ ∗ b ] is the pullback of Pic C S / B S [1 + τ ∗ ] along the R -point b , the lemma follows. Proposition 5.12. Let R be a discrete valuation ring in which N is a unit. Let K = Frac R and let ord K : K × ։ Z be the normalized discrete valuation. Let b ∈ B ( R ) and suppose that ord K ∆( b ) ≤ . Let J b , P b , P ∨ b and E b be the Néron models of J b , P b , P ∨ b and E b respectively. Then:1. The special fibres of the R -groups J b , P b , P ∨ b and E b are connected.2. If ord K ∆( b ) = 1 , the special fibre of the quasi-finite étale R -group scheme P b [2] has order .Proof. If ord K ∆( b ) = 0 , all abelian varieties in question have good reduction so the proposition holds. Thusfor the remainder of the proof we assume that ord K ∆( b ) = 1 . The choice of N implies that ∆ equals ∆ E · ∆ ˆ E up to a unit in Z [1 /N ] (Property 3 of §5.1). This allows us to consider two separate cases, the first casebeing (ord K ∆ E ( b ) , ord K ∆ ˆ E ( b )) = (1 , and the second case being (ord K ∆ E ( b ) , ord K ∆ ˆ E ( b )) = (0 , . Let R and B be the closed subschemes of C b and E b given by intersecting them with the line { y = 0 } ⊂ P R usingEquations (5.1.1) and (5.1.2) respectively. Then the morphism C b → E b restricts to a finite étale morphism C b − R → E b − B . We recall that ∆ ˆ E ( b ) = 4 p ( b ) + 27 p ( b ) .Case 1. In this case the discriminant of E b has valuation . By Tate’s algorithm (see [Sil94, Lemma IV.9.5(a)]),this implies that E b is regular and its special fibre has a unique singularity, which is a node. Since ord K ∆ ˆ E ( b ) = 0 , the singular point of E b,k is not contained in B hence it lifts to two distinct singularpoints of C b,k which are also nodes. Since C b → E b is étale outside B , C b is regular and its special fibrehas two nodal singular points which are swapped by the involution τ b : C b → C b .Case 2. Now E b is smooth over R so the singular points of C b are contained in R . Since ord K ∆ ˆ E ( b ) = 1 , thediscriminant of the polynomial x + p x + p has valuation . Since , ∈ R × , the unique multipleroot of the reduction of this polynomial lies in k ; let α ∈ R be a of this root. Using the substitution x x − α , the curve C b is given by the equation y + a xy + a y = x + a x + a x + a , (5.3.1)39or some a i ∈ R with ord K a ≥ and ord K a ≥ . Since the discriminant of the cubic polynomial onthe right hand side of (5.3.1) has valuation , the formula for such a discriminant shows that a is auniformizer and a ∈ R × . Since E b is smooth over R , we have a ∈ R × . Therefore C b is regular and itsspecial fibre contains a unique nodal singularity. (For this last claim, see [Liu02, Exercise 7.5.7(b)].)We conclude that in both cases E b and C b are regular. Since E b can be identified with the smooth locus of E b [Sil94, Theorem IV.9.1], the special fibre of E b is connected. By Lemma 5.11, the special fibres of J b and P b are connected. Because of the isomorphism P ∨ b ≃ P ˆ b (Theorem 3.14 and its spreading out: Property10 of §5.1) and the fact that ∆( b ) and ∆(ˆ b ) are equal up to a unit in R , the connectedness of the specialfibre of P ∨ b follows from that of P b .For the second part of the lemma, it suffices to prove that the special fibre of P b is an extension of an ellipticcurve by a rank torus. This follows from the fact that the special fibres of J b and E b are semiabelianvarieties of toric rank and respectively in the first case and of toric rank and in the second case.We proceed with a finer analysis of Néron models of Prym varieties, only necessary to obtain a lower boundin Theorem 1.1. If A/K is an abelian variety with Néron model A /R we write A ◦ for the identity component of A , obtained by removing the connected components of A k not containing the identity section. Recallfrom Lemma 5.10 that we may view P b as a closed subgroup scheme of J b . Definition 5.13. Let b ∈ B ( R ) with ∆( b ) = 0 . We say b is admissible if J ◦ b ∩ P b = P ◦ b or equivalently, J ◦ b ∩ P b has connected fibres. The reason for introducing admissibility is Lemma 5.17. It seems unlikely that every b ∈ B ( R ) with ∆( b ) = 0 is admissible, but we have not found a counterexample. Our first goal is establishing a sufficient conditionfor admissibility, Proposition 5.16. This we achieve with the help of the following two lemmas. Lemma 5.14. Let b ∈ B ( R ) with ∆( b ) = 0 . Let ˜ C be a regular model of C b , i.e. a regular, proper, flat R -scheme whose generic fibre is isomorphic to C b . Suppose that the involution τ b of C b extends to an involution τ b of ˜ C . Let Pic C /R → Spec R be the identity component of the Picard scheme of ˜ C /R . Then b is admissibleif (and only if ) the special fibre of Pic C /R [1 + τ ∗ b ] is connected.Proof. Since C b has a K -rational point ∞ , the special fibre of ˜ C has an irreducible component of degree .Therefore by a theorem of Raynaud [BLR90, §9.5, Theorem 4(b)], we have an isomorphism J ◦ b ≃ Pic C /R .This isomorphism intertwines the involutions τ ∗ b on both sides, because these involutions are the uniqueextensions of their restriction to the generic fibre. By Lemma 5.10, we see that J ◦ b ∩ P b ≃ Pic C /R [1 + τ ∗ b ] .Since the generic fibre of J ◦ b ∩ P b equals P b which is connected, the equivalence of the lemma follows fromthe definition of admissibility.For the statement of the next lemma, recall [DG70, Expose VI A , Theoreme 5.4.2] that the category of finitetype commutative group schemes over a field is abelian. Lemma 5.15. Let → A → B → C → be a short exact sequence of finite type commutative group schemes over k . Let τ be an involution of A , B and C whose action is compatible with the above sequence. Suppose that either (1) the quotient A τ / (1 + τ ) A s trivial, or (2) A τ / (1 + τ ) A is finite étale and C [1 + τ ] is connected. Then the following sequence is shortexact: → A [1 + τ ] → B [1 + τ ] → C [1 + τ ] → . Proof. We consider A, B and C as sheaves on the big fppf site of Spec k . The long exact sequence in Z / Z -group cohomology of sheaves applied to the Z / Z -action − τ shows that the following sequence is exact: → A [1 + τ ] → B [1 + τ ] → C [1 + τ ] δ −→ A τ / (1 + τ ) A. (Alternatively, the exactness of this sequence is a statement that can be formulated in any abelian category.Since it is true in the category of R -modules for any ring R , it remains true in our setting.) If A τ / (1 + τ ) A is trivial, the lemma is proven. If A τ / (1 + τ ) A is finite étale and C [1 + τ ] is connected, then δ = 0 sincethere are no nonconstant maps from a connected scheme to a finite étale k -scheme.Recall that (up to a unit in R ) the discriminant ∆ factors as ∆ E · ∆ ˆ E . The proof of Proposition 5.12 showsthat every b ∈ B ( R ) with ord K ∆( b ) ≤ is admissible. We will need the stronger: Proposition 5.16. Let b ∈ B ( R ) with ∆( b ) = 0 and ord K ∆ E ( b ) ≤ . Then b is admissible.Proof. Since Néron models commute with the formation of strict henselization and completion [BLR90, §7.2,Theorem 1(b)], we may assume that R is complete and its residue field k is separably closed. We distinguishcases according to the value of ord K ∆ E ( b ) .Case ord K ∆ E ( b ) = 0 .If C b is regular, b is admissible by Lemma 5.11. We may therefore assume that C b has a non-regular point P ∈ C b . Since E b is R -smooth, P is the unique non-regular point and lies in the ramification locus of themorphism C b → E b . By the same reasoning as Case 2 in the proof of Proposition 5.12, we may assume afterchanging variables x x − α that C b is given by the equation y + a xy + a y = x + a x + a x + a (5.3.2)for some a i ∈ R with ord K a ≥ and ord K a ≥ , and P corresponds to the origin in the special fibre.Again by the smoothness of E b /R , we see that a ∈ R × . Therefore the completed local ring of P k in C b,k isisomorphic to k [[ x, y ]] / ( y − ( x + a x )) . So C b,k has one singular point which is a node or a cusp.Consider the sequence of proper birational morphisms · · · → C → C → C := C b , where for i ≥ weinductively define C i +1 → C i as the composition of the blowup C ′ i → C i of the non-regular locus and thenormalization C i +1 → C ′ i . By a result of Lipman [Liu02, Theorem 8.3.44], there exists an n ≥ such thatthe scheme C n is regular; we denote this scheme by ˜ C . The morphism ˜ C → C b does not depend on n and wecall it the canonical desingularization of C b . Since this process is canonical, the involution τ b of C b lifts to aninvolution of ˜ C .Let X be the closure of C b,k \ { P } in ˜ C k and let Y → X be its normalization. The composite Y → C b,k isalso the normalization of C b,k . Since C b,k has arithmetic genus and resolving a cusp or node decreases thegenus by , the smooth curve Y has genus . The involution τ b of C b,k uniquely lifts to an involution τ b of Y .Moreover the composite morphism Y → X → C b,k → E b,k is a double cover of a smooth genus- curve by asmooth genus- curve hence has two branch points. Therefore the Prym variety Pic Y/k [1 + τ ∗ b ] of this coveris connected and one-dimensional: the connectedness follows from [Mum74, §2, Property (vi)] (in particularthe description of ‘ ker ψ ’ there) combined with [Mum74, §3, Lemma 1].41e have an exact sequence [BLR90, §9.2, Corollary 11] → G → Pic C k /k → Pic Y/k → , where G is a smooth commutative algebraic group of dimension which is an extension of an abelianvariety by a connected linear algebraic group, hence connected. Since Pic C k /k [1 + τ ∗ b ] is two-dimensionaland Pic Y/k [1 + τ ∗ b ] is one-dimensional, G [1 + τ ∗ b ] must be one-dimensional hence equal to G itself. Therefore τ ∗ b | G = − Id G and so G τ ∗ b / (1 + τ ∗ b ) G = G [2] , which is finite étale (note that is invertible in k ). Since Pic Y/k [1 + τ ∗ b ] is connected, Lemma 5.15(2) shows that the following sequence is exact: → G → Pic C k /k [1 + τ ∗ b ] → Pic Y/k [1 + τ ∗ b ] → . Since the outer terms of the sequence are connected, the same is true for the middle term. Therefore b isadmissible by Lemma 5.14.Case ord K ∆ E ( b ) = 1 .By assumption, the curve E b /R is regular and its special fibre has a unique singularity, which is a node[Sil94, Lemma IV.9.5(a)]; let Q ∈ E b be this point. Recall that the morphism f : C b → E b is branched overthe closed subscheme ( y = 0) and étale over the complement. We distinguish two cases.• Suppose that Q is contained in the branch locus of f . Then Q uniquely lifts to P ∈ C b and C b /R issmooth outside P . We may then assume after changing variables x x − α that C b is given by theequation y + a xy + a y = x + a x + a x + a , where ord K a ≥ and ord K a ≥ . Since E b /R is not smooth at Q , we have ord K a ≥ . On theother hand, E b is regular at Q so a is a uniformizer for R . Therefore C b is also regular at P . So C b isregular everywhere, hence b is admissible by Lemma 5.11.• Suppose that Q is not contained in the branch locus of f . Since C b → E b is étale above Q , this pointlifts to two regular nodal points P , P of C b . Therefore the curve C b is regular outside ( y = 0) ⊂ C b,k .Let ˜ C → C b be the canonical desingularization, described in the second paragraph of the first case ofthe proof. The involution τ b of C b lifts to an involution of ˜ C . Let X → ˜ C k be the partial normalizationof the special fibre, given by normalizing the nodes corresponding to P and P . Then τ b also lifts toan involution of X . We have a τ ∗ b -equivariant exact sequence → G m × G m → Pic C k /k → Pic X/k → , where τ ∗ acts on G m × G m by interchanging the two factors. Since Pic C k /k [1 + τ ∗ ] is two-dimensionaland ( G m × G m )[1 + τ ∗ b ] is one-dimensional, Pic X/k [1 + τ ∗ ] = Pic X/k by dimension reasons. By Lemma5.15(1) we obtain an exact sequence → G m → Pic C k /k [1 + τ ∗ ] → Pic X/k → . Since the outer terms are connected, the same is true for the middle term hence b is admissible byLemma 5.14.The reason for introducing admissibility is the following lemma, which is a key ingredient for Proposition5.19. 42 emma 5.17. Let b ∈ B ( R ) be admissible. Then the following commutative diagram has exact rows: P ◦ b J ◦ b E ◦ b P b J b E b Proof. The exactness of the bottom row follows from the exact sequence → P b → J b → E b → and[BLR90, §7.5, Proposition 3(a)], noting that there exists an injection E b ֒ → J b such that the composite E b → J b → E b is multiplication by (Property 1 of §3.3). To verify exactness of the top row, note thatagain by [BLR90, §7.5, Proposition 3(a)] the image of J b → E b contains E ◦ b . Hence the top row is exact at E ◦ b . Since b is admissible, it is also exact at J ◦ b . Finally because P b → J b is a closed immersion, the sameholds for P ◦ b → J ◦ b so the top row is exact at P ◦ b too. Remark 5.18. If b is not admissible, the top row of the above commutative diagram fails to be exact at J ◦ b . If A/K is an abelian variety with Néron model A /R , define the component group of A as Φ A := A k / A ◦ k , afinite étale group scheme over k . Proposition 5.19. Let b ∈ B ( R ) with ∆( b ) = 0 . Suppose that b is admissible and that E b = E ◦ b . Then themorphism ρ : P b → P ∨ b induces an isomorphism of component groups Φ P b ∼ −→ Φ P ∨ b .Proof. Since ˆ ρ ◦ ρ = [2] and ρ ◦ ˆ ρ = [2] , it will suffice to prove that the restriction to -primary parts Φ P b [2 ∞ ] → Φ P ∨ b [2 ∞ ] is an isomorphism of finite étale group schemes. By definition, ρ is given by thecomposite of P b → J b with J b → P ∨ b . So it will suffice to prove that the morphisms Φ P b [2 ∞ ] → Φ J b [2 ∞ ] and Φ J b [2 ∞ ] → Φ P ∨ b [2 ∞ ] are isomorphisms.By Lemma 5.17 and the snake lemma, we obtain an exact sequence → Φ P b → Φ J b → Φ E b . Since Φ E b istrivial by assumption, Φ P b → Φ J b is an isomorphism of finite étale group schemes. Since Grothendieck’spairing on component groups is perfect on l -primary parts when l is invertible in k [Ber01, Theorem 7], thefinite étale group schemes Φ P b [2 ∞ ] , Φ J b [2 ∞ ] and Φ P ∨ b [2 ∞ ] have the same order.By the same reasoning as the proof of Lemma 5.17 using the fact that J ◦ b ∩ E b = E ◦ b = E b , the exact sequence → E b → J b → P ∨ b → induces a commutative diagram with exact rows: E ◦ b J ◦ b P ∨ , ◦ b E b J b P ∨ b The snake lemma gives an exact sequence → Φ E b → Φ J b → Φ P ∨ b . Since Φ E b is trivial, the morphism offinite étale k -group schemes Φ J b → Φ P ∨ b is injective. Since the -primary parts have the same order, theinduced morphism Φ J b [2 ∞ ] → Φ P ∨ b [2 ∞ ] is an isomorphism, proving the proposition.The following important corollary will be the one that we will use later on.43 orollary 5.20. Let p be a prime not dividing N . Let b ∈ B ( Z p ) with ∆( b ) = 0 and p ∤ ∆ ˆ E ( b ) . Thenthe image of the ˆ ρ -descent map P b ( Q p ) / ˆ ρ ( P ∨ b ( Q p )) → H ( Q p , P ∨ b [ˆ ρ ]) coincides with the subset of unramifiedclasses H nr ( Q p , P ∨ b [ˆ ρ ]) .Proof. Since ∆ ˆ E ( b ) and ∆ E (ˆ b ) coincide up to a unit in Z p , we see that p ∤ ∆ E (ˆ b ) . By Proposition 5.16, ˆ b isadmissible and by Tate’s algorithm [Sil94, Lemma IV.9.5(a)], E ˆ b = E ◦ ˆ b . Therefore by Proposition 5.19 andTheorem 3.14, ˆ ρ induces an isomorphism Φ P ∨ b ∼ −→ Φ P b . Under these circumstances, the claim of the corollaryis well-known and follows essentially from Lang’s theorem; see [Čes16, Proposition 2.7(d)]. In this section we analyse the orbits of V and V ⋆ over points in B ( Z p ) and B ⋆ ( Z p ) of square-free discriminant.This will be the first step in proving Theorem 5.4 and will be used in the proofs of Theorems 8.5 and 8.8when applying the square-free sieve. Lemma 5.21. Let R be a discrete valuation ring with residue field k in which N is a unit. Let K = Frac R and let ord K : K × ։ Z be the normalized discrete valuation. Let x ∈ V ( R ) with b = π ( x ) ∈ B ( R ) andsuppose that ord K ∆( b ) = 1 . Then the reduction x k of x in V ( k ) is regular and G (¯ k ) -conjugate to σ ( b ) k . Inaddition the R -group scheme Z G ( x ) is quasi-finite étale and has special fibre of order .Proof. We are free to replace R by a