aa r X i v : . [ m a t h . N T ] S e p CAMPANA POINTS AND POWERFUL VALUES OF NORMFORMS
SAM STREETER
Abstract.
We give an asymptotic formula for the number of weak Campanapoints of bounded height on a family of orbifolds associated to norm forms forGalois extensions of number fields. From this formula we derive an asymptoticfor the number of elements with m -full norm over a given Galois extension of Q . We also provide an asymptotic for Campana points on these orbifolds whichillustrates the vast difference between the two notions, and we compare this tothe Manin-type conjecture of Pieropan, Smeets, Tanimoto and V´arilly-Alvarado. Contents
1. Introduction 12. Background 43. The norm torus 84. Heights and indicator functions 135. Weak Campana points 166. Campana points 287. Comparison to Manin-type conjecture 33References 351.
Introduction
The theory of Campana points is of growing interest in arithmetic geometrydue to its ability to interpolate between rational and integral points. Two com-peting notions of Campana points can be found in the literature, both extendinga definition of “orbifold rational points” for curves within Campana’s theory of“orbifoldes g´eom´etriques” in [7], [8], [9] and [10]. They capture the idea of rationalpoints which are integral with respect to a weighted boundary divisor . These twonotions have been termed
Campana points and weak Campana points in the re-cent paper [27] of Pieropan, Smeets, Tanimoto and V´arilly-Alvarado, in which theauthors initiate a systematic quantitative study of points of the former type onsmooth Campana orbifolds and prove a logarithmic version of Manin’s conjecturefor Campana points on vector group compactifications. The only other quantita-tive results in the literature are found in [5], [31], [6], [26] and [32], and the formerthree of these indicate the close relationship between Campana points and m -fullsolutions of equations. (Given m ∈ Z ≥ , we say that n ∈ Z \ { } is m -full if allprimes in the prime decomposition of n have multiplicity at least m .) Mathematics Subject Classification.
In this paper, we bring together the perspectives in the above papers and providethe first result for Campana points on singular orbifolds. As observed in [27, § (cid:0) P d − K , (cid:0) − m (cid:1) Z ( N ω ) (cid:1) , where N ω is a norm form associated to a K -basis ω of aGalois extension of number fields E/K of degree d ≥ m ∈ Z ≥ if d isnot prime. When K = Q , we derive from the result for weak Campana points anasymptotic for the number of elements of E of bounded height with m -full normover Q . We compare the result for Campana points to a conjecture of Pieropan,Smeets, Tanimoto and V´arilly-Alvarado [27, Conj. 1.1, p. 3].1.1. Results.Theorem 1.1.
Let
E/K be a Galois extension of number fields of degree d ≥ ,and let m ≥ be an integer which is coprime to d if d is not prime. Let ω be a K -basis of E . Denote by ∆ ω m the Q -divisor (cid:0) − m (cid:1) Z ( N ω ) of P d − K for N ω the normform corresponding to ω . Let H denote the anticanonical height function on P d − K from Definition 4.4. Then there exists an explicit finite set S ( ω ) ⊂ Val( K ) suchthat, for any finite set of places S ⊃ S ( ω ) , the number N (cid:0)(cid:0) P d − K , ∆ ω m (cid:1) , H, B, S (cid:1) ofweak Campana O K,S -points of height at most B on the orbifold (cid:0) P d − K , ∆ ω m (cid:1) withrespect to the model P d − O K,S of P d − K has the asymptotic formula N (cid:0)(cid:0) P d − K , ∆ ω m (cid:1) , H, B, S (cid:1) ∼ c ( ω , m, S ) B m (log B ) b ( d,m ) − for some explicit positive constant c ( ω , m, S ) , where b ( d, m ) = 1 d (cid:18)(cid:18) d + m − d − (cid:19) − (cid:18) m − d − (cid:19)(cid:19) . Note 1.2. If ω is a relative integral basis of E/K , then S ( ω ) = S ∞ , the set ofarchimedean places of K , in Theorem 1.1 (see Remark 4.3).Each rational point P ∈ P d − ( Q ) possesses precisely two sets of coordinates in Z d prim = { ( x , . . . , x d − ) ∈ Z d : gcd( x , . . . , x d − ) = 1 } . Interpreting H and N ω asfunctions on this set, we immediately obtain the following result. Corollary 1.3.
Taking K = Q and letting ω be an integral basis with the notationand hypotheses of Theorem 1.1, we have { x ∈ Z d prim : H ( x ) ≤ B, N ω ( x ) is m -full } ∼ c ( ω , m, S ∞ ) B m (log B ) b ( d,m ) − . Arithmetically special (e.g. prime, square-free) values of norm forms are a topicof long-standing interest in number theory (see e.g. [13], [21]).Campana points are only defined and studied for smooth orbifolds (i.e. smoothvarieties for which the orbifold divisor has strict normal crossings support) in [27].In order to study the Campana points of (cid:0) P d − K , ∆ ω m (cid:1) , which is smooth only when d = 2, we must first generalise the definition of Campana points, which we do inSection 2.1. Using the same strategy employed in the proof of Theorem 1.1, wethen derive an asymptotic for the number of Campana points on (cid:0) P d − K , ∆ ω m (cid:1) . AMPANA POINTS 3
Theorem 1.4.
With the notation and hypotheses of Theorem 1.1, denote by e N (cid:0)(cid:0) P d − K , ∆ ω m (cid:1) , H, B, S (cid:1) the number of Campana O K,S -points on (cid:0) P d − K , ∆ ω m (cid:1) ofheight at most B with respect to H for some finite set of places S ⊃ S ( ω ) . Thenthere exists an explicit positive constant e c ( ω , m, S ) such that e N (cid:0)(cid:0) P d − K , ∆ ω m (cid:1) , H, B, S (cid:1) ∼ e c ( ω , m, S ) B m . Remark 1.5.
It is not clear if the exponent of the logarithm in Theorem 1.1 ad-mits a geometric interpretation as it does in Theorem 1.4 (cf. [27, Conj. 1.1, p. 3]).
Acknowledgements.
Parts of this work were completed during the program“Reinventing Rational Points” at the Institut Henri Poincar´e and the conference“Topics in Rational and Integral Points” at the Universit¨at Basel; the authorwould like to thank the organisers of both events for their hospitality. Thanks goto the authors of [27] for their helpful feedback and to Daniel Loughran for hisguidance as an expert and mentor.
Conventions.
Algebra.
We take N = Z ≥ . We denote by R ∗ the group of units of a ring R .Given a group G , we denote by 1 G the identity element of G , and for any n ∈ N ,we set G [ n ] = { g ∈ G : g n = 1 G } . For any perfect field F , we fix an algebraicclosure F and set G F = Gal (cid:0) F /F (cid:1) . Given a topological group G , we denote by G ∧ = Hom( G, S ) its group of continuous characters. A monomial in the variables x , . . . , x n is a product x a . . . x a n n , ( a , . . . , a n ) ∈ Z n ≥ . For any n ∈ N , we denoteby µ n the group of n th roots of unity and by S n the symmetric group of order n . Geometry.
We denote by P nR the projective n -space over the ring R . We omit thesubscript if the ring R is clear. Given a homogeneous polynomial f ∈ R [ x , . . . , x n ],we denote by Z ( f ) = Proj R [ x , . . . , x n ] / ( f ) the zero locus of f viewed as a closedsubscheme of P nR . A variety over a field F is a geometrically integral separatedscheme of finite type over F . Given a variety X defined over F and an extension E/F , we denote by X E = V × Spec F Spec E the base change of X over E , andwe write X = X × Spec F Spec F . When F = K and E = K v for a numberfield K and a place v of K , we write X v = X K v . Given a field F , we define G m,F = Spec F [ x , x ] / ( x x − F if the field is clear. Number theory.
Given an extension of number fields
L/K with K -basis ω = { ω , . . . , ω d − } , we write N ω ( x , . . . , x d − ) = N L/K ( x ω + · · · + x d − ω d − ) for theassociated norm form. We denote by Val( K ) the set of valuations of a numberfield K , and we denote by S ∞ the set of archimedean valuations. For v ∈ Val( K ),we denote by O v the maximal compact subgroup of K v . For a finite set of places S containing S ∞ , we denote by O K,S = { α ∈ K : α ∈ O v for all v S } thering of algebraic S -integers of K . We write O K = O K,S ∞ . For v ∈ Val( K ) non-archimedean, we denote by π v and q v a uniformiser for the residue field of K v andthe size of the residue field of K v respectively. If v | ∞ , then we set log q v = 1. Foreach v ∈ Val( K ), we choose the absolute value | x | v = | N K v / Q p ( x ) | p for the unique p ∈ Val( Q ) with v | p and the usual absolute value | · | p on Q p . We normaliseour Haar measures dx v on each K v as Tate does in [11, Ch. XV]. We denote by A K = bQ O v v ∈ Val( K ) K v the adele ring of K with the restricted product topology. SAM STREETER Background
Campana points.
In this section we define Campana orbifolds, Campanapoints and weak Campana points, generalising the definitions in [27, § Definition 2.1. A Campana orbifold over a field F is a pair ( X, D ǫ ) consisting ofa proper, normal variety X over F and an effective Weil Q -divisor D ǫ = X α ∈A ǫ α D α on X , where the D α are prime divisors and ǫ α = 1 − m α for some m α ∈ Z ≥ ∪ {∞} (by convention, we take ∞ = 0). We define the support of the Q -divisor D ǫ to be D red = X α ∈A D α . We say that (
X, D ǫ ) is smooth if X is smooth and D red has strict normal crossings.Let ( X, D ǫ ) be a Campana orbifold over a number field K . Let S ⊂ Val( K ) bea finite set containing S ∞ . Definition 2.2. A model of ( X, D ǫ ) over O K,S is a pair ( X , D ǫ ), where X is a flatproper model of X over O K,S (i.e. a flat proper O K,S -scheme with X (0) ∼ = X ) and D ǫ = P α ∈A ǫ α D α for D α the Zariski closure of D α in X .Define D red = P α ∈A D α . Denote by D α v , α v ∈ A v the irreducible componentsof D red over Spec O v . We write α v | α if D α v ⊂ D α .Let ( X , D ǫ ) be a model for ( X, D ǫ ) over O K,S . Given P ∈ X ( K ) and a place v S , we get an induced point P v ∈ X ( O v ) by the valuative criterion of properness[14, Thm. II.4.7, p. 101]. Define X ◦ = X \ D red . Definition 2.3.