discrete valuation ring R ′ containing R such that any uniformizer in R is also a uniformizer in R ′ . Therefore we may assume that R is complete and k algebraically closed.Let x k = y s + y n be the Jordan decomposition of x k ∈ V ( k ) as a sum of its semisimple and nilpotent parts.Let h ,k = z h ( y s ) and h ,k = image(Ad( y s )) . Then h k = h ,k ⊕ h ,k , where Ad( x k ) acts nilpotently on h ,k and invertibly on h ,k . By Hensel’s lemma, this decomposition lifts to an Ad( x ) -invariant decompositionof free R -modules h R = h ,R ⊕ h ,R , where Ad( x ) acts topologically nilpotently on h ,R and invertibly on h ,R . There exists a unique closed subgroup L ⊂ H R with Lie algebra h ,R such that L is R -smooth withconnected fibres; this follows from an argument identical to the proof of [Lag20, Lemma 4.19]. Moreover theconstruction of L shows that L k = Z H ( y s ) .The proof of Proposition 5.12 shows that the curve C b,k either has one node, or two nodes swapped by τ .Therefore the affine surface S /k cut out by the equation z + y + p ( b ) xy + p ( b ) y = x + p ( b ) x + p ( b ) in A k either has one ordinary double point, or two such double points swapped by ( x, y, z ) ( x, − y, − z ) .The surface S is the fibre above b k ∈ B ( k ) of a semi-universal deformation of a simple singularity of type F ,in the sense of [Slo80, §6.2]. The results of [Slo80, §6.6] (in particular Propositions 2, 3 and the subsequentremark) imply that the derived group of L has type A and the centre Z ( L ) of L has rank . Moreover therestriction θ L of θ to L is a stable involution, in the sense that for each geometric point of Spec R thereexists a maximal torus of L on which θ acts as − , by [Tho13, Lemma 2.4]. There is an isomorphism L /Z ( L ) ≃ PGL inducing an isomorphism h derR, ≃ h R, / z ( h R, ) ≃ sl ,R under which θ L corresponds to theinvolution ξ = Ad ( diag (1 , − . The lemma now follow easily from explicit calculations in sl ,R identical to[Lag20, Lemma 4.19], which we omit.The following proposition and its corollary describe orbits in V of square-free discriminant. Their proofs areidentical to the proofs of [Lag20, Proposition 4.20 and Corollary 4.21], using Proposition 5.12 and Lemma5.21; they will be omitted. Proposition 5.22. Let R be a discrete valuation ring in which N is a unit. Let K = Frac R and let ord K : K × ։ Z be the normalized discrete valuation. Let b ∈ B ( R ) and suppose that ord K ∆( b ) ≤ . Then: . If x ∈ V b ( R ) , then Z G ( x )( K ) = Z G ( x )( R ) .2. The natural map α : G ( R ) \ V b ( R ) → G ( K ) \ V b ( K ) is injective and its image contains η b ( P b ( K ) / P b ( K )) .3. If further R is complete and has finite residue field then the image of α equals η b ( P b ( K ) / P b ( K )) . Corollary 5.23. Let X be a Dedekind scheme in which N is a unit with function field K . For everyclosed point p of X write ord p : K × ։ Z for the normalized discrete valuation of p . Let b ∈ B ( X ) be amorphism such that ord p (∆( b )) ≤ for all p . Let P ∈ P b ( K ) / P b ( K ) and let η b ( P ) ∈ G ( K ) \ V b ( K ) bethe corresponding orbit from Proposition 5.3. Then the object of GrLieE K,b , corresponding to η b ( P ) usingProposition 5.8, uniquely extends to an object of GrLieE X,b . We now consider orbits of square-free discriminant in the representation V ⋆ . We will only need to considerthe case of Z p ; the representation-theoretic input of the following proposition has already been establishedby Bhargava and Shankar [BS15a]. Proposition 5.24. Let p be a prime number not dividing N and let b ∈ B ( Z p ) such that ∆( b ) = 0 and p ∤ ∆ ˆ E ( b ) . Let b ⋆ = Q ( b ) ∈ B ⋆ ( Z p ) . Then:1. If x ∈ V ⋆b ⋆ ( Z p ) , then Z G ⋆ ( x )( Q p ) = Z G ⋆ ( x )( Z p ) .2. The natural map α : G ⋆ ( Z p ) \ V ⋆b ⋆ ( Z p ) → G ⋆ ( Q p ) \ V ⋆b ⋆ ( Q p ) is injective and its image equals η ⋆b ( P b ( Q p ) / ˆ ρ ( P ∨ b ( Q p ))) .Proof. Let E ′ / Q p be the elliptic curve with Weierstrass equation y = x + I ( Q ( b )) / x − J ( Q ( b )) / , where Q : B S → B ⋆S is the resolvent binary quartic map from §4.2 and I, J are the invariants of a binary quarticform of (4.2.1) and (4.2.2). By Equation (3.4.2), Proposition 4.10 and the choice of N , ˆ E b and E ′ arequadratic twists, where the twisting is given by an element of Z [1 /N ] × . We have chosen E ′ so that by[BS15a, Theorem 3.2], there exists an injection η ′ : E ′ ( Q p ) / E ′ ( Q p ) ֒ → G ⋆ ( Q p ) \ V ⋆b ⋆ ( Q p ) with the propertythat the composite E ′ ( Q p ) / E ′ ( Q p ) η ′ −→ G ⋆ ( Q p ) \ V ⋆b ⋆ ( Q p ) ֒ → H ( Q p , ˆ E b [2]) = H ( Q p , E ′ [2]) coincides withthe -descent map. (The second map comes from Proposition 4.12.)Since p is a unit in Z [1 /N ] and the discriminant of ˆ E b is not divisible by p , the same is true for thediscriminant of E ′ . Therefore the Tamagawa number of E ′ is [Sil94, Lemma IV.9.5(a)], hence the imageof E ′ ( Q p ) / E ′ ( Q p ) η ′ −→ G ⋆ ( Q p ) \ V ⋆b ⋆ ( Q p ) ֒ → H ( Q p , ˆ E b [2]) coincides with the subgroup of unramified classes H nr ( Q p , ˆ E b [2]) ⊂ H ( Q p , ˆ E b [2]) [BPS16, Lemma 7.1].Again by the fact that the discriminant of E ′ is square-free, [BS15a, Proposition 3.18] implies that Z G ⋆ ( x )( Q p ) = Z G ⋆ ( x )( Z p ) , that α is injective and that the image of the composite G ⋆ ( Z p ) \ V ⋆b ⋆ ( Z p ) → G ⋆ ( Q p ) \ V ⋆b ⋆ ( Q p ) ֒ → H ( Q p , ˆ E b [2]) is H nr ( Q p , ˆ E b [2]) . By the construction of η ⋆b in the proof of Theorem 4.14, it remains toprove that this subset of H ( Q p , ˆ E b [2]) coincides with the image of the ˆ ρ -descent map P b ( Q p ) / ˆ ρ ( P ∨ ( Q p )) → H ( Q p , P ∨ b [ˆ ρ ]) transported along the isomorphism H ( Q p , P ∨ b [ˆ ρ ]) ≃ H ( Q p , ˆ E b [2]) afforded by Corollary 3.15.Since the latter isomorphism preserves the unramified classes (which only depend on the Galois module ˆ E b [2] ≃ P ∨ b [ˆ ρ ] ), this follows from Corollary 5.20. V In this subsection we prove Theorem 5.4. Our strategy is closely modelled on the strategy of proving[Lag20, Theorem 4.1]: we deform to the case of square-free discriminant using a Bertini type theorem over Z p and using the compactified Prym variety. The following proposition and its proof are very similar to[Lag20, Corollary 4.23]. It establishes the existence of a deformation with good properties.45 roposition 5.25. Let p be a prime number not dividing N . Let b ∈ B ( Z p ) with ∆( b ) = 0 and Q ∈ P b ( Q p ) .Then there exists a morphism X → Z p that is of finite type, smooth of relative dimension and withgeometrically integral fibres, together with a point x ∈ X ( Z p ) satisfying the following properties.1. There exists a morphism ˜ b : X → B Z p with the property that ˜ b ( x ) = b and that the discriminant ∆(˜ b ) ,seen as a map X → A Z p , is not identically zero on the special fibre and is square-free on the genericfibre of X .2. Write X rs for the open subscheme of X where ∆(˜ b ) does not vanish. Then there exists a morphism ˜ Q : X rs → P lifting the morphism X rs → B rs Z p satisfying ˜ Q ( x Q p ) = Q .Proof. We apply [Lag20, Proposition 4.22] to the compactified Prym variety P → B introduced in §3.5. In§5.1 we have spread out P to a scheme P → B S with similar properties. (Recall that S = Z [1 /N ] .) Define D to be the pullback of { ∆ = 0 } ⊂ B Z p along P Z p → B Z p . Since the latter morphism is proper, we may extend Q ∈ P b ( Q p ) ⊂ P b ( Q p ) to an element of P b ( Z p ) , still denoted by Q . We now claim that the triple ( P Z p , D , Q ) satisfies the assumptions of [Lag20, Proposition 4.22]. Indeed, the properties of P Z p follow from Proposition3.24. (Or rather the analogous properties obtained by spreading out in §5.1.) Moreover P F p is not containedin D since ∆ is nonzero mod p by our assumptions on N . Since P Z p is B Z p -flat, D is a Cartier divisor. Sincethe smooth locus of P Q p → B Q p has complement of codimension at least two and { ∆ = 0 } ⊂ B Q p is reduced,the scheme D Q p is reduced too. Finally Q Q p 6∈ D Q p since b has nonzero discriminant.We obtain a closed subscheme X ֒ → P Z p satisfying the conclusions of [Lag20, Proposition 4.22]. Write x ∈ X ( Z p ) for the section corresponding to Q , e b for the restriction of P Z p → B Z p to X and e Q for therestriction of the inclusion X ֒ → P Z p to X rs . Then the tuple ( X , x, e b, e Q ) satisfies the conclusion of theproposition.We now give the proof of Theorem 5.4 which is very closely modelled on [Lag20, §4.5]. We keep theassumptions and notation of Proposition 5.25 and assume that we have made a choice of ( X , x, e b, e Q ) satisfyingthe conclusions of that proposition. The strategy is to extend the orbit η b ( Q ) (which corresponds to thepoint x Q p ) to larger and larger subsets of X .Let y ∈ X be a closed point of the special fibre with nonzero discriminant having an affine open neighbourhoodcontaining x Q p . Let R be the semi-local ring of X at x Q p and y . Since every projective module of constantrank over R is free, we can apply Proposition 5.3 to obtain an orbit η e b ( e Q ) ∈ G ( R ) \ V e b ( R ) . This orbit spreadsout to an element of G ( U ) \ V e b ( U ) , where U ⊂ X is an open subset containing x Q p and intersecting thespecial fibre nontrivially. Under the bijection of Proposition 5.8, this defines an object A of GrLieE U , e b such that the pullback of A along the point x Q p ∈ U ( Q p ) corresponds to the orbit η b ( Q ) .Let U = X Q p . By Corollary 5.23, the restriction of A to U ∩ U extends to an object A of GrLieE U , e b . Wecan glue A and A to an object A of GrLieE U , e b , where U = U ∪ U . The complement of U has dimensionzero since X F p is irreducible. By Lemma 5.9, A extends to an object A of GrLieE X , e b . Let A ∈ GrLieE Z p ,b denote the pullback of A along the point x ∈ X ( Z p ) . Since H ( Z p , G ) is trivial by [Mil80, III.3.11(a)] andLang’s theorem, Propositions 5.6 and 5.8 implies that A determines an element of G ( Z p ) \ V b ( Z p ) mappingto η b ( Q ) in G ( Q p ) \ V b ( Q p ) . This completes the proof of Theorem 5.4.We conclude this subsection by stating a consequence for orbits of Z . We will need the following lemma,whose proof is identical to that of [RT, Proposition 5.7]. Write E := B ( Z ) ∩ B rs ( Q ) and E p := B ( Z p ) ∩ B rs ( Q p ) .46 emma 5.26. Let p be a prime (not necessarily coprime to N ) and let b ∈ E p . Then there exists an integer n ≥ and an open compact neighbourhood W p ⊂ E p of b such that for all b ∈ W p and for all y ∈ J p n · b ( Q p ) ,the orbit η p n · b ( y ) ∈ G ( Q p ) \ V p n · b ( Q p ) of Theorem 4.5 has a representative in V p n · b ( Z p ) . Corollary 5.27. Let b ∈ E . Then for each prime p dividing N we can find an open compact neighbourhood W p of b in E p and an integer n p ≥ with the following property. Let M = Q p | N p n p . Then for all b ∈ E ∩ (cid:16)Q p | N W p (cid:17) and for all y ∈ Sel ( P M · b ) , the orbit η M · b ( y ) ∈ G ( Q ) \ V M · b ( Q ) contains an element of V M · b ( Z ) .Proof. The group G has class number : G ( A ∞ ) = G ( Q ) · G ( b Z ) (Proposition 6.1). Therefore an orbit v ∈ G ( Q ) \ V ( Q ) has a representative in V ( Z ) if and only if for every prime p the associated G ( Q p ) -orbit hasa representative in V ( Z p ) . The corollary follows from combining Theorem 5.4 and Lemma 5.26. V ⋆ Proof of Theorem 5.5. Let A ∈ P b ( Q p ) be an element giving rise to a G ⋆ ( Q p ) -orbit η ⋆b ( A ) in V ⋆b ⋆ ( Q p ) . Byconstruction of η ⋆b we have a commutative diagram: P b ( Q p ) / P b ( Q p ) G ( Q p ) \ V b ( Q p ) P b ( Q p ) / ˆ ρ ( P ∨ b ( Q p )) G ⋆ ( Q p ) \ V ⋆b ⋆ ( Q p ) η b Q η ⋆b The diagram shows that the orbit η ⋆b ( A ) is the image of the orbit η b ( A ) under the map Q . By Theorem5.4, η b ( A ) has an integral representative v ∈ V ( Z p ) . Therefore, the element Q ( v ) ∈ V ⋆ ( Z p ) is an integralrepresentative of η ⋆b ( A ) .Again we state the following global corollary which follows from Lemma 5.26. Corollary 5.28. Let b ∈ E . Then for each prime p dividing N we can find an open compact neighbourhood W p of b in E p and an integer n p ≥ with the following property. Let M = Q p | N p n p . Then for all b ∈ E ∩ (cid:16)Q p | N W p (cid:17) with b ⋆ = Q ( b ) and for all y ∈ Sel ˆ ρ ( P ∨ M · b ) , the orbit η ⋆M · b ( y ) ∈ G ⋆ ( Q ) \ V ⋆M · b ⋆ ( Q ) containsan element of V ⋆M · b ⋆ ( Z ) .Proof. For each p dividing N , let W p ⊂ E p be an open compact neighbourhood of b and n p ≥ be an integersatisfying the conclusion of Lemma 5.26. Let M = Q p | N n p , let b ∈ E ∩ (cid:16)Q p | N W p (cid:17) and let y ∈ Sel ˆ ρ ( P ∨ M · b ) with corresponding orbit G ⋆ ( Q ) · v = η ⋆M · b ( y ) . The orbit G ⋆ ( Q p ) · v lies in the image of η ⋆M · b so by an argumentsimilar to the proof of Theorem 5.5, it is of the form G ⋆ ( Q p ) · Q ( w p ) for some w p ∈ V ( Q p ) that lies in theimage of η M · b . By Lemma 5.26 we may assume that w p ∈ V ( Z p ) . Therefore G ⋆ ( Q p ) · v has a representativein V ⋆ ( Z p ) for every prime p . Since G ⋆ = PGL has class number one, G ⋆ ( A ∞ ) = G ⋆ ( b Z ) G ⋆ ( Q ) so G ⋆ ( Q ) · v has a representative in V ⋆ ( Z ) . 47 Counting integral orbits in V In this section we will apply the counting techniques of Bhargava to provide estimates for the integral orbitsof bounded height in our representation ( G , V ) . Recall that B = Spec Z [ p , p , p , p ] and that π : V → B denotes the morphism of taking invariants. Forany b ∈ B ( R ) we define the height of b by the formulaht ( b ) := sup | p i ( b ) | /i . We define ht ( v ) = ht ( π ( v )) for any v ∈ V ( R ) . We have ht ( λ · b ) = | λ | ht ( b ) for all λ ∈ R and b ∈ B ( R ) . If A is a subset of V ( R ) or B ( R ) and X ∈ R > we write A Let dt, dn, dk be Haar measures on T ( R ) ◦ , N ( R ) , K respectively. Then the assignment f Z t ∈ T ( R ) ◦ Z n ∈ N ( R ) Z k ∈ K f ( tnk ) dk dn dt = Z t ∈ T ( R ) ◦ Z n ∈ N ( R ) Z k ∈ K f ( ntk ) δ G ( t ) − dk dn dt defines a Haar measure on G ( R ) . 48e now fix Haar measures on the groups T ( R ) ◦ , K and N ( R ) , as follows. We give T ( R ) ◦ the measure pulledback from the isomorphism Q β ∈ S G β : T ( R ) ◦ → R > , where R > gets its standard Haar measure d × λ = dλ/λ .We give K is probability Haar measure. Finally we give N ( R ) the unique Haar measure dn such that theHaar measure on G ( R ) from Lemma 6.2 coincides with dg .Next we introduce measures on V and B . Let ω V be a generator for the free rank one Z -module of left-invariant top differential forms on V . Then ω V is uniquely determined up to sign and it determines Haarmeasures dv on V ( R ) and V ( Q p ) for every prime p . We define the top form ω B = dp ∧ dp ∧ dp ∧ dp on B . It defines measures db on B ( R ) and B ( Q p ) for every prime p . Lemma 6.3. There exists a unique rational number W ∈ Q × with the following property. Let k/ Q be afield extension, let c a Cartan subalgebra of h k contained in V k , and let µ c : G k × c → V k be the natural actionmap. Then µ ∗ c ω V = W ω G ∧ π | ∗ c ω B .Proof. The proof is identical to that of [Tho15, Proposition 2.13]. Here we use the fact that the sum of theinvariants equals the dimension of the representation: Q V . Lemma 6.4. Let W ∈ Q × be the constant of Lemma 6.3. Then:1. Let V ( Z p ) rs = V ( Z p ) ∩ V rs ( Q p ) and define a function m p : V ( Z p ) rs → R ≥ by the formula m p ( v ) = X v ′ ∈ G ( Z p ) \ ( G ( Q p ) · v ∩ V ( Z p )) Z G ( v )( Q p ) Z G ( v )( Z p ) . (6.2.1) Then m p ( v ) is locally constant.2. Let B ( Z p ) rs = B ( Z p ) ∩ B rs ( Q p ) and let ψ p : V ( Z p ) rs → R ≥ be a bounded, locally constant functionwhich satisfies ψ p ( v ) = ψ p ( v ′ ) when v, v ′ ∈ V ( Z p ) rs are conjugate under the action of G ( Q p ) . Then wehave the formula Z v ∈ V ( Z p ) rs ψ p ( v )d v = | W | p vol ( G ( Z p )) Z b ∈ B ( Z p ) rs X g ∈ G ( Q p ) \ V b ( Z p ) m p ( v ) ψ p ( v ) Z G ( v )( Q p ) d b. (6.2.2) Proof. The proof is identical to that of [RT18, Proposition 3.3], using Lemma 6.3. Let K ⊂ G ( R ) be the maximal compact subgroup fixed in §6.2. For any c ∈ R > , define T c := { t ∈ T ( R ) ◦ |∀ β ∈ S G , β ( t ) ≤ c } . A Siegel set is, by definition, any subset S ω,c := ω · T c · K , where ω ⊂ N ( R ) is a compactsubset and c > . Proposition 6.5. 