Let P ∈ X ◦ ( K ) and take a place v S . For each α v ∈ A v , wedefine the local intersection multiplicity n α v ( D α v , P ) of D α v and P at v to be thecolength of the non-zero ideal P ∗ v D α v ⊂ O v . We then define n v ( D α , P ) = X α v | α n α v ( D α v , P ) , and we define the total intersection number of D ǫ and P at v to be n v ( D ǫ , P ) = X α ∈A ǫ α n v ( D α , P ) . We are now ready to define weak Campana points and Campana points. Bothnotions arise from [1], with the former appearing in its current form in [2, § Definition 2.4.
We say that P ∈ X ◦ ( K ) is a weak Campana O K,S -point of ( X , D ǫ )if the following holds: if v S is a place of K , then either n v ( D ǫ , P ) = 0 or n v ( D ǫ , P ) ≤ X α ∈A n v ( D α , P ) ! − , i.e. X α ∈A m α n v ( D α , P ) ≥ . AMPANA POINTS 5
We denote the set of weak Campana O K,S -points of ( X , D ǫ ) by ( X , D ǫ ) w ( O K,S ). Definition 2.5.
We say that P ∈ X ◦ ( K ) is a Campana O K,S -point of ( X , D ǫ ) ifthe following implications hold for all places v S of K and for all α ∈ A .(i) If ǫ α = 1 (meaning m α = ∞ ), then n v ( D α , P ) = 0.(ii) If ǫ α = 1 and n v ( D α , P ) >
0, then for all α v | α , we have n α v ( D α v , P ) ≥ − ǫ α , i.e. n α v ( D α v , P ) ≥ m α . We denote the set of Campana O K,S -points of ( X , D ǫ ) by ( X , D ǫ )( O K,S ). Remark 2.6.
Informally, weak Campana points are rational points P ∈ X ◦ ( K )which, upon reduction modulo any place v S , either do not lie on D red or lie on D α with multiplicity at least m α on average over α . Similarly, Campana pointsare rational points P ∈ X ◦ ( K ) which, upon reduction modulo any place v S ,either do not lie on D red or lie on each v -adic irreducible component of each D α with multiplicity either 0 or at least m α . Lemma 2.7.
Let ( X, D ǫ ) be a smooth Campana orbifold over a number field K which is Kawamata log terminal (i.e. ǫ α < for all α ∈ A ), and let ( X , D ǫ ) bea model of ( X, D ǫ ) over O K,S with X smooth over O K,S and D red a relative strictnormal crossings divisor in X / O K,S as defined in [17, § . Then the definition ofCampana points on ( X , D ǫ ) above coincides with the one in [27, § .Proof. Since D red is a relative strict normal crossings divisor, each irreduciblecomponent D α is smooth over O K,S . In particular, its base change over Spec O v is smooth for any v S , so the divisors D α v , α v | α are disjoint. Then, for anyrational point P ∈ X ◦ ( K ), the reduction of P at the place v can lie on at mostone of the divisors D α v , α v | α , so n v ( D α , P ) = P α v | α n α v ( D α v , P ) is either 0 or m α if and only if each n α v ( D α v , P ), α v | α , is either 0 or m α . (cid:3) Remark 2.8.
If one were to apply the definitions given in [27, § (cid:0) P d − K , ∆ ω m (cid:1) of Theorem 1.1, which is singular for all d ≥
3, then the weak Campanapoints and the Campana points would be the same, but the asymptotic of Theorem1.1 differs to [27, Conj. 1.1, p. 3] for d ≥ Toric varieties.Definition 2.9. An (algebraic) torus over a field F is an algebraic group T over F such that T ∼ = G nm for some n ∈ N . The splitting field of a torus T over a field F is defined to be the smallest Galois field extension E of F for which T E ∼ = G nm . Definition 2.10. A toric variety is a smooth projective variety X equipped witha faithful action of an algebraic torus T such that there is an open dense orbitcontaining a rational point. SAM STREETER
Definition 2.11.
Let T be a torus over a field F . The character group of T is X ∗ (cid:0) T (cid:1) = Hom (cid:0) T , G m (cid:1) . Then X ∗ ( T ) = X ∗ (cid:0) T (cid:1) G F is the collection of char-acters of T which are defined over F . The cocharacter group of T is X ∗ ( T ) =Hom( X ∗ ( T ) , Z ). We let X ∗ ( T ) R = X ∗ ( T ) ⊗ Z R and X ∗ ( T ) R = X ∗ ( T ) ⊗ Z R . Definition 2.12.
An algebraic torus T over a field F is anisotropic if it has trivialcharacter group over F , i.e. X ∗ ( T ) = 0 . Let T be a torus over a number field K with splitting field E . Set T ∞ = Q v |∞ T v .For v ∈ Val( K ), let T ( O v ) denote the maximal compact subgroup of T ( K v ). Definition 2.13.
Let v ∈ Val( K ) and w ∈ Val( E ) with w | v .For v ∤ ∞ with ramification degree e v in E/K , define the mapsdeg
T,v : T ( K v ) → X ∗ ( T v ) , t v [ χ v v ( χ v ( t v ))]and deg T,E,v = e v deg T,v .For v | ∞ , define the mapsdeg T,v : T ( K v ) → X ∗ ( T v ) R , t v [ χ v log | χ v ( t v ) | v ]and deg T,E,v = [ E w : K v ] deg T,v .Finally, define the mapsdeg T = X v ∈ Val( K ) log q v deg T,v , deg T,E = X v ∈ Val( K ) log q w deg T,E,v . Lemma 2.14. [4, § Let v ∈ Val( K ) , and let f be either deg T,v or deg T,E,v .(i) If v is non-archimedean, then we have the exact sequence → T ( O v ) → T ( K v ) f −→ X ∗ ( T v ) . The image of f is open and of finite index. Further, if v is unramified in E , then f is surjective.(ii) If v is archimedean, then we have the short exact sequence → T ( O v ) → T ( K v ) f −→ X ∗ ( T v ) R → . Further, f admits a canonical section.(iii) Letting g be either deg T or deg T,E and denoting its kernel by T ( A K ) , wehave the split short exact sequence → T ( A K ) → T ( A K ) g −→ X ∗ ( T ) R → , hence we have an isomorphism T ( A K ) ∼ = T ( A K ) × X ∗ ( T ) R . (2.1) Definition 2.15.
Let χ be a character of T ( A K ). We say that χ is automorphic if it is trivial on T ( K ). We say that χ is unramified at v ∈ Val( K ) if χ v is trivialon T ( O v ), and we say that it is unramified if it is unramified at every v ∈ Val( K ).The canonical sections of the maps T ( K v ) deg T,v −−−→ X ∗ ( T v ) R for each v | ∞ fromLemma 2.14(ii) induce a canonical section of the composition T ( A K ) → Y v |∞ T ( K v ) → X ∗ ( T ∞ ) R , AMPANA POINTS 7 which in turn induces a “type at infinity map” T ( A K ) ∧ → X ∗ ( T ∞ ) R . (2.2)Defining K T = Q v T ( O v ), the splitting (2.1) for g = deg T induces a map (cid:0) T ( A K ) /T ( K ) K T (cid:1) → X ∗ ( T ∞ ) R which has finite kernel and image a lattice of codimension rank X ∗ ( T ) (see [4,Lem. 4.52, p. 96]). Note 2.16.
When T is anisotropic, we have T ( A K ) = T ( A K ) by Lemma 2.14(iii),and then we see from the above that there is a map( T ( A K ) /T ( K ) K T ) → X ∗ ( T ∞ ) R with finite kernel and image a lattice of full rank.2.3. Hecke characters.Definition 2.17. A Hecke character for K is an automorphic character of G m,K .Each Hecke character χ has a conductor q ( χ ) ∈ N (see [18, § χ at the non-archimedean places of K . Definition 2.18.
A Hecke character is principal if it is trivial on G m,K ( A K ) .By Lemma 2.14(iii), χ is principal if and only if χ = k · k iθ for some θ ∈ R ,where k · k denotes the adelic norm map, i.e. k · k : A ∗ K → S , ( x v ) v → Y v ∈ Val( K ) | x v | v . Definition 2.19.
The (Hecke) L -function L ( χ, s ) of a Hecke character χ is L ( χ, s ) = Y v (cid:18) − χ v ( π v ) q sv (cid:19) − , where the product is taken over all places v ∤ ∞ at which χ is unramified.The Dedekind zeta function of K is ζ K ( s ) = L (1 , s ) . Given a Hecke character χ for a number field L and w ∈ Val( L ), we denote by L w ( χ, s ) the local factor at w for the Euler product of L ( χ, s ), i.e. L w ( χ, s ) = ( − χ w ( π w ) q sw if w ∤ ∞ and χ is unramified at w, . When working over the field L ⊃ K , we define L v ( χ, s ) for each v ∈ Val( K ) by L v ( χ, s ) = Y w | v L w ( χ, s ) . Theorem 2.20. [15, § The L -function of a Hecke character χ admits a mero-morphic continuation to C . If χ = k · k iθ for some θ ∈ R , then this continuationadmits a single pole of order at s = 1 + iθ . Otherwise, it is holomorphic. SAM STREETER
Definition 2.21.
Let ψ be a character of Q v |∞ K ∗ v . The restriction of ψ to each R > ⊂ K ∗ v is of the form x
7→ | x | iκ v for some κ v ∈ R . We define k ψ k = max v |∞ | κ v | . Lemma 2.22. [20, Lem. 3.1, p. 2561]
Let χ be a non-principal Hecke characterof K , let C be a compact subset of Re s ≥ and let ε > . Then L ( χ, s ) ≪ ε,C q ( χ )(1 + k χ ∞ k ) ε , ( s − ζ K ( s ) ≪ ε,C , s ∈ C. Definition 2.23.