1. For every ω ⊂ N ( R ) and c > , the set { γ ∈ G ( Z ) | γ · S ω,c ∩ S ω,c = ∅} is finite.2. We can choose ω ⊂ N ( R ) and c > such that G ( Z ) · S ω,c = G ( R ) . roof. The first part follows from the Siegel property [Bor69, Corollaire 15.3]. By [PR94, Theorem 4.15], thesecond part is reduced to proving that G ( Q ) = P ( Q ) · G ( Z ) . This follows from [Bor66, §6, Lemma 1(b)], usingthat (in the terminology of that paper) the lattice V is special with respect to the pinning ( T , P , { X α } ) .Now fix ω ⊂ N ( R ) and c > so that S ω,c satisfies the conclusions of Proposition 6.5. By enlarging ω , wemay assume that S ω,c is semialgebraic. We drop the subscripts and for the remainder of §6 we write S forthis fixed Siegel set. The set S will serve as a fundamental domain for the action of G ( Z ) on G ( R ) .A G ( Z ) -coset of G ( R ) may be represented more than once in S , but by keeping track of the multiplicities thiswill not cause any problems. The surjective map ϕ : S → G ( Z ) \ G ( R ) has finite fibres and if g ∈ S we define µ ( g ) := ϕ − ( ϕ ( g )) . The function µ : S → N is uniformly bounded by µ max := { γ ∈ G ( Z ) | γ S ∩ S = ∅} and has semialgebraic fibres. By pushing forward measures via ϕ , we obtain the formula Z g ∈ S µ ( g ) − dg = vol ( G ( Z ) \ G ( R )) . (6.3.1)We now construct special subsets of V rs ( R ) which serve as our fundamental domains for the action of G ( R ) on V rs ( R ) . By the same reasoning as in [Tho15, §2.9], we can find open subsets L , . . . , L k of { b ∈ B rs ( R ) | ht ( b ) = 1 } and sections s i : L i → V ( R ) of the map π : V → B satisfying the following properties:• For each i , L i is connected and semialgebraic and s i is a semialgebraic map with bounded image.• Set Λ = R > . Then we have an equality V rs ( R ) = k [ i =1 G ( R ) · Λ · s i ( L i ) . (6.3.2)If v ∈ s i ( L i ) let r i = Z G ( v )( R ) ; this integer is independent of the choice of v . We record the followingchange-of-measure formula, which follows from Lemma 6.3. Lemma 6.6. Let φ : V ( R ) → C be a continuous function of compact support and i ∈ { , . . . , k } . Let G ⊂ G ( R ) be a measurable subset and let m ∞ ( v ) be the cardinality of the fibre of the map G × Λ × L i → V ( R ) , ( g, λ, l ) g · λ · s ( l ) above v ∈ V ( R ) . Then Z v ∈ G · Λ · s i ( L i ) f ( v ) m ∞ ( v ) dv = | W | Z b ∈ Λ · L i Z g ∈ G f ( g · s i ( b )) dg db, where W ∈ Q × is the scalar of Lemma 6.3. V For any G ( Z ) -invariant subset A ⊂ V ( Z ) , define N ( A, X ) := X v ∈ G ( Z ) \ A We have N ( V ( Z ) sirr ∩ V ( R ) sol , X ) = | W | G ( Z ) \ G ( R )) vol ( B ( R ) In the above notation, let ( L, s, r ) be ( L I , s I , r I ) for some I ⊂ { , . . . , k } . Then N ( G ( R ) · Λ · s ( L ) ∩ V ( Z ) sirr , X ) = | W | r vol ( G ( Z ) \ G ( R )) vol((Λ · L ) Given X ≥ , n ∈ N ( R ) , t ∈ T ( R ) and λ ∈ Λ , define B ( n, t, λ, X ) := ( ntλG · s ( L )) There exists δ > such that N ( S ( α ) sirr , X ) = O ( X − δ ) . Having dealt with the cuspidal region, we may now count lattice points in the main body using the followingproposition [BW14, Theorem 1.3], which strengthens a well-known result of Davenport [Dav51].53 roposition 6.11. Let m, n ≥ be integers, and let Z ⊂ R m + n be a semialgebraic subset. For T ∈ R m ,let Z T = { x ∈ R n | ( T, x ) ∈ Z } , and suppose that all such subsets Z T are bounded. Then for any unipotentupper-triangular matrix u ∈ GL n ( R ) , we have Z T ∩ u Z n ) = vol( Z T ) + O (max { , vol( Z T,j } ) , where Z T,j runs over all orthogonal projections of Z T to any j -dimensional coordinate hyperplane (1 ≤ j ≤ n − . Moreover, the implied constant depends only on Z . Proposition 6.12. Let A = V ( Z ) ∩ ( G ( R ) · Λ · s ( L )) . Then N ( A \ S ( α ) , X ) = | W | r vol ( G ( Z ) \ G ( R )) vol((Λ · L ) Choose a generator for the weight space V α for each α ∈ Φ V and let k·k denote the supremum normof V ( R ) with respect to this choice of basis. Since the set ω · G · s ( L ) is bounded, we can choose a constant J > such that k v k ≤ J for all v ∈ ω · G · s ( L ) . Let F ( n, t, λ, X ) = { v ∈ B ( n, t, λ, X ) | v α = 0 } . If F ( n, t, λ, X ) ∩ V ( Z ) = ∅ , there exists an element v ∈ B ( n, t, λ, X ) such that k v α k ≥ , hence λα ( t ) ≥ /J .We estimate A \ S ( α )) ∩ B ( n, t, λ, X )] = V ( Z ) ∩ F ( n, t, λ, X )] using Proposition 6.11. An element v ∈ F ( n, t, λ, X ) has weight { h ∈ G | v ∈ ntλh · s ( L )) } , and F ( n, t, λ, X ) is partitioned into finitely manybounded semialgebraic subsets of constant weight. Moreover we have an equality of (weighted) volumes vol( F ( n, t, λ, X )) = vol( B ( n, t, λ, X )) . If M ⊂ Φ V then the volume of the projection of F ( n, t, λ, X ) to V ( M )( R ) is bounded above by O (cid:16) λ − M Q α ∈ Φ V \ M α ( t ) (cid:17) . Since Q α ∈ Φ V α ( t ) = 1 and α is maximal, weconclude by Proposition 6.11 that the number of weighted elements of [( A \ S ( α )) ∩ B ( n, t, λ, X )] is givenby: ( if λα ( t ) < /J, vol( B ( n, t, λ, X )) + O ( λ α ( t ) − ) otherwise.By Lemma 6.9, we obtain N ( A \ S ( α ) , X ) = 1 r Z Xλ = K − Z t ∈ T c ,α ( t ) ≥ /λJ Z n ∈ ω (cid:0) vol( B ( n, t, λ, X )) + O ( λ α ( t ) − ) (cid:1) µ ( nt ) − δ G ( t ) − dn dt d × λ. The integral of the second summand is easily seen to be o ( X ) . On the other hand, the integral of the firstsummand is r Z g ∈ S vol (cid:0) ( g · Λ · G · s ( L )) The following proposition is proven in §6.9. Proposition 6.13. Let V alred denote the subset of almost Q -reducible elements v ∈ V ( Z ) with v S ( α ) .Then N ( V alred , X ) = o ( X ) . 54e now finish the proof of Proposition 6.8. Again let A = V ( Z ) ∩ ( G ( R ) · Λ · s ( L )) . Then N ( A sirr , X ) = N ( A sirr \ S ( α ) , X ) + N ( S ( α ) sirr , X ) The second term on the right-hand-side is o ( X ) by Proposition 6.10, and N ( A sirr \ S ( α ) , X ) = N ( A \ S ( α ) , X ) + o ( X ) by Proposition 6.13. Using Proposition 6.12, we obtain N ( A sirr , X ) = | W | r vol ( G ( Z ) \ G ( R )) vol((Λ · L ) We now introduce a weighted version of Theorem 6.7. If w : V ( Z ) → R is a function and A ⊂ V ( Z ) is a G ( Z ) -invariant subset we define N w ( A, X ) := X v ∈ G ( Z ) \ A ht ( v ) Let w : V ( Z ) → R be defined by finitely many congruence conditions. Then N w ( V ( Z ) sirr ∩ V ( R ) sol , X ) = µ w | W | G ( Z ) \ G ( R )) vol ( B ( R ) Let w : V ( Z ) → [0 , be an acceptable function. Then N w ( V ( Z ) irr ∩ V sol ( R ) , X ) ≤ | W | Y p Z V ( Z p ) w p ( v )d v ! vol ( G ( Z ) \ G ( R )) vol ( B ( R ) This inequality follows from Theorem 6.14; the proof is identical to the first part of the proof of[BS15a, Theorem 2.21]. 55 .9 Estimates on reducibility and stabilizers In this subsection we give the proof of Proposition 6.13 and the following proposition which will be usefulin §8. Proposition 6.16. Let V bigstab denote the subset of strongly Q -irreducible elements v ∈ V ( Z ) with Z G ( v )( Q ) > . Then N ( V bigstab , X ) = o ( X ) . By the same reasoning as [BG13, §10.7] it will suffice to prove Lemma 6.17 below, after having introducedsome notation.Let N be the integer of §5.1 and let p be a prime not dividing N . We define V alredp ⊂ V ( Z p ) to be the set ofvectors whose reduction mod p is almost F p -reducible. We define V bigstabp ⊂ V ( Z p ) to be the set of vectors v ∈ V ( Z p ) such that p | ∆( v ) or the image ¯ v of v in V ( F p ) has nontrivial stabilizer in G ( F p ) . Lemma 6.17. We have lim Y → + ∞ Y N
The proof is very similar to the proof of [Lag20, Lemma 5.7] which is in turn based on the proof of[RT, Proposition 6.9]. We first treat the case of V alredp . We have the formula Z V alredp dv = 1 V ( F p ) { v ∈ V ( F p ) | v is almost F p -reducible } . (6.9.1)Since V ( F p ) = V rs ( F p ) + O ( p ) , it suffices to prove that there exists a nonnegative δ < with theproperty that V rs ( F p ) { v ∈ V rs ( F p ) | v is almost F p -reducible } < δ for all p large enough. If b ∈ B rs ( F p ) , Proposition 5.2 and the triviality of H ( F p , G ) (Lang’s theorem)show that V b ( F p ) is partitioned into ( F p , P b [2]) many orbits, each of size G ( F p ) / P b [2]( F p ) . Since P b [2]( F p ) = P b ( F p ) / P b ( F p ) = ( F p , P b [2]) , we have V rs ( F p ) = G ( F p ) B rs ( F p ) . Moreover byCorollary 4.13 (or rather a similar statement for Z [1 /N ] -algebras, which continues to hold by the sameproof), an orbit corresponding to an element of H ( F p , P b [2]) is almost F p -reducible if and only if its imagein H ( F p , ˆ E b [2]) is trivial. Therefore the left-hand-side of (6.9.1) equals B rs ( F p ) X b ∈ B rs ( F p ) (cid:16) H ( F p , P b [2]) → H ( F p , ˆ E b [2]) (cid:17) P b [2]( F p ) . (6.9.2)We have (cid:16) H ( F p , P b [2]) → H ( F p , ˆ E b [2]) (cid:17) ≤ ( F p , E b [2]) by Corollary 3.15. Since ( F p , E b [2]) = E b [2] , the quantity (6.9.2) is bounded above by B rs ( F p ) X b ∈ B rs ( F p ) E b [2]( F p ) P b [2]( F p ) . (6.9.3)56ach summand in (6.9.3) is the inverse of an integer; let η p be the proportion of b ∈ B rs ( F p ) where thissummand equals . Then the quantity (6.9.3) is ≤ η p + (1 − η p ) / / η p / . So it suffices to prove that η p → η for some η < . In the notation of Proposition 3.4, let C ⊂ W ζ E be the subset of elements such that ζ )Λ / w / ζ ) w = 1 . Then [Ser12, Proposition 9.15] applied to the W ζ E -torsor t rs → B rs from Proposition 3.4 implies that B rs ( F p ) (cid:26) b ∈ B rs ( F p ) | E b [2]( F p ) P b [2]( F p ) = 1 (cid:27) = C W ζ E + O ( p − / ) . Since C , this implies that η < .Next we briefly treat the case of V bigstabp , referring to [Lag20, Lemma 5.7] for more details. By a similarargument to the one above, it suffices to find an element w ∈ W ζ E with (cid:0) (Λ / ζ (cid:1) w = 0 . This can be achievedby taking a Coxeter element of W E fixed by ζ : the end of the proof of [Lag20, Lemma 5.7] shows that such anelement has no nonzero fixed vector on Λ / hence the same is true for its restriction to the ζ -fixed points.An example of such a Coxeter element is w w w w w w , using Bourbaki notation [Bou68, Planche V] forlabelling the simple roots of E .We explain why Lemma 6.17 implies Propositions 6.13 and 6.16. We first claim that if v ∈ V ( Z ) with b = π ( v ) is almost Q -reducible, then for each prime p not dividing N the reduction of v in V ( F p ) is almost F p -reducible. Indeed, either ∆( b ) = 0 in F p (in which case v is almost F p -reducible), or p ∤ ∆( b ) and Q ( v ) is G ⋆ ( Q ) -conjugate to Q ( σ ( b )) . In the latter case Proposition 5.24 implies that Q ( v ) is G ⋆ ( Z p ) -conjugate to Q ( σ ( b )) , so their reductions are G ⋆ ( F p ) -conjugate, proving the claim. By a congruence version of Proposition6.12, for every subset L ⊂ B ( R ) considered in Proposition 6.8 and for every Y > we obtain the estimate: N ( V alred ∩ G ( R ) · Λ · s ( L ) , X ) ≤ C Y N is a constant independent of Y . By Lemma 6.17, the product of the integrals converges to zeroas Y tends to infinity, so N ( V alred ∩ G ( R ) · Λ · s ( L ) , X ) = o ( X ) . Since this holds for every such subset L ,we obtain Proposition 6.13.Note that we have not used Theorem 6.7 in this argument, but we may use it now to prove Proposition6.16. Again the reduction of an element of V bigstab modulo p lands in V bigstabp if p does not divide N , byProposition 5.22. Since lim X → + ∞ N ( V bigstab , X ) /X is O ( Q N
Let M ∈ C be a subset such that V ( M )( Q ) contains strongly Q -irreducible elements.Then there exists a subset M ⊂ Φ V \ M and a function f : M → R ≥ satisfying the following conditions: • We have P α ∈ M p ( α ) < M . • For each i = 1 , . . . , we have P α ∈ Φ + G n i ( α ) − P α ∈ M n i ( α ) + P α ∈ M p ( α ) n i ( α ) > . The proof of Proposition 6.18 will be given after some useful lemmas. We will use the notation of Table 3to label the elements of Φ V . Lemma 6.19. Let M ∈ C and suppose that V ( M )( Q ) sirr = ∅ . Then M ⊂ { , , , , , , , , , , } and { , } 6⊂ M .Proof. Let M be such a subset. Suppose that ∈ M . Since M ∈ C we have { , , , , , , , } ⊂ M .By Lemma 2.15, this implies that V ( M )( Q ) sirr = ∅ , contradiction. The same argument involving the otherthree subsets of Lemma 2.15 shows that M , { , } 6⊂ M and M . Therefore M is containedin the subset of α ∈ Φ V with the property that α , α and α , which is easily checked to be { , , , , , , , , , , } .For the reader’s convenience we give the Hasse diagram of the subset { , , , , , , , , , , } with respectto the partial ordering on Φ V . 12 43 75 6 89 10 13 We say a subset M ∈ C is good if there exists a subset M ⊂ Φ V \ M and a function f : M → R ≥ satisfyingthe conclusions of Proposition 6.18. The following lemma is a slight generalization of [RT, Lemma 6.6]; itsproof is identical. 58 ′ M ′ Weights f ′ : M ′ → R ≥ , , , , , 10 4 , 12 2 , , , − , , , , , , , , 14 4 , , , − , , )1 , , , , , , , , , 13 9 , , − , − , , , , )1 , , , , , , , , , 13 10 , , − , − , − , , , )1 , , , , , , , , , , , − , − , − , − , , , ) Table 5: Examples of good M ′ Lemma 6.20. Suppose that M ′ , M ′′ ∈ C with M ′′ ⊂ M ′ , that M ′ ⊂ Φ V \ M ′ , and that there exists a function f ′ : M ′ → R ≥ satisfying the conditions of Proposition 6.18. If there exists a function g : ( M ′ \ M ′′ ) → M ′ such that1. α ≥ g ( α ) for all α ∈ M ′ \ M ′′ ,2. f ′ ( α ) − g − ( α ) ≥ for all α ∈ M ′ ,then any M ∈ C such that M ′′ ⊂ M ⊂ M ′ is good.Proof. Given such a subset M , define f : M ′ → R by f ( α ) = f ′ ( α ) − g − ( α ) ∩ ( M ′ \ M )] . The secondcondition on g implies that f takes values in R ≥ . We have X α ∈ M ′ f ( α ) = X α ∈ M ′ f ′ ( α ) − M ′ \ M ] < M . Moreover X α ∈ Φ + G \ M α + X α ∈ M ′ f ( α ) α = X α ∈ Φ + G \ M ′ α + X α ∈ M ′ f ′ ( α ) α + X α ∈ M ′ \ M ( α − g ( α )) . The first term on the right-hand-side has positive coordinates with respect to S G by assumption on f ′ andthe second term has nonnegative coordinates by the first condition on g .Table 5 gives examples of M ′ ∈ C together with a subset M ′ ⊂ Φ V \ M ′ and a function f ′ : M ′ → R ≥ satisfying the conclusions of Proposition 6.18. The third column denotes the coordinates of P α ∈ Φ V \ M ′ α with respect to the basis S G . The validity this table can be easily checked in conjunction with Table 3. Forexample, checking the second row amounts to checking that the nonnegative reals ( v , v , v ) = (0 , , ) have the property that v + v + v < and that the vector (2 v + 2 v − v + 4 , v + 8 , v − v + v + 4 , − v + v + v − has strictly positive entries. Proof of Proposition 6.18. Let M ∈ C be such a subset. By Lemma 6.19, M ⊂ { , , , , , , , , , , } and { , } 6⊂ M . We prove that M is good by considering various cases together with the information ofTable 5. If M ≤ then M = { } , { , } or { , } so taking M = 3 and f (3) = 1 / shows that M isgood. We may assume for the remainder of the proof that M ≥ , which implies that { , } ⊂ M .Case 1: M and M . Then { , } ⊂ M ⊂ { , , , , , } . We apply Lemma 6.20 with ( M ′ , M ′ , f ′ ) given by the first row of Table 5, M ′′ = { , } and g : ( M ′ \ M ′′ ) → M ′ given by , , , .59ase 2: M and ∈ M . Then M by Lemma 6.19, hence M = { , , , , , } . The second rowof Table 5 shows that M is good.Case 3: ∈ M and M . Then { , , } ⊂ M ⊂ { , , , , , , , , , } . We apply Lemma 6.20with ( M ′ , M ′ , f ′ ) given by the third row of Table 5, M ′′ = { , , } and g : ( M ′ \ M ′′ ) → M ′ given by , , 9; 7 11; 8 , , .Case 4: ∈ M and ∈ M . Lemma 6.19 then implies that M . If ∈ M , then M = { , , , , , , , , , } , which is good by the fourth row of Table 5. If M , then { , , , , , , } ⊂ M ⊂ { , , , , , , , , } . We apply Lemma 6.20 with ( M ′ , M ′ , f ′ ) given by the fifth row of Table 5, M ′′ = { , , , , , , } and g : ( M ′ \ M ′′ ) → M ′ given by ; . V ⋆ In this section we count integral orbits in the representation V ⋆ . Since V ⋆ is essentially