Let
E/K be Galois, let χ be a Hecke character for E and let g ∈ Gal(
E/K ). We define the (Galois) twist of χ by g to be the character χ g : A ∗ E → S , ( t w ) w χ (cid:16) ( g w ( t w )) gw (cid:17) . Here, gw denotes the place of E obtained by the action of g on Val( E ), and g w : E w → E gw is the induced map on completions. One may easily verify that χ g is trivial on E ∗ , hence it is also a Hecke character for E .3. The norm torus
In this section, we fix an extension of number fields
L/K of degree d ≥ E and a K -basis ω = { ω , . . . , ω d − } . We write N ω ( x , . . . , x d − )for the norm form corresponding to ω , and G = Gal( E/K ). From the equality N ω ( x ω + · · · + x d − ω d − ) = Y g ∈ G/ Gal(
E/L ) ( x g ( ω ) + · · · + x d − g ( ω d − )) , (3.1)we see that N ω is irreducible over K and has splitting field E . We denote by T the norm torus T ω = P d − K \ Z ( N ω ). As noted in [20, § P d − K is a toric varietywith respect to T , and T ∼ = R L/K G m / G m is anisotropic. Since its boundary is Z ( N ω ), its splitting field is E . We have the short exact sequence0 → G m → R L/K G m → T → . (3.2) Note 3.1.
The isomorphisms T ( A K ) ∼ = A ∗ L / A ∗ K and T ( K ) ∼ = L ∗ /K ∗ follow fromHilbert’s Theorem 90 [23, Prop. IV.3.5, p. 281] by applying Galois cohomologyto (3.2). They allow us to interpret an automorphic character χ of T as a Heckecharacter for L , and we will do so frequently. In fact, distinct automorphic char-acters of T correspond to distinct Hecke characters of L by [3, Cor. 1.4.16, p. 606]and [3, Thm. 3.1.1, p. 619]. Since T ( K v ) ∼ = (cid:16)Q w | v L ∗ w (cid:17) /K ∗ v for each v ∈ Val( K ),we see that, if χ is unramified at v , then it is unramified as a Hecke character atall w | v . In particular, if χ is unramified at v and v is unramified in L/K , then Q w | v χ w ( π w ) = 1, since π v is a uniformiser for L w for each w | v .3.1. Geometry.
In this section we study fan-theoretic objects relating to T . Webegin by describing the fan Σ ⊂ X ∗ (cid:0) T (cid:1) R associated to the equivariant compact-ification P d − K of T and the associated piecewise-linear function ϕ Σ (see [3, § AMPANA POINTS 9
Denoting by l ( x ) , . . . , l d − ( x ) ∈ E [ x ] the E -linear factors of N ω ( x ), we have the E -isomorphismΦ : T = P d − \ d − [ i =0 Z ( l i ) ∼ −→ G d − m = P d − \ d − [ j =0 Z ( x j ) , [ x , . . . , x d − ] [ l ( x ) , . . . , l d − ( x )] . By [16, § P d − E as a compactification of G d − m,E is thefan whose r -dimensional cones are generated by the r -dimensional subsets of { e ′ , . . . , e ′ d − } for 0 ≤ r ≤ d −
1, where e ′ i ∈ X ∗ (cid:0) G d − m (cid:1) ∼ = Hom (cid:0) G m , G d − m (cid:1) isdefined by e ′ i : R ∗ → G d − m ( R ) , t [ x ,i ( t ) , . . . , x d − ,i ( t )] , where in turn x j,i ( t ) = ( t if i = j, Definition 3.2.
Set e i = Φ − ◦ e ′ i for i = 0 , . . . , d −
1, and define Σ to bethe fan whose r -dimensional cones are generated by the r -dimensional subsetsof { e , . . . , e d − } for 0 ≤ r ≤ d − P d − E as a compactification of T E .Also, we see that P d − i =0 e i = 0 and that { e , . . . , e d − } is the dual of the basis { m , . . . , m d − } of X ∗ (cid:0) T (cid:1) , where m i ( x ) = l i ( x ) l ( x ) for i = 1 , . . . , d −
1. By [16, § P d − K = P d − E /G of T over K .By [3, Prop. 1.2.12, p. 597], the line bundle L ( ϕ Σ ) associated to the function ϕ Σ : X ∗ (cid:0) T (cid:1) R → R (see [3, Prop. 1.2.9, p. 597]) defined by ϕ Σ ( e i ) = 1 for all i = 0 , . . . , d − − K P d − .By [3, Prop. 1.3.11, p. 601], G acts transitively on Σ(1) = {h e i , . . . , h e d − i} (since Pic P d − K ∼ = Z ). For v ∈ Val( K ) non-archimedean, let G v denote the asso-ciated decomposition subgroup of G . By the proof of [3, Thm. 3.1.3, p. 619], the G v -orbits of Σ(1) are in bijection with the places of L over v , and the length ofthe G v -orbit corresponding to a place w | v is its inertia degree.We now show that the action of G on Σ(1) is compatible with its action on the E -linear factors of N ω . Denote by ∗ the action of G , and set l g ( i ) = g ∗ l i . Lemma 3.3.
For all g ∈ G and i = 0 , . . . , d − , we have g ∗ e i = e g ( i ) . Proof.
Let g ∈ G . It suffices to show that( g ∗ e i )( m j ) = e g ( i ) ( m j ) (3.3)for all i ∈ { , . . . , d − } and j ∈ { , . . . , d − } . Note that, for any i, j, k ∈{ , . . . , d − } , we have e i (cid:18) l j l k (cid:19) = δ ij − δ ik , (3.4)where δ ij is the Kronecker delta symbol, defined by δ ij = ( i = j, Then (3.3) becomes δ ig − ( j ) − δ ig − (0) = δ g ( i ) j − δ g ( i )0 , which clearly holds. (cid:3) Proposition 3.4.
Let v ∈ Val( K ) be non-archimedean with ramification degree e v in E/K , and let
Σ(1) = [ w | v Σ w (1) denote the decomposition of Σ(1) = {h e i , . . . , h e d − i} into G v -orbits. For each w | v , let n w be the sum of the elements of Σ w (1) and let f w ( x ) be the productof the linear factors in the G v -orbit of { l , . . . , l d − } corresponding to Σ w (1) byLemma 3.3. Then the map deg T,E,v : T ( K v ) → X ∗ ( T v ) is given by t v e v X w | v v ( f w ( t v ))deg f w n w . Proof.
The image of t v in X ∗ ( T v ) ∼ = X ∗ (cid:0) T (cid:1) G v under deg T,E,v is the cocharacter ϕ t v : X ∗ ( T v ) → Z , λ e v v ( λ ( t v )) . We first show that { n w : w | v } spans X ∗ (cid:0) T (cid:1) G v . Given g ∈ G and σ = P d − i =0 a i e i ,we have g ∗ σ = P d − i =0 a g − ( i ) e i , so g ∗ σ = σ if and only if there exists r g ∈ Z suchthat a i = a g − ( i ) + r g for all i ∈ { , . . . , d − } . Setting s = G , we have a i = a g s ( i ) = a g s − ( i ) + r g = · · · = a i + sr g , hence r g = 0. We deduce that σ ∈ Σ G v if and only if a i = a j for all e i , e j in the same G v -orbit, so the result follows. Moreover, we observe that P w | v a w n w = P w | v b w n w if and only if there exists r ∈ Z such that b w = a w + r for all w | v , since there isa unique expression for σ ∈ X ∗ (cid:0) T (cid:1) in the form σ = P d − i =0 c i e i where c d = 0.Now, write ϕ t v = X w | v α w n w . Define µ i ∈ X ∗ (cid:0) T (cid:1) and λ w ∈ X ∗ (cid:0) T (cid:1) G v for all i ∈ { , . . . , d − } and all w | v by µ i ( x ) = l di N ω ( x ) , λ w ( x ) = Y e i ∈ Σ w µ i ( x ) = f w ( x ) d N ω ( x ) deg f w . By (3.4), we have e i ( µ j ) = ( d − i = j, − . Then, setting d w = deg f w , we see that n w ( λ w ′ ) = ( dd w − d w if w = w ′ , − d w d w ′ otherwise , so we deduce that e v v ( λ w ( t v )) = dd w α w − d w X w ′ | v d w ′ α w ′ (3.5) AMPANA POINTS 11 for all w | v . On the other hand, we have e v v ( λ w ( t v )) = e v dv ( f w ( t v )) − e v d w X w ′ | v v ( f w ′ ( t v )) . (3.6)Set β w = d w α w − e v v ( f w ( t v )). Combining (3.5) and (3.6), we obtain dβ w = d w X w ′ | v β w ′ , hence β w ′ = d w ′ d w β w for all w | v , w ′ | v . Since K v ∼ = E G v w for any w | v , it followsthat d w | v ( f w ( t v )), so β w ∈ d w Z for all w | v . We deduce that there exists aninteger n ∈ Z such that, for all w | v , we have β w = d w n , hence α w = e v v ( f w ( t v ))deg f w + n. Since P w | v n w = P i e i = 0, we conclude that ϕ t v = e v X w | v v ( f w ( t v ))deg f w n w . (cid:3) We now study polynomials introduced by Batyrev and Tschinkel in [3, § Definition 3.5.
Let v ∈ Val( K ) be non-archimedean, and let Σ(1) = S li =1 Σ i (1)be the decomposition of Σ(1) into G v -orbits. Let d i be the cardinality of Σ i (1).For each Σ i (1), define an independent variable u i . Let σ ∈ Σ G v , and let Σ i (1) ∪· · · ∪ Σ i k (1) be the set of 1-dimensional faces of σ . We define the rational function R σ,v ( u , . . . , u l ) = u d i . . . u d k i k (cid:0) − u d i (cid:1) . . . (cid:16) − u d k i k (cid:17) , and we define the polynomial Q Σ ,v ( u , . . . , u l ) by Q Σ ,v ( u , . . . , u l ) (cid:0) − u d (cid:1) . . . (cid:16) − u d l l (cid:17) = X σ ∈ Σ Gv R σ,v ( u , . . . , u l ) . Proposition 3.6.
For all non-archimedean valuations v ∈ Val( K ) , we have Q Σ ,v ( u , . . . , u l ) = 1 − u d . . . u d l l . Proof.
Observe that the G v -invariant cones in Σ are precisely those cones generatedby a set of 1-dimensional cones of the form Σ i (1) ∪· · ·∪ Σ i k (1) for some i , . . . , i k ∈{ , . . . , l } pairwise distinct with k < l . From this observation, we deduce that Q Σ ,v ( u , . . . , u l ) (cid:0) − u d (cid:1) . . . (cid:16) − u d l l (cid:17) = l − X k =1 X i ,...,i k ∈{ ,...,l } pairwise distinct u d i . . . u d k i k (cid:0) − u d i (cid:1) . . . (cid:16) − u d k i k (cid:17) . In particular, we see that Q Σ ,v ( u , . . . , u l ) = X ( t ,...,t l ) ∈{ , } l l Y i =1 (cid:0) t i + (1 − t i ) u d i i (cid:1) − u d . . . u d l l , so it suffices to prove that X ( t ,...,t l ) ∈{ , } l l Y i =1 (cid:0) t i + (1 − t i ) u d i i (cid:1) = 1 . (3.7)Splitting the sum into two smaller sums for t = 0 and t = 1, we obtain X ( t ,...,t l ) ∈{ , } l l Y i =1 (cid:0) t i + (1 − t i ) u d i i (cid:1) = (cid:0) u d + (cid:0) − u d (cid:1)(cid:1) X ( t ,...,t n ) ∈{ , } l − l Y i =2 (cid:0) t i + (1 − t i ) u d i i (cid:1) = X ( t ,...,t n ) ∈{ , } l − l Y i =2 (cid:0) t i + (1 − t i ) u d i i (cid:1) . Repeating this process for each variable t , . . . , t l , we deduce (3.7). (cid:3) Haar measures and volume.
Let ω be an invariant d -form on T . By aclassical construction (see [12, § ω gives rise to a Haar measure | ω | v on T ( K v ) for each v ∈ Val( K ). In [24, § c v = ( L v (cid:0) X ∗ (cid:0) T (cid:1) , (cid:1) − if v ∤ ∞ , v | ∞ . Here, L v (cid:0) X ∗ (cid:0) T (cid:1) , s (cid:1) is the local factor at v of the Artin L -function L (cid:0) X ∗ (cid:0) T (cid:1) , s (cid:1) .Defining µ v = c − v | ω | v , the product of the µ v converges to give a Haar measure µ on T ( A K ), which is independent of ω by the product formula. Note 3.7.
From the short exact sequence (3.2), we obtain L (cid:0) X ∗ (cid:0) T (cid:1) , s (cid:1) = ζ L ( s ) ζ K ( s ) . Lemma 3.8.
With respect to the Haar measure µ , we have vol( T ( A K ) /T ( K )) = d Res s =1 ζ L ( s )Res s =1 ζ K ( s ) . Proof.
By [24, § (cid:0) T ( A K ) /T ( K ) (cid:1) = | Pic T || X ( T ) | L (cid:0) X ∗ (cid:0) T (cid:1) , (cid:1) , where X ( T ) is the Tate-Shafarevich group of T , i.e. X ( T ) = ker H ( K, T ) → Y v ∈ Val( K ) H ( K v , T ) . By [29, Prop. 8.3, p. 58] and [20, Cor. 4.6, p. 2568], the rationality of T implies that X ( T ) is trivial. Further, we have Pic T ∼ = Z /d Z (see [14, Prop. II.6.5(c), p. 133]).Since ζ K ( s ) and ζ L ( s ) both have a simple pole at s = 1, we have L (cid:0) X ∗ (cid:0) T (cid:1) , (cid:1) = Res s =1 ζ L ( s )Res s =1 ζ K ( s ) . Finally, as T is anisotropic, we have T ( A K ) = T ( A K ). (cid:3) AMPANA POINTS 13 Heights and indicator functions
In this section we define functions which allow us to use harmonic analysis tostudy weak Campana points. Let
L/K be an extension of number fields with K -basis ω = { ω , . . . , ω d − } and Galois closure E/K . For any i, j ∈ { , . . . , d − } and g ∈ G = Gal( E/K ), write ω i · ω j = d − X k =0 a ijk ω k , g ( ω i ) = d − X k =0 b gk ω k , d − X k =0 c k ω k . Definition 4.1.
We define S ( ω ) to be the minimal set of places of K containing S ∞ such that a ijk , b gk , c k ∈ O v for all v S ( ω ), i, j, k ∈ { , . . . , d − } and g ∈ G . Remark 4.2.
By (3.1), the S ( ω )-integrality of all the a ijk and b gk implies that N ω is defined over O K,S ( ω ) , while the S ( ω )-integrality of all the c k implies that thecoefficients of N ω are not all divisible by some α ∈ O K,S ( ω ) \ O ∗ K,S ( ω ) . Since N ω isirreducible over K , we therefore deduce that N ω is irreducible over O K,S ( ω ) , hencethe Zariski closure of Z ( N ω ) in P d − O K,S ( ω ) is Proj O K,S ( ω ) [ x , . . . , x d − ] / ( N ω ). Remark 4.3.
For ω a relative integral basis, it is clear that S ( ω ) = S ∞ , sinceevery algebraic integer is expressible as an O K -linear combination of elements ofa relative integral basis, and O K is closed under multiplication and conjugation.From now on, we fix the model (cid:16) P d − O K,S ( ω ) , D ω m (cid:17) for (cid:0) P d − K , ∆ ω m (cid:1) , where D ω m = (cid:0) − m (cid:1) Proj O K,S ( ω ) [ x , . . . , x d − ] / ( N ω ). We denote by D ω red the support of D ω m .4.1. Definitions.Definition 4.4. [3, § v of K , we define the local height function H v : T ( K v ) → R > , t v e ϕ Σ ( deg T,E,v ( t v ) ) log q v . We then define the global height function H : T ( A K ) → R > , ( t v ) v Y v ∈ Val( K ) H v ( t v ) . Definition 4.5.
For each place v S ( ω ), define the function H ′ v : T ( K v ) → R > , x max {| x i | dv }| N ω ( x ) | v . Remark 4.6.
Note that H ′ v ( x ) ≥ x ∈ T ( K v ). Indeed, one may alwaysselect v -adic coordinates x i such that max {| x i | v } = 1, and N ω has coefficients in O v by Remark 4.2, so, by the strong triangle inequality, we have | N ω ( x ) | v ≤ Lemma 4.7.
For all but finitely many places v S ( ω ) , we have H ′ v = H v .Proof. Note that H ′ v is the local Weil function associated to the basis of globalsections of − K P d − consisting of all monomials of degree d in [3, Def. 2.1.1, p. 606].It is well-known (see [12, § (cid:3) Definition 4.8.
We define the finite set S ′ ( ω ) = S ( ω ) ∪ { v S ( ω ) : H ′ v = H v } ∪ { v ∈ Val( K ) : E/K is ramified at v } . Definition 4.9.
For each place v S ( ω ), define the local indicator function φ m,v : T ( K v ) → { , } , t v ( H ′ v ( t v ) = 1 or H ′ v ( t v ) ≥ q mv , . Setting φ m,v = 1 for v ∈ S ( ω ), we then define the global indicator function φ m : T ( A K ) → { , } , ( t v ) v Y v ∈ Val( K ) φ m,v ( t v ) . Remark 4.10.
Let v S ( ω ) be a non-archimedean place of K . Since H ′ v iscontinuous with discrete image in R > , its level sets are clopen. It follows that φ m,v is continuous for all v ∈ Val( K ). Also, since φ m,v ( T ( O v )) = 1 for all v S ′ ( ω )by Lemma 2.14(i), we see that φ m is well-defined and continuous on T ( A K ). Lemma 4.11.
The weak Campana O K,S ( ω ) -points of (cid:0) P d − K , ∆ ω m (cid:1) are precisely therational points t ∈ T ( K ) such that φ m ( t ) = 1 .Proof. Take v S ( ω ), and let t , . . . , t d − be a set of O v -coordinates for t ∈ T ( K )with at least one t i ∈ O ∗ v . Then we have H ′ v ( t ) = 1 | N ω ( t , . . . , t d − ) | v = q v ( N ω ( t ,...,t d − )) v = q n v ( D ω red ,t ) v . (cid:3) Invariant subgroups.
For this section, let L = E be Galois over K . Lemma 4.12.
For all v S ( ω ) and x, y ∈ T ( K v ) , we have H ′ v ( x · y ) ≤ H ′ v ( x ) H ′ v ( y ) . Proof.
Choose sets of projective coordinates { x , . . . , x d − } and { y , . . . , y d − } for x and y respectively. Note that( x ω + · · · + x d − ω d − ) · ( y ω + · · · + y d − ω d − ) = ( z ω + · · · + z d − ω d − ) , where, for a ijk ∈ O v as in Definition 4.1, we have z k = d − X i =0 d − X j =0 a ijk x i y j . Using N ω ( x · y ) = N ω ( x ) N ω ( y ) and the strong triangle inequality, we deduce that H ′ v ( x · y ) = max {| P d − i =0 P d − j =0 a ijk x i y j | dv }| N ω ( x · y ) | v ≤ | N ω ( x ) | v | N ω ( y ) | v max {| a ijk | dv } max {| x i | dv } max {| y j | dv }≤ H ′ v ( x ) H ′ v ( y ) . (cid:3) Lemma 4.13.
For any place v S ( ω ) , the level set K v = { t v ∈ T ( K v ) : H ′ v ( t v ) = 1 } is a subgroup of T ( O v ) . AMPANA POINTS 15
Proof.
From Proposition 3.4 and Lemma 2.14(i), it is clear that H ′ v ( t v ) = 1 implies t v ∈ T ( O v ), so K v ⊂ T ( O v ). It is also clear that H ′ v (1) = 1, and closure undermultiplication follows from Lemma 4.12 and Remark 4.6. It only remains to verifythat x ∈ K v implies x − ∈ K v . Let x ∈ K v , and choose coordinates x , . . . , x d − with max {| x i | dv } = 1. Since H ′ v ( x ) = 1, we must have | N ω ( x ) | v = 1. Note that( x ω + · · · + x d − ω d − ) − = 1 N ω ( x , . . . , x d − ) Y g ∈ Gg =1 G ( x g ( ω ) + · · · + x g ( ω )) . Recursively applying Lemma 4.12, we obtain H ′ v (cid:0) x − (cid:1) ≤ Y g ∈ Gg =1 G H ′ v ( g ( x )) . By Remark 4.6, it suffices to show that, for any g ∈ G , we have H ′ v ( g ( x )) = 1.Since N ω ( g ( x )) = N ω ( x ), it suffices by Remark 4.6 to show that max {| g ( x ) i | v } ≤ b gk ∈ O v since v S ( ω ), see Definition 4.1. (cid:3) Corollary 4.14.
For every place v S ( ω ) , the function H ′ v is K v -invariant.Proof. Take x ∈ K v , and let y ∈ T ( K v ). Then by Lemma 4.12, we have H ′ v ( x · y ) ≤ H ′ v ( x ) H ′ v ( y ) = H ′ v ( y ) , while on the other hand, since x − ∈ K v by Lemma 4.13, we have H ′ v ( y ) = H ′ v (cid:0) x − · ( x · y ) (cid:1) ≤ H ′ v (cid:0) x − (cid:1) H ′ v ( x · y ) = H ′ v ( x · y ) , so we conclude that H ′ v ( x · y ) = H ′ v ( y ). (cid:3) Lemma 4.15.
For each place v S ( ω ) , the functions H v and φ m,v are both K v -invariant and on K v . Further, K v is compact, open and of finite index in T ( O v ) .Moreover, when v S ′ ( ω ) , we have K v = T ( O v ) .Proof. Let v S ( ω ). By [3, Thm. 2.1.6(i), p. 608], H v is T ( O v )-invariant, hencetrivial and invariant on all of T ( O v ). By Corollary 4.14, the function φ m,v is K v -invariant; since φ m,v (1) = 1, it is also trivial on K v .Now, since K v = (cid:0) H ′ v | T ( O v ) (cid:1) − ( { } ), it is open. Since the cosets of an opensubgroup form an open cover of a topological group, any open subgroup of acompact topological group is closed and of finite index. Then K v ⊂ T ( O v ) isclosed, hence compact, and of finite index. Finally, we note that, when v S ′ ( ω ),we have H ′ v = H v , and H − v ( { } ) = T ( O v ) by Lemma 2.14(i), so K v = T ( O v ). (cid:3) Definition 4.16.
For each v ∈ S ( ω ), set K v = T ( O v ). Let K = Q v ∈ Val( K ) K v ,and let U be the group of automorphic characters of T which are trivial on K .4.3. Height zeta function and Fourier transforms.Definition 4.17.
For Re s ≫
0, we define the height zeta function Z m : C → C , s X x ∈ P d − ( K ) φ m ( x ) H ( x ) s . Definition 4.18.
Let f : T ( A K ) → C be a continuous function given as a productof local factors f v : T ( K v ) → C such that f ( T ( O v )) = 1 for all but finitely manyplaces v ∈ Val( K ). For each place v ∈ Val( K ) and each character χ of T ( A K ), wedefine the local Fourier transform of χ v with respect to f v to be b H v ( f v , χ v ; − s ) = Z T ( K v ) f v ( t v ) χ v ( t v ) H v ( t v ) s dµ v for all s ∈ C for which the integral exists. We then define the global Fouriertransform of χ with respect to f to be b H ( f, χ ; − s ) = Y v ∈ Val( K ) b H v ( f v , χ v ; − s ) = Z T ( A K ) f ( t ) χ ( t ) H ( t ) s dµ. Weak Campana points
In this section we prove Theorem 1.1. Fix an extension of number fields
L/K of degree d with K -basis ω , set T = T ω as in Section 3 and let m ∈ Z ≥ .5.1. Strategy.
Following [20] and [3], we will apply a Tauberian theorem [3,Thm. 3.3.2, p. 624] to our height zeta function Z m ( s ) in order to find an asymp-totic for the number of weak Campana points of bounded height. By loc. cit., it suffices to show that Z m ( s ) is absolutely convergent for Re s > m and that Z m ( s ) (cid:0) s − m (cid:1) b ( d,m ) admits an extension to a holomorphic function on Re s ≥ m which is not zero at s = m . In order to do this, we will apply the version of thePoisson summation formula given by Bourqui [4, Thm. 3.35, p. 64]. Formally ap-plying this version with G = T ( A K ), H = T ( K ), dg = dµ , dh the discrete measureon T ( K ) and F ( t ) = φ m ( t ) H ( t ) s for some s ∈ C with Re s > m gives Z m ( s ) = 1vol( T ( A K ) /T ( K )) X χ ∈ ( T ( A K ) /T ( K )) ∧ b H ( φ m , χ ; − s ) . (5.1)5.2. Analytic properties of Fourier transforms.Lemma 5.1.
For any place v ∈ Val( K ) , any character χ v of T ( K v ) and any ǫ > , the local Fourier transform b H v ( φ m,v , χ v ; − s ) is absolutely convergent and isbounded uniformly (in terms of ǫ and v ) on Re s ≥ ǫ .Proof. Let Re s ≥ ǫ . Since | b H v ( φ m,v , χ v ; − s ) | ≤ Z T ( K v ) (cid:12)(cid:12)(cid:12)(cid:12) φ m,v ( t v ) χ v ( t v ) H v ( t v ) s (cid:12)(cid:12)(cid:12)(cid:12) dµ v ≤ b H v (1 , − ǫ ) , it suffices to prove that b H v (1 , − ǫ ) is convergent. For v | ∞ , this follows from [3,Prop. 2.3.2, p. 614], so assume that v ∤ ∞ . The following argument is essentiallythe one in [3, Rem. 2.2.8, p. 613], but we fill in the details for the sake of clarity.Since H v and dµ v are T ( O v )-invariant and R T ( O v ) dµ v = 1, we have b H v (1 , − ǫ ) = Z T ( O v ) H v ( t v ) ǫ dµ v = X t v ∈ T ( K v ) /T ( O v ) H v (cid:0) t v (cid:1) ǫ . AMPANA POINTS 17
Now, by Lemma 2.14(i), T ( K v ) /T ( O v ) can be identified with a sublattice of finiteindex in X ∗ ( T v ), and this sublattice coincides with X ∗ ( T v ) when v is unramifiedin L/K . Then we see that, interpreting H v as a function on X ∗ ( T v ), we have X t v ∈ T ( K v ) /T ( O v ) H v (cid:0) t v (cid:1) ǫ ≤ X n v ∈ X ∗ ( T v ) H v ( n v ) ǫ , and the proof of [3, Thm. 2.2.6, p. 611] and Proposition 3.6 give X n v ∈ X ∗ ( T v ) H v ( n v ) ǫ = (cid:18) − q dǫv (cid:19) Y w | v (cid:18) − q ǫw (cid:19) − , so we deduce that b H v (1 , − ǫ ) is convergent, and this concludes the proof. (cid:3) Lemma 5.2.
For any v ∈ Val( K ) , the local Fourier transform b H v ( φ m,v , − s ) isnon-trivial for all s ∈ R > .Proof. The proof is analogous to the proof of [20, Lem. 5.1, p. 2575]. (cid:3)
Lemma 5.3.
Let L = E be a Galois extension of K . For any place v ∈ Val( K ) ,let χ v be a character of T ( K v ) which is non-trivial on K v . Then b H v ( φ m,v , χ v ; − s ) = 0 . Proof.
Since φ m,v and H v are K v -invariant, the result follows by character orthog-onality. (cid:3) Corollary 5.4.
Let L = E be a Galois extension of K , and let χ be an automor-phic character of T . If χ
6∈ U for U as in Definition 4.16, then b H ( φ m , χ ; − s ) = 0 . Lemma 5.5.
Let v ∤ ∞ be a non-archimedean place of K unramified in L/K , andlet χ be an automorphic character of T which is unramified at v . Then we have b H v (1 , χ v ; − s ) = (cid:18) − q dsv (cid:19) Y w | v (cid:18) − χ w ( π w ) q sw (cid:19) − = L v ( χ, s ) ζ K,v ( ds ) − . Proof.
The result follows from [3, Thm. 2.2.6, p. 611] and Proposition 3.6. (cid:3)
Definition 5.6.
Given a vector u = ( u , . . . , u r ) ∈ N r , define f r,n,u ( x . . . , x r ) tobe the sum of all monomials of degree n in the r variables x , . . . , x r with respectto the weighting induced by u , i.e. f r,n,u ( x , . . . , x r ) = X P ri =1 u i a i = na i ∈ Z ≥ ∀ i x a . . . x a r r . Set f r,n ( x , . . . , x r ) = f r,n,u ( x , . . . , x r )for u = (1 , . . . , ∈ N r . Proposition 5.7.
Let v S ′ ( ω ) be a non-archimedean place of K , and let χ ∈ U .Let w , . . . , w r ∈ Val( L ) be the places of L over v . Let u i be the inertia degree of w i over v for each i = 1 , . . . , r . Set c χ,v,n = f r,n,u ( χ v ( π w ) , . . . , χ w r ( π w r )) . Then, for Re s > , we have b H v ( φ m,v ; χ v ; − s ) = 1 + ∞ X n = m c χ,v,n − c χ,v,n − d q nsv . Proof.
Let s ∈ C with Re s >
0. As χ ∈ U and v S ′ ( ω ), it follows that χ isunramified at v . Then, expanding geometric series, we have L v ( χ, s ) = r Y i =1 (cid:18) − χ w i ( π w i ) q u i sv (cid:19) − = 1 + ∞ X n =1 c χ,v,n q nsv , so, by Lemma 5.5, we obtain b H v (1 , χ v ; − s ) = 1 + ∞ X n =1 c χ,v,n − c χ,v,n − d q nsv . On the other hand, we may write b H v (1 , χ v ; − s ) = Z T ( K v ) χ v ( t v ) H v ( t v ) s dµ v = ∞ X n =0 q nsv Z H v ( t v )= q nv χ v ( t v ) dµ v , so, comparing these expressions, we see for n ≥ c χ,v,n − c χ,v,n − d = Z H v ( t v )= q nv χ v ( t v ) dµ v . Since v S ′ ( ω ), we have φ m,v ( t v ) = 1 if and only if H v ( t v ) = 1 or H v ( t v ) ≥ q mv , so b H v ( φ m,v , χ v ; − s ) = 1 + ∞ X n = m c χ,v,n − c χ,v,n − d q nsv . (cid:3) Regularisation.
Now that we have expressions for the local Fourier trans-forms at all but finitely many places, our goal is to find “regularisations” for theglobal Fourier transforms, i.e. functions expressible as Euler products whose con-vergence is well-understood and whose local factors approximate the local Fouriertransforms well (as expansions in q v ) at all but finitely many places. As in [3], [20]and [27], we will construct our regularisations from L -functions. Proposition 5.8.
Let G be a subgroup of S d which acts freely and transitively on { , . . . , d } , and let m ≥ be a positive integer. Let S m act upon G m by permutationof coordinates, and let G act on G m /S m by right multiplication of every elementof a representative m -tuple. Set S ( G, m ) = ( G m /S m ) /G .(i) If d is coprime to m , then we have f d,m ( x , . . . , x d ) = X ( g ,...,g m ) ∈ S ( G,m ) d X i =1 x g ( i ) . . . x g m ( i ) . (5.2) (ii) If d is prime and m = kd , then we have f d,m ( x , . . . , x d ) + ( d − x k . . . x kd = X ( g ,...,g m ) ∈ S ( G,m ) d X i =1 x g ( i ) . . . x g m ( i ) . (5.3) AMPANA POINTS 19
Proof.
Let d and m be coprime with m ≥
2. For ( g , . . . , g m ) ∈ G m , set φ ( g ,...,g m ) ( x , . . . , x d ) = d X i =1 x g ( i ) . . . x g m ( i ) . First, we claim that, if ( g , . . . , g m ) = ( h , . . . , h m ) in S ( G, m ), then φ ( g ,...,g m ) ( x , . . . , x d ) = φ ( h ,...,h m ) ( x , . . . , x d ) . From this, it will follow that the sum on the right-hand side of (5.2) is well-definedand contains every monomial of degree m at least once. Note that ( g , . . . , g m ) =( h , . . . , h m ) if and only if { h , . . . , h m } = { g g, . . . , gg m } as multisets for some g ∈ G . If the coordinates of ( h , . . . , h m ) ∈ G m are a permutation of those of( g , . . . , g m ) ∈ G m , then x h ( i ) . . . x h m ( i ) = x g ( i ) . . . x g m ( i ) for any i ∈ { , . . . , d } , thus φ ( g ,...,g m ) ( x , . . . , x d ) = φ ( h ,...,h m ) ( x , . . . , x d ) . If ( h , . . . , h m ) = ( g g, . . . , g m g ) for some g ∈ G , then for any i ∈ { , . . . , d } , wehave x g g ( i ) . . . x g m g ( i ) = x g ( j ) . . . x g m ( j ) for the unique j ∈ { , . . . , d } with g ( i ) = j , so φ ( g ,...,g m ) ( x , . . . , x d ) = φ ( g g,...,g m g ) ( x , . . . , x d ) = φ ( h ,...,h m ) ( x , . . . , x d ) . The claim now follows. It now suffices to prove that φ ( g ,...,g m ) ( x , . . . , x d ) and φ ( h ,...,h m ) ( x , . . . , x d ) share no common summand if ( g , . . . , g m ) = ( h , . . . , h m ),and that no monomial appears twice in any φ ( g ,...,g m ) ( x , . . . , x d ), as then everydegree- m monomial appears at most once on the right-hand side of (5.2).Since each φ ( g ,...,g m ) ( x , . . . , x d ) is invariant under the right action of G on( g , . . . , g m ) and reordering of the g i , we may take g = 1 G without loss of gen-erality. Suppose that φ (1 G ,g ,...,g m ) ( x , . . . , x d ) and φ (1 G ,h ,...,h m ) ( x , . . . , x d ) share acommon summand, i.e. x i x g ( i ) . . . x g m ( i ) = x j x h ( j ) . . . x h m ( j ) for some i, j ∈ { , . . . , d } . This is equivalent to the equality of multisets { i, g ( i ) , . . . , g m ( i ) } = { j, h ( j ) , . . . , h m ( j ) } . If i = j , we have { g ( i ) , . . . , g m ( i ) } = { h ( j ) , . . . , h m ( j ) } = { h ( i ) , . . . , h m ( i ) } , and by the freeness of the action of G on { , . . . , d } , we have ( g , . . . , g m ) =( h , . . . , h m ) up to reordering, i.e. the m -tuple (1 G , h , . . . , h m ) is a permutationof the m -tuple (1 G , g , . . . , g m ). If i = j , we may take g ( i ) = j without loss ofgenerality (note that g = 1 G in this case), and we obtain the equality of multisets { j, h ( j ) , . . . , h m ( j ) } = { g ( i ) , h g ( i ) , . . . , h m g ( i ) } . Once again, by freeness of the action of G on { , . . . , d } , we get that { , g , . . . , g m } = { g , h g , . . . , h m g } as multisets, but then we see that, up to permuting coordinates, we have(1 G , g , . . . , g m ) = (1 G g , h g , . . . , h m g ) , i.e. the m -tuples belong to the same orbit under the right action of G . It followsthat φ ( g ,...,g m ) ( x , . . . , x d ) and φ ( h ,...,h m ) ( x , . . . , x d ) share no common summandif ( g , . . . , g m ) = ( h , . . . , h m ) in S ( G, m ) and that a monomial appears twice in φ ( g ,...,g m ) ( x , . . . , x d ) if and only if we have { G , g , . . . , g m } = { G g r , g g r , . . . , g m g r } (5.4)as multisets for some r ∈ { , . . . , m } with g r = 1 G . Without loss of generality wemay take r = 2 in (5.4), so g = 1 G and it becomes { G , g , . . . , g m } = { g , g , g g , . . . , g m g } . This is equivalent to the multiset { G , g , . . . , g m } being closed under right mul-tiplication by g . In particular, it must contain all powers of g . Let n be theorder of g . Then n | d by Lagrange’s theorem, and the multiset { , g , . . . , g m } contains the set { G , g , g , . . . , g n − } . Further, it must contain all of the sets { g s , g s g , g s g . . . , g s g n − } for s = 3 , . . . , m . The sets corresponding to g s and g s are not disjoint if and only if g s = g t g s for some t ∈ N , in which case theyare equal, so { G , g , . . . , g m } as a multiset can be written as a disjoint union ofsets of size n . Then n | m , but n | d and d and m are coprime, hence g = 1 G ,contradicting the assumption g = 1 G . We conclude that no monomial appearstwice in φ ( g ,...,g m ) ( x , . . . , x d ). Hence we have proved the first result.Let now d = p a prime, and let m = kp for some k ∈ N . By the above, the sumon the right-hand side of (5.3) contains every degree- kp monomial in the variables x , . . . , x p at least once, no two summands of the outer sum share a commonsummand and a monomial appears twice in φ ( G ,g ...,g kp )( x , . . . , x p ) if and only if { G , g , . . . , g kp } = r [ i =1 { h i , h i g, . . . , h i g n − } as sets for some elements h , . . . , h r of G and some non-identity element g ∈ G .Since G is of prime order, such an element g generates G , so( g , . . . , g kp ) = (1 G , g, . . . , g p − , . . . , G , g, . . . , g p − ) . (5.5)Further, the right-hand side of (5.5) is independent of the choice of g . Letting g bea generator of G , we conclude that φ (1 G ,g,...,g p − ,..., G ,g,...,g p − ) ( x , . . . , x p ) is the onlyone of the polynomials φ ( g ,...,g kp )( x , . . . , x p ), ( g , . . . , g kp ) ∈ S ( G, kp ) in which amonomial appears twice. In this polynomial, the only monomial which appears is x k . . . x kp , and so it appears p times. (cid:3) Remark 5.9.
It follows immediately from Proposition 5.8 that S ( G, m ) = ( d (cid:0) d + m − d − (cid:1) if d and m are coprime , d (cid:0)(cid:0) d + m − d − (cid:1) − (cid:1) + 1 if d is prime and d divides m ,since the number of polynomials of degree m in d variables is (cid:0) d + m − d − (cid:1) .For the rest of this section, let L = E be Galois over K with Galois group G ,and assume that m is coprime to d if d is not prime. AMPANA POINTS 21
Lemma 5.10.
Let v S ′ ( ω ) be a non-archimedean place which is totally split in E/K , let χ ∈ U and define F ′ m,χ,v ( s ) = Q ( g ,...,g m ) ∈ S ( G,m ) L v ( χ g . . . χ g m , ms ) Q ( h ,...,h m − d ) ∈ S ( G,m − d ) L v ( χ h . . . χ h m − d , ms ) . Then we have F ′ m,χ,v ( s ) = Y ( g ,...,g m ) ∈ S ′ ( G,m ) L v ( χ g . . . χ g m , ms ) , where S ′ ( G, m ) = { ( g , . . . , g m ) ∈ S ( G, m ) : { g , . . . , g m } ≤ d − } . .Proof. Let G = { g , . . . , g d } . First, we show that every factor of the denominatorappears on the numerator. Let ( h , . . . , h m − d ) ∈ S ( G, m − d ). Then we claim that L v (cid:0) χ h . . . x h m − d , ms (cid:1) = L v (cid:0) χ h . . . x h m − d χ g . . . χ g d , ms (cid:1) . Since G acts freely and transitively on the places w , . . . , w d of E over v , we have { χ g w i ( π w i ) , . . . , χ g d w i ( π w i ) } = { χ w ( π w ) , . . . , χ w d ( π w d ) } for any i = 1 , . . . , d . Since χ v is trivial on K v = T ( O v ), we have from Note 3.1that Q di =1 χ w i ( π w i ) = χ v (1) = 1, so ( χ g . . . χ g d ) w ( π w ) = 1 for all w | v . Then theequality follows.It now suffices to show that, for ( h , . . . , h m − d ) = (cid:0) h ′ , . . . , h ′ m − d (cid:1) , we have( h , . . . , h m − d , g , . . . , g d ) = (cid:0) h ′ , . . . , h ′ m − d , g , . . . , g d (cid:1) . If not, then { h ′ , . . . , h ′ m − d , g , . . . , g d } = { gh , . . . , gh m − d , gg , . . . , gg d } as mul-tisets for some g ∈ G . Since we have { gg , . . . , gg d } = { g , . . . , g d } , this im-plies that { h ′ , . . . , h ′ m − d } = { gh , . . . , gh m − d } as multisets, but then we have( h , . . . , h m − d ) = (cid:0) h ′ , . . . , h ′ m − d (cid:1) , which is false. (cid:3) Remark 5.11.
It follows from the proof of Lemma 5.10 that S ′ ( G, m ) = S ( G, m ) − S ( G, m − d ). Combining this with Remark 5.9, we obtain S ′ ( G, m ) = 1 d (cid:18)(cid:18) d + m − d − (cid:19) − (cid:18) m − d − (cid:19)(cid:19) = b ( d, m ) . Note that the term d (cid:0) m − d − (cid:1) only appears when d ≤ m . Definition 5.12.
For all χ ∈ U , Re s > v ∤ ∞ , set F m,χ,v ( s ) = Y ( g ,...,g m ) ∈ S ′ ( G,m ) L v ( χ g . . . χ g m , ms ) , G m,χ,v ( s ) = b H v ( φ m,v , χ v ; − s ) F m,χ,v ( s ) , and define F m,χ ( s ) = Y v ∤ ∞ F m,χ,v ( s ) = Y ( g ,...,g m ) ∈ S ′ ( G,m ) L ( χ g . . . χ g m , ms ) , (5.6) G m,χ ( s ) = Y v ∤ ∞ G m,χ,v ( s ) . For any non-archimedean place v ∤ ∞ , write b H v ( φ m,v , χ v ; − s ) = ∞ X n =0 a χ,v,n q nsv , where a χ,v,n = R H v ( t v )= q nv φ m,v ( t v ) χ v ( t v ) dµ v , and write F m,χ,v ( s ) = 1 + ∞ X n =1 b χ,v,mn q mnsv for the expansion of F m,χ,v ( s ) as a multidimensional geometric series in q msv , so G m,χ,v ( s ) = P ∞ n =0 a χ,v,n q nsv P ∞ n =1 b χ,v,mn q mnsv = ∞ X n =0 d χ,v,n q nsv , where d χ,v,n is defined for all n ≥ d χ,v,n = a χ,v,n − ⌊ nm ⌋ X r =1 b χ,v,mr d χ,v,n − mr . (5.7)In particular, we have d χ,v,n = a χ,v,n for 0 ≤ n ≤ m − Corollary 5.13.
For v S ′ ( ω ) a non-archimedean place, we have d χ,v, = 1 and d χ,v,n = 0 for all n ∈ { , . . . , m } .Proof. Let c χ,v,n be as defined in Proposition 5.7. Since v S ′ ( ω ), we have from loc. cit. that a χ,v, = 1, a χ,v,n = 0 for n ∈ { , . . . , m − } and a χ,v,m = c χ,v,m − c χ,v,m − d . Then, by (5.7), we see that d χ,v, = a χ,v, = 1 and d χ,v,n = a χ,v,n = 0 for1 ≤ n ≤ m −
1. Further, we obtain d χ,v,m = a χ,v,m − b χ,v,m d χ,v, = c χ,v,m − c χ,v,m − d − b χ,v,m , so, to complete the proof, it suffices to show that b χ,v,m = c χ,v,m − c χ,v,m − d .Since E/K is Galois, all of the places w , . . . , w r of E over v share a commoninertia degree d v . Since χ v ( T ( O v )) = 1, it is unramified as a Hecke character atall of the w i (see Note 3.1), and for any g , . . . , g m ∈ G , so is χ g . . . χ g m . Then L v ( χ g . . . χ g m , ms ) = r Y i =1 (cid:18) − ( χ g . . . χ g m ) w i ( π w i ) q d v msv (cid:19) − = 1 + 1 q d v msv r X i =1 ( χ g . . . χ g m ) w i ( π w i ) + O q ( d v m +1) sv ! . (5.8)First, suppose that v is totally split in E/K . Then (5.8) gives L v ( χ g . . . χ g m , ms ) = 1 + φ ( g ,...,g m ) ( χ w ( π w ) , . . . , χ w d ( π w d )) q msv + O q ( m +1) sv ! . Since G acts freely and transitively on the w i , it follows from Proposition 5.8 andLemma 5.10 that b χ,v,m = c χ,v,m − c χ,v,m − d , and so d χ,v,m = 0.Now assume that v is not totally split in E/K . If gcd( d, m ) = 1, then c χ,v,m = c χ,v,m − d = 0 as d v | d implies d v ∤ m . If d is prime, then v is inert and wehave b H v ( φ m,v , χ v ; − s ) = 1 since T ( K v ) = T ( O v ). Then, in either case, we have c χ,v,m − c χ,v,m − d = 0, and (5.8) implies that b χ,v,m = 0, hence d χ,v,m = 0. (cid:3) AMPANA POINTS 23
Corollary 5.14.
For any χ ∈ U , we have b H ( φ m ; χ ; − s ) = Y v |∞ b H v (1 , χ v ; − s ) F m,χ ( s ) G m,χ ( s ) , where G m,χ ( s ) is holomorphic and uniformly bounded with respect to χ for Re s ≥ m and G m, (cid:0) m (cid:1) = 0 . In particular, b H ( φ m , χ ; − s ) possesses a holomorphic con-tinuation to the line Re s = m , apart from possibly at s = m . When χ = 1 , theright-hand side has a pole of order b ( d, m ) at s = m .Proof. By construction, b H ( φ m ; χ ; − s ) = Q v |∞ b H v (1 , χ v ; − s ) F m,χ ( s ) G m,χ ( s ). Wewill prove the stronger result that G m,χ ( s ) is holomorphic on Re s > m +1 anduniformly bounded with respect to both χ and ǫ on Re s ≥ m +1 + ǫ for all ǫ > v ∤ ∞ and s ∈ C with Re s = σ ≥ ǫ for some ǫ >
0, we have ∞ X n =0 (cid:12)(cid:12)(cid:12)(cid:12) a χ,v,n q nsv (cid:12)(cid:12)(cid:12)(cid:12) = ∞ X n =0 q nσv (cid:12)(cid:12)(cid:12)(cid:12)Z H v ( t v )= q nv φ m,v ( t v ) χ v ( t v ) dµ v (cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X n =0 q nσv Z H v ( t v )= q nv | φ m,v ( t v ) χ v ( t v ) | dµ v = Z T ( K v ) (cid:12)(cid:12)(cid:12)(cid:12) φ m,v ( t v ) χ v ( t v ) H v ( t v ) s (cid:12)(cid:12)(cid:12)(cid:12) dµ v , so, by Lemma 5.1, the series P ∞ n =0 a χ,v,n q nsv is absolutely convergent and bounded bya constant depending only on ǫ and v . Now, for any N ∈ N , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =0 a χ,v,n q nsv − N X n =0 a χ,v,n q nsv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X n =0 (cid:12)(cid:12)(cid:12)(cid:12) a χ,v,n q nǫv (cid:12)(cid:12)(cid:12)(cid:12) + N X n =0 (cid:12)(cid:12)(cid:12)(cid:12) a χ,v,n q nǫv (cid:12)(cid:12)(cid:12)(cid:12) , from which it follows that P ∞ n =0 a χ,v,n q nsv is also uniformly convergent, hence thefunction b H v ( φ m,v , χ v ; − s ) is holomorphic on Re s >
0. Then, we note that F m,χ,v ( s )is clearly holomorphic on Re s >
0, and we have1 | F m,χ,v ( s ) | = Y ( g ,...,g m ) ∈ S ′ ( G,m ) (cid:12)(cid:12) L v ( χ g ...g m , ms ) − (cid:12)(cid:12) ≤ (cid:18) q ǫv (cid:19) db ( d,m ) , hence G m,χ,v ( s ) is holomorphic on Re s > ǫ and v on Re s ≥ ǫ .To conclude the result, it suffices to prove that there exists N ∈ N such that Y q v >N G m,χ,v ( s )is holomorphic and uniformly bounded with respect to χ on Re s ≥ m +1 + ǫ forall ǫ >
0. Let v S ′ ( ω ) be non-archimedean, and let Re s = σ ≥ m +1 + ǫ . From b H v ( φ m,v , χ v ; − s ) = (cid:18) − q dsv (cid:19) L v ( χ, s ) , and the definition of F m,χ,v ( s ), we have | a χ,v,n | ≤ d n , | b χ,v,n | ≤ ( b ( d, m ) d ) n . Then, by (5.7), it follows inductively that we have | d χ,v,n | ≤ n ( b ( d, m ) d ) n ≤ (2 b ( d, m ) d ) n . (5.9)Choose N > (2 b ( d, m ) d ) σ so that, for all places v ∤ ∞ with q v > N , we have v S ′ ( ω ). Now, any normally convergent infinite product is holomorphic (see[28, § Q q v >N G m,χ,v ( s ) converges normally if and only if X q v >N ∞ X n = m +1 d χ,v,n q nσv , converges, so, by (5.9), we need only check convergence of X q v >N q ( m +1) σv , which is clear. Then G m,χ ( s ) is holomorphic on Re s > m +1 . Further, for Re s ≥ m +1 + ǫ , we have the bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y q v >N G m,χ,v ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Y q v >N ∞ X n = m +1 b ( d, m ) dq m +1 + ǫv ! n ! , which is uniform with respect to χ . Now, as a convergent infinite product, G m, (cid:0) m (cid:1) is zero if and only if G m, ,v (cid:0) m (cid:1) = b H v ( φ m,v , − m ) F m, ,v ( m ) = 0 for some place v ∤ ∞ . However, b H v (cid:0) φ m,v , − m (cid:1) = 0 by Lemma 5.2, so we conclude that G m, (cid:0) m (cid:1) = 0. The orderof the pole of the right-hand side being b ( d, m ) follows from Theorem 2.20, since F m, ( s ) = ζ E ( ms ) b ( d,m ) . (cid:3) Note 5.15.
In constructing the regularisation F m,χ ( s ), one must ensure that b H v ( φ m,v , χ v ; − s ) F m,χ,v ( s ) = 1 + O q ( m +1) sv ! for all non-archimedean places v with q v is sufficiently large. As seen above, therestrictions on d , m and E ensure that this is automatic for all such places whichare not totally split, i.e. we only need to approximate the local Fourier transformat totally split places not in S ′ ( ω ). Without these restrictions, one might have toapproximate the local Fourier transform at places of more than one splitting typesimultaneously, and to do this would require a new approach.Before applying our key theorems, we give one more result, which will be usedin order to move from the Poisson summation formula to the Tauberian theorem. Lemma 5.16. [20, Lem. 5.9, p. 2577]
Choose an R -vector space norm k · k on X ∗ ( T ∞ ) R and let L ⊂ X ∗ ( T ∞ ) R be a lattice. Let C be a compact subset of Re s ≥ m and let g : X ∗ ( T ∞ ) R × C → C be a function. If there exists ≤ δ < d − such that | g ( ψ, s ) | ≪ C (1 + k ψ k ) δ AMPANA POINTS 25 for all ψ ∈ X ∗ ( T ∞ ) R and s ∈ C , then the sum X ψ ∈L g ( ψ, s ) Y v |∞ b H v (1 , ψ ; − s ) is absolutely and uniformly convergent on C . Theorem 5.17.
Let Ω m ( s ) = Z m ( s ) (cid:18) s − m (cid:19) b ( d,m ) . Then Ω m ( s ) admits an extension to a holomorphic function on Re s ≥ m .Proof. Let s ∈ C with Re s > m . Combining the formal application (5.1) of thePoisson summation formula with Lemma 3.8 and Corollary 5.4 gives Z m ( s ) = Res s =1 ζ K ( s ) d Res s =1 ζ E ( s ) X χ ∈U b H ( φ m , χ ; − s ) . (5.10)By Corollary 5.14, the function φ m ( t ) H ( t ) s is L for Re s > m . To show that thisapplication is valid, we apply Bourqui’s criterion [4, Cor. 3.36, p. 64], by which itsuffices to show that the right-hand side of (5.10) is absolutely convergent, φ m ( t ) H ( t ) s iscontinuous and there exists an open neighbourhood U ⊂ T ( A K ) of the origin andstrictly positive constants C and C such that for all u ∈ U and all t ∈ T ( A K ),we have C (cid:12)(cid:12)(cid:12)(cid:12) φ m ( t ) H ( t ) s (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) φ m ( ut ) H ( ut ) s (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12)(cid:12)(cid:12) φ m ( t ) H ( t ) s (cid:12)(cid:12)(cid:12)(cid:12) . We may take U = K , and continuity is clear. It only remains to prove the absoluteconvergence. We will prove the stronger result that X χ ∈U b H ( φ m , χ ; − s ) (cid:18) s − m (cid:19) b ( d,m ) is absolutely and uniformly convergent on any compact subset C of the half-planeRe s ≥ m , which will both verify validity of the application and prove the theorem.Since K ⊂ K T is of finite index, the map (2.2) yields a homomorphism U → X ∗ ( T ∞ ) R , χ χ ∞ , with finite kernel N and image L a lattice of full rank. We obtain X χ ∈U b H ( φ m , χ ; − s ) (cid:18) s − m (cid:19) b ( d,m ) = X ψ ∈L Y v |∞ b H v (1 , ψ ; − s ) X χ ∈U χ ∞ = ψ Y v ∤ ∞ b H v ( φ m,v , χ v ; − s ) (cid:18) s − m (cid:19) b ( d,m ) , where the inner sum is finite. Then, for s ∈ C , we have X χ ∈U b H ( φ m , χ ; − s ) (cid:18) s − m (cid:19) b ( d,m ) ≪ X ψ ∈L Y v |∞ (cid:12)(cid:12)(cid:12) b H v (1 , ψ ; − s ) (cid:12)(cid:12)(cid:12) X χ ∈U χ ∞ = ψ Y v ∤ ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b H v ( φ m,v , χ v ; − s ) (cid:18) s − m (cid:19) b ( d,m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Now, for χ ∈ U , we deduce from the proof of Corollary 5.14 that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y v ∤ ∞ b H v ( φ m,v , χ v ; − s ) (cid:18) s − m (cid:19) b ( d,m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F m,χ ( s ) (cid:18) s − m (cid:19) b ( d,m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . In order to deduce the result from Lemma 5.16, it suffices to prove that, for each ψ ∈ L and some constant 0 ≤ δ < d − , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X χ ∈U χ ∞ = ψ F m,χ ( s ) (cid:18) s − m (cid:19) b ( d,m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ C (1 + k ψ k ) δ for k · k as in Definition 2.21. As K ⊂ K T is of finite index, there exists a constant Q > q ( χ ) < Q for all χ ∈ U . Since F m,χ ( s ) is a product of b ( d, m ) L -functions of Hecke characters evaluated at ms , it follows from Lemma 2.22 that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X χ ∈U χ ∞ = ψ F m,χ ( s ) (cid:18) s − m (cid:19) b ( d,m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ ε,C |N | · Q ε (1 + k ψ k ) ǫ for all for all ε > s ∈ C . The result now follows from Lemma 5.16. (cid:3) The leading constant.
In order to apply [3, Thm. 3.3.2, p. 624] and deduceTheorem 1.1 from Theorem 5.17, it only remains to show that Ω m (cid:0) m (cid:1) = 0. Definition 5.18.
Let U G be the subgroup of G -invariant elements of U , and set U = ( U [ m ] if d = 2 , U G ∩ U [ m ] otherwise. Lemma 5.19.
For any Galois extension of number fields
E/K , the subgroup U [ m ] ≤ ( T ( A K ) /T ( K )) ∧ is finite. In particular, U is a finite subgroup of U .Proof. By class field theory, U may be interpreted as a subset of Gal (cid:16) E ab S ′ ( ω ) /E (cid:17) ,where E ab S ′ ( ω ) is the maximal S ′ ( ω )-unramified abelian extension of E , hence U [ m ]is in bijection with a subset of Hom (cid:16) Gal (cid:16) E ab S ′ ( ω ) /E (cid:17) , µ m (cid:17) , which is finite. (cid:3) Lemma 5.20. (i) The characters χ ∈ U contributing to the pole of Z m ( s ) of order b ( d, m ) at s = m are precisely the characters χ ∈ U such that G m,χ (cid:0) m (cid:1) = 0 . AMPANA POINTS 27 (ii) Suppose that d = 2 . If d and m are coprime, then U = { } . If d is primeand m is a multiple of d , then U = { χ ∈ U : χ d = 1 } .Proof. From Theorem 2.20 and Corollary 5.14, χ ∈ U contributes to the poleof Z m ( s ) at s = m if and only if each factor of F m,χ ( s ) in (5.6) equals ζ E ( ms )and G m,χ (cid:0) m (cid:1) = 0. Denoting by ψ the Hecke character associated to χ , thismeans precisely that G m,χ (cid:0) m (cid:1) = 0 and, for each ( g , . . . , g m ) ∈ S ′ ( G, m ), we have( ψ g . . . ψ g m ) v = 1 for all v ∤ ∞ , which is equivalent by strong approximation [11,Thm., p. 67] to ψ g . . . ψ g m = 1. By Note 3.1, this holds if and only if χ g . . . χ g m = 1 for all ( g , . . . , g m ) ∈ S ′ ( G, m ). (5.11)To conclude the first part, it only remains to show that (5.11) holds if and only if χ ∈ U . Taking ( g , . . . , g m ) = (1 , . . . ,
1) in (5.11), we obtain χ m = 1. If d = 2,then S ′ ( G, m ) = (1 , . . . , g , . . . , g m ) =( g, , . . . ,
1) for any g ∈ G , we obtain χ m − χ g = 1, so χ m = 1 and χ = χ g for all g ∈ G . Conversely, if χ m = 1 and χ = χ g for all g ∈ G , then (5.11) holds.Let now d = 2, χ ∈ U , v S ′ ( ω ) and w | v . We have ψ w ( π w ) = ψ gw ( π w ) = ψ gw ( π gw ) for all g ∈ G . Since Q w | v ψ w ( π w ) = 1 (see Note 3.1) and G acts tran-sitively on the places of E over v , we obtain ψ dw ( π w ) = 1, hence χ d = 1. On theother hand, ψ mw ( π w ) = 1. For d and m coprime, we conclude that ψ w = 1 for all w | v , so ψ = 1 by strong approximation, hence χ = 1. (cid:3) Proposition 5.21.
The limit Ω m (cid:18) m (cid:19) = lim s → m (cid:18) s − m (cid:19) b ( d,m ) X χ ∈U b H ( φ m , χ ; − s ) is non-zero.Proof. We have X χ ∈U b H ( φ m , χ ; − s ) = X χ ∈U Z T ( A K ) φ m ( t ) χ ( t ) H ( t ) s dµ = Z T ( A K ) φ m ( t ) H ( t ) s X χ ∈U χ ( t ) dµ. Let t ∈ T ( A K ). Note that, if there exists χ ′ ∈ U with χ ′ ( t ) = 1, then X χ ∈U χ ( t ) = X χ ∈U χχ ′ ( t ) = χ ′ ( t ) X χ ∈U χ ( t ) , so P χ ∈U χ ( t ) = 0. Then we have X χ ∈U b H ( φ m , χ ; − s ) = |U | Z T ( A K ) U ,φm H ( t ) s dµ, where T ( A K ) U ,φ m = { t ∈ T ( A K ) : φ m ( t ) = χ ( t ) = 1 for all χ ∈ U } . For any χ ∈ U and non-archimedean place v S ′ ( ω ), comparing the expressionsof b H v ( φ m,v , χ v ; − s ) and F m,χ,v ( s ) = F m, ,v ( s ) in Definition 5.12 gives Z H v ( t v )=1 χ v ( t v ) dµ v = Z H v ( t v )=1 dµ v , Z H v ( t v )= q mv χ v ( t v ) dµ v = Z H v ( t v )= q mv dµ v , so χ v ( t v ) = 1 for all χ ∈ U whenever H v ( t v ) = 1 or H v ( t v ) = q mv .For each place v S ′ ( ω ), define the continuous function θ m,v : T ( K v ) → { , } , t v ( H ′ v ( t v ) = 1 or H ′ v ( t v ) = q mv , . Letting θ m,v be the indicator function of K v for v ∈ S ′ ( ω ), we define the function θ m : T ( A K ) → { , } , θ m (( t v ) v ) = Y v ∈ Val( K ) θ m,v ( t v ) . By the above, we deduce that T ( A K ) θ m ⊂ T ( A K ) U ,φ m , where T ( A K ) θ m = { t ∈ T ( A K ) : θ m ( t ) = 1 } . Then, by comparing limits along the real line, we see that it suffices to prove thatlim s → m (cid:18) s − m (cid:19) b ( d,m ) b H ( θ m , − s ) = 0 . It is easily seen that for any non-archimedean place v S ′ ( ω ), we have b H v ( θ m,v , − s ) = 1 + a ,v,m q msv for a χ,v,n as in Definition 5.12, so, as in Corollary 5.14, we may deduce that b H ( θ m , − s ) = ζ E ( ms ) b ( d,m ) G m ( s )for G m ( s ) a function holomorphic on Re s ≥ m . It also follows that G m (cid:0) m (cid:1) = 0,since b H v (cid:0) θ m,v , − m (cid:1) = 0 analogously to Lemma 5.2. Then the result follows. (cid:3) Corollary 5.22.
We have Ω (cid:18) m (cid:19) = Res s =1 ζ K ( s ) d Res s =1 ζ E ( s ) lim s → m (cid:18) s − m (cid:19) b ( d,m ) X χ ∈U b H ( φ m , χ ; − s ) = 0 . Proof of Theorem 1.1.
Since Ω (cid:0) m (cid:1) = 0, the result for S = S ( ω ) follows from [3,Thm. 3.3.2, p. 624] and Theorem 5.17, taking c ( ω , m