the space of binaryquartic forms, the methods here will be very similar to the ones employed by Bhargava and Shankar [BS15a]. The following lemma is presumably well-known; it generalizes the observation that E ( Q p ) /nE ( Q p )) = | n | − p E ( Q p )[ n ] if E/ Q p is an elliptic curve. It follows from local duality theorems. Lemma 7.1. Let k = R or Q p for some p and let K be a finite extension of k . Write | · | k : k × → R > forthe normalized absolute value of k . Let A be an abelian variety over K with dual A ∨ . Let λ : A → A ∨ be aself-dual isogeny. Then the degree of λ is a square number m for some m ∈ Z ≥ . Consider the quantity c ( λ ) := A ∨ ( K ) /λ ( A ( K ))) A [ λ ]( K ) . Then c ( λ ) = | m | − [ K : k ] k .Proof. The selfduality of λ implies that there is a perfect alternating pairing A [ λ ] × A [ λ ] → G m,K , which implies that the degree of λ is a square number m for some m ∈ Z ≥ . This pairing induces a pairingon Galois cohomology H ( K, A [ λ ]) × H ( K, A [ λ ]) → H ( K, G m ) ֒ → Q / Z which is also perfect and alternating and the image of the descent map A ∨ ( K ) /λ ( A ( K )) → H ( K, A [ λ ]) is amaximal isotropic subspace [PR12, Proposition 4.10]. This implies that (cid:18) A ∨ ( K ) λ ( A ( K )) (cid:19) = ( K, A [ λ ]) . (7.1.1)By the local Euler characteristic formula [Mil86, Theorems I.2.8 and I.2.13], we obtain the equality ( K, A [ λ ]) = ( K, A [ λ ]) ( K, A [ λ ]) | m | − K : Q p ] k . (7.1.2)We have H ( K, A [ λ ]) = A [ λ ]( K ) and local Tate duality implies that H ( K, A [ λ ]) ≃ H ( K, A [ λ ]) ∨ ≃ A [ λ ]( K ) ∨ ≃ A [ λ ]( K ) too. The lemma follows from combining Equations (7.1.1) and (7.1.2).60 orollary 7.2. Let k = R or Q p for some p . If b ∈ B rs ( k ) , then in the notation of Lemma 7.1 the quantities c ( ρ : P b → P ∨ b ) and c (ˆ ρ : P ∨ b → P b ) coincide and equal / if k = R , if k = Q , else . Proof. We only treat the case of c ( ρ ) , the case of c (ˆ ρ ) being analogous. The isogeny ρ is self-dual by Lemma3.7 and its kernel is isomorphic to E b [2] , which has order . Apply Lemma 7.1. We discuss objects and notions analogous to those of §6.1 and §6.2 in the context of the representation V ⋆ .Recall from §4.2 that we have defined the isomorphism Q : B → B ⋆ which was spread out to an isomorphism B S → B ⋆S in §5.1. We will use Q to transport definitions of B S to B ⋆S . For example, we set ∆ ⋆ := ∆ ◦ Q − , ∆ ⋆E := ∆ E ◦ Q − , ∆ ⋆ ˆ E := ∆ ˆ E ◦ Q − , all elements of S [ B ⋆ ] . Furthermore we define B ⋆, rs := B ⋆S [(∆ ⋆ ) − ] , andwe define V ⋆, rs as the preimage of B ⋆, rs under π ⋆ : V ⋆ → B ⋆ . The S -schemes B ⋆, rs and V ⋆, rs have generic fibre B ⋆, rs and V ⋆, rs .For any b ∈ B ⋆ ( R ) we define the height of b by the formulaht ( b ) = ht ( Q − ( b )) , where the height on B ( R ) is defined in §6.1. We define ht ( v ) = ht ( π ⋆ ( v )) for any v ∈ V ⋆ ( R ) . If A is a subsetof V ⋆ ( R ) or B ⋆ ( R ) and X ∈ R > we write A There exists a constant W ∈ Q × with the following properties:1. Let V ⋆ ( Z p ) rs = V ⋆ ( Z p ) ∩ V ⋆, rs ( Q p ) and define a function m p : V ⋆ ( Z p ) rs → R ≥ by the formula m p ( v ) = X v ′ ∈ G ⋆ ( Z p ) \ ( G ⋆ ( Q p ) · v ∩ V ⋆ ( Z p )) Z G ⋆ ( v )( Q p ) Z G ⋆ ( v )( Z p ) . (7.2.1) Then m p ( v ) is locally constant.2. Let B ⋆ ( Z p ) rs = B ⋆ ( Z p ) ∩ B ⋆, rs ( Q p ) and let ψ p : V ⋆ ( Z p ) rs → R ≥ be a bounded, locally constant functionwhich satisfies ψ p ( v ) = ψ p ( v ′ ) when v, v ′ ∈ V ⋆ ( Z p ) rs are conjugate under the action of G ⋆ ( Q p ) . Thenwe have the formula Z v ∈ V ⋆ ( Z p ) rs ψ p ( v )d v = | W | p vol ( G ⋆ ( Z p )) Z b ∈ B ⋆ ( Z p ) rs X g ∈ G ⋆ ( Q p ) \ V ⋆b ( Z p ) m p ( v ) ψ p ( v ) Z G ⋆ ( v )( Q p ) d b. (7.2.2)61 roof. The proof is the same as that of [RT18, Proposition 3.3], using the fact that the sum of the weightsof the G m -action on V ⋆ equals , the sum of the invariants of B ⋆ .We henceforth fix a constant W ∈ Q × satisfying the properties of Lemma 7.3.We now construct special subsets of V ⋆, rs ( R ) which serve as our fundamental domains for the action of G ⋆ ( R ) on V ⋆, rs ( R ) . We have to slightly alter the fundamental sets used in [BS15a] because the discriminant we usehere to define B ⋆, rs ( R ) is larger than the discriminant used by Bhargava and Shankar.In the notation of [BS15a, §2.1], if i = 0 , , , − , let V ⋆ ( R ) ( i ) ⊂ V ⋆ ( R ) be the subset of elements ( b , b , q ) where the binary quartic form q has − i real roots, and is positive/negative definite if i = 2+ / − respectively. For each such i , an open semialgebraic subset K ( i ) ⊂ { b ∈ B ⋆ ( R ) | b = b = 0 and ht ( b ) = 1 } and a section s ( i ) : K ( i ) → V ⋆ ( R ) ( i ) of π ⋆ is given in [BS15a, Table 1], with the property that (if Λ = R > ): { v ∈ V ⋆ ( R ) | b = b = 0 and ∆ ⋆ ˆ E ( π ⋆ ( v )) = 0 } = G i =0 , , ± G ⋆ ( R ) · Λ · s ( i ) ( L ( i ) ) . Let M ( i ) ⊂ V ⋆ ( R ) be the subset of elements v = ( b , b , q ) satisfying ht ( v ) = 1 and q ∈ Λ · K ( i ) . Let L , . . . , L k be the connected components of π ⋆ ( M ( i ) ) ∩ B ⋆, rs ( R ) for all i = 0 , , ± ; they are also the connectedcomponents of { b ∈ B ⋆, rs ( R ) | ht ( b ) = 1 } .By construction, the open subsets L , . . . , L k of { b ∈ B ⋆, rs ( R ) | ht ( b ) = 1 } come equipped with sections s i : L i → V ⋆, rs ( R ) of π ⋆ : V ⋆ ( R ) → B ⋆ ( R ) and satisfy the following properties, which follow from the corre-sponding properties of K ( i ) :• For each i , L i is connected and semialgebraic and s i is a semialgebraic map with bounded image.• Set Λ = R > . Then we have an equality V ⋆, rs ( R ) = k G i =1 G ⋆ ( R ) · Λ · s i ( L i ) . (7.2.3)If v ∈ s i ( L i ) let r i = Z G ⋆ ( v )( R ) ; this integer is independent of the choice of v .We record the following change-of-measure formula, whose proof is identical to the proof of [RT18, Lemma3.7]. Lemma 7.4. Let φ : V ⋆ ( R ) → C be an almost everywhere continuous function of compact support and i ∈ { , . . . , r } . Let G ⊂ G ⋆ ( R ) be a measurable subset such that the fibre of the map G × Λ × L i → V ⋆ ( R ) , ( g, λ, l ) g · λ · s ( l ) has cardinality m ∞ ( v ) for v ∈ V ⋆ ( R ) . Then Z v ∈ G · Λ · s i ( L i ) f ( v ) m ∞ ( v ) dv = | W | Z b ∈ Λ · L i Z g ∈ G f ( g · s i ( b )) dg db, where W ∈ Q × is the scalar of Lemma 7.3. V ⋆ For any G ⋆ ( Z ) -invariant subset A ⊂ V ⋆ ( Z ) and function w : V ⋆ ( Z ) → R , define N w ( A, X ) := X v ∈ G ⋆ ( Z ) \ A Let w : V ⋆ ( Z ) → R be a function defined by finitely many congruence conditions. Then N w ( V ⋆ ( Z ) irr ∩ V ⋆ ( R ) sol , X ) = µ w | W | G ⋆ ( Z ) \ G ⋆ ( R )) vol( B ( R ) For every b ∈ B ⋆, rs ( R ) and v ∈ V ⋆b ( R ) we have equalities (cid:0) G ⋆ ( R ) \ V ⋆b ( R ) sol (cid:1) / Z G ⋆ ( v )( R ) = P b ( R ) / ˆ ρ ( P ∨ b ( R ))) / P ∨ b [ˆ ρ ]( R ) = 1 / , where the first follows from the definition of R -solubility and Lemma 4.9, and the second from Corollary 7.2.By an argument identical to that of [Lag20, Lemma 5.5], the subset V ⋆ ( R ) sol is open and closed in V ⋆, rs ( R ) .Using the decomposition (7.2.3) and discarding those sections which do not contain R -soluble elements, itsuffices to prove that for each L i we have N w ( G ⋆ ( R ) · Λ · s i ( L i ) ∩ V ⋆ ( Z ) irr , X ) = µ w | W | r i vol ( G ⋆ ( Z ) \ G ⋆ ( R )) vol((Λ · L i ) For a prime p not dividing N , let W p ( V ⋆ ) denote the subset of v ∈ V ⋆ ( Z ) irr such that p | ∆ ⋆ ˆ E ( v ) . For any M > N we have lim X →∞ N ( ∪ p>M W p ( V ⋆ ) , X ) /X = O (1 / log M ) (7.3.1) where the implied constant is independent of M . p a G ⋆ ( Z p ) -invariant function w p : V ⋆ ( Z p ) → [0 , with the followingproperties:• The function w p is locally constant outside a closed subset of V ⋆ ( Z p ) of measure zero.• For p sufficiently large and not dividing N , we have w p ( v ) = 1 for all v ∈ V ⋆ ( Z p ) such that p ∤ ∆ ⋆ ˆ E ( v ) and ∆ ⋆ ( v ) = 0 .In this case we can define a function w : V ⋆ ( Z ) → [0 , by the formula w ( v ) = Q p w p ( v ) if ∆ ⋆ ( v ) = 0 and w ( v ) = 0 otherwise. Call a function w : V ⋆ ( Z ) → [0 , defined by this procedure acceptable . Theorem 7.7. Let w : V ⋆ ( Z ) → [0 , be an acceptable function. Then N w ( V ⋆ ( Z ) irr ∩ V ⋆ ( R ) sol , X ) = | W | ∞ Y p Z V ⋆ ( Z p ) w p ( v )d v ! vol ( G ⋆ ( Z ) \ G ⋆ ( R )) vol ( B ( R ) Our definition of an acceptable function slightly differs from the one the one employed in [BS15a, §2.7],since we only require that for sufficiently large primes p , w p ( v ) = 1 if p ∤ ∆ ⋆ ˆ E ( v ) and ∆ ⋆ ( v ) = 0 . Let S ⊂ V ⋆ ( Z ) be the subset of b with ∆ ⋆ ( b ) = 0 . Bearing in mind that N ( S, X ) = o ( X ) and the closure of S in V ⋆ ( Z p ) is of measure zero, the proof of the theorem is identical to that of [BS15a, Theorem 2.21], usingProposition 7.6. Remark 7.8. We obtain an equality in Theorem 7.7 whereas in Theorem 6.15 we only obtain an upperbound. This is because the proof of Theorem 7.7 relies on the uniformity estimate for ∆ ∗ ˆ E of Proposition 7.6.However, a uniformity estimate for integral orbits in V with respect to the discriminant ∆ is not known tohold. In this section we combine all the previous results and prove the main theorems of the introduction. Tocalculate the average size of the ρ -Selmer group in §8.2, we reduce it to calculating the average size of the ˆ ρ -Selmer group using the bigonal construction (Theorem 3.14). To make this reduction step precise, weconsider the effect of changing the parameter space by an automorphism in §8.1. This section is based on a remark of Poonen and Stoll [PS14, Remark 8.11]. Let n ≥ be an integer and B = A n Z be affine n -space with coordinates x , . . . , x n . Suppose that B is equipped with a G m -action suchthat λ · x i = λ d i x i for some set of positive weights d ≤ · · · ≤ d n ; let d be their sum. Definition 8.1. Let T be a subset of B ( R ) × Q p B ( Z p ) of the form T ∞ × Q p T p . We say T is a generalizedbox if • T ∞ ⊂ B ( R ) is open, bounded and semialgebraic. • For each prime number p , T p ⊂ B ( Z p ) is open and compact and for all but finitely many p we have T p = B ( Z p ) . f in addition T ∞ is a product of intervals ( a , b ) × · · · × ( a n , b n ) , we say T is a box . Let T be a generalized box. We define E T, Let φ : B → B be a G m -equivariant morphism such that φ Q : B Q → B Q is an isomorphism.Let T be a generalized box of B . Then φ ( T ) is a generalized box, φ ( E T, Let f, C and φ be as above and • ∈ {∅ , ≤} . Suppose that Eq • ( f, C ) holds for all boxes of B . Then Eq • ( f ◦ φ, C ) holds for all generalized boxes of B .Proof. We only consider the case of Eq ≤ ( f, C ) , the case of Eq( f, C ) being analogous. By approximating theinfinite component using rectangles, Eq ≤ ( f, C ) holds for all generalized boxes of B . If T is a generalized boxthen by Lemma 8.2, φ ( T ) is a generalized box with the same volume as T and φ ( E T, Most orbit-counting results using the geometry-of-numbers methods as employed in §6 arevalid for any generalized box, with the same proof. Proposition 8.3 shows that for these counting results, thechoice of homogeneous coordinates of B is irrelevant. For example, consider the family of elliptic curves y + p xy + p y = x + p x + p . (8.1.2)65 fter applying a homogeneous change of coordinates we obtain the family ( y + p x + p ) = x + p x + p . (8.1.3) The results of [BS15a, BS15b, BS13a, BS13b] are valid for any box of A p ,p ) hence trivially for any box of A p ,p ,p ,p ) parametrizing elliptic curves in Family (8.1.3). Proposition 8.3 shows that these results remainvalid for any generalized box for the elliptic curves in Family (8.1.2) too. ρ -Selmer group Recall that E ⊂ B ( Z ) denotes the subset of elements b with ∆( b ) = 0 . We say a subset F ⊂ E is defined byfinitely many congruence conditions if it is the preimage of a subset of B ( Z /M Z ) under the mod M reductionmap E → B ( Z /M Z ) . Theorem 8.5. Let F ⊂ E be a subset defined by finitely many congruence conditions. Then lim X →∞ F Let b ∈ F . Then we can find for each prime p dividing N an open compact neighbourhood W p of b in E p with the following property. Let F W = F ∩ (cid:16)Q p | N W p (cid:17) . Then we have lim X →∞ P b ∈F W , ht ( b ) Choose sets W p and integers n p ≥ for p | N satisfying the conclusion of Corollary 5.28. We assumeafter shrinking the W p that they satisfy W p ⊂ F p . If p does not divide N , set W p = F p and n p = 0 . Let M = Q p p n p .For v ∈ V ⋆ ( Z ) with b ⋆ = π ⋆ ( v ) and Q − ( b ⋆ ) = b ∈ B ( Q ) , define w ( v ) ∈ Q ≥ by the following formula: w ( v ) = (cid:16)P v ′ ∈ G ⋆ ( Z ) \ ( G ⋆ ( Q ) · v ∩ V ⋆ ( Z )) Z G ⋆ ( v ′ )( Q ) Z G ⋆ ( v ′ )( Z ) (cid:17) − if b ∈ p n p · W p and G ⋆ ( Q p ) · v ∈ η ⋆b ( P b ( Q p ) / ˆ ρ ( P b ( Q p ))) for all p, otherwise.Define w ′ ( v ) by the formula w ′ ( v ) = Z G ⋆ ( v )( Q ) · w ( v ) . Corollaries 4.15 and 5.28 imply that if b ∈ M · F W ,non-identity elements in the ˆ ρ -Selmer group of P ∨ b correspond bijectively to G ⋆ ( Q ) -orbits in V ⋆b ⋆ ( Q ) thatintersect V ⋆ ( Z ) nontrivially, that are Q -irreducible and that are soluble at R and Q p for all p . In otherwords, we have the formula: X b ∈F W ht ( b ) 1) = X b ∈ M ·F W ht ( b ) 1) = N w ′ ( V ⋆ ( Z ) irr ∩ V ⋆ ( R ) sol , M · X ) . (8.2.1)Since the number of G ⋆ ( Z ) -orbits of v ∈ V ⋆ ( Z ) 1) = lim X → + ∞ X − N w ( V ⋆ ( Z ) irr ∩ V ⋆ ( R ) sol , M · X ) . (That is, the limit on the left-hand-side exists if and only if the limit on the right-hand-side exists, and inthat case their values coincide.) By Theorem 7.7 and the estimate vol( B ( R ) Sel ρ P b . Theorem 8.7. Let F ⊂ E be a subset defined by finitely many congruence conditions. Then the averagesize of Sel ρ P b for b ∈ F , when ordered by height, exists and equals .Proof. In the notation of §8.1, Theorem 8.5 remains valid for any box of B , by an identical proof. Since Sel ρ P ˆ b ≃ Sel ˆ ρ P ∨ b (Theorem 3.14), the theorem follows from Proposition 8.3 applied to the automorphism χ : B → B . -Selmer group If b ∈ B rs ( Q ) , let Sel ♮ P b ⊂ Sel P b be the subset of elements whose image under the embedding Sel P b ֒ → G ( Q ) \ V b ( Q ) is strongly Q -irreducible (as defined in §6.4). By Corollary 4.13, it coincides with the subset of Sel P b whose image in Sel ˆ ρ P ∨ b under ρ is nontrivial. Theorem 8.8. Let F ⊂ E be a subset defined by finitely many congruence conditions (see §8.2). Then theaverage size of Sel ♮ P b for b ∈ F , when ordered by height, is bounded above by .Proof. The proof is very similar to that of Theorem 8.5, using the results of §4.1, §5.4 and §6. We give abrief sketch. Again it suffices to prove that for each b ∈ F and for every prime p dividing N , we can findan open compact neighbourhood W p of b in F p such that the average size of Sel ♮ P b is bounded above by in the family F W := F ∩ (cid:16)Q p | N W p (cid:17) . Choose sets W p ⊂ F p and integers n p ≥ for p | N satisfyingthe conclusion of Corollary 5.27. Set W p = F p and n p = 0 if p does not divide N . Let M = Q p p n p . For v ∈ V ( Z ) with π ( v ) = b , define w ( v ) ∈ Q ≥ by the following formula: w ( v ) = (cid:16)P v ′ ∈ G ( Z ) \ ( G ( Q ) · v ∩ V ( Z )) Z G ( v ′ )( Q ) Z G ( v ′ )( Z ) (cid:17) − if b ∈ p n p · W p and G ( Q p ) · v ∈ η b ( P b ( Q p ) / P b ( Q p ) for all p, otherwise.Then Corollaries 4.6 and 5.27 and Proposition 6.16 imply that X b ∈F W ht ( b ) Theorem 8.9. Let F ⊂ E be a subset defined by finitely many congruence conditions. Then the averagesize of Sel P b for b ∈ F , when ordered by height, is bounded above by . Remark 8.10. For every b ∈ B rs ( Q ) we have an exact sequence ˆ E b [2]( Q ) → Sel ρ P b → Sel P b → Sel ˆ ρ P ∨ b . Moreover an easy Hilbert irreducibility argument shows that the average size of E b [2]( Q ) for b ∈ F is .We conclude that the average size of Sel P b equals the sum of the average sizes of Sel ρ P b (which is byTheorem 8.7) and Sel ♮ P b , provided the latter quantity exists. References [AK80] Allen B. Altman and Steven L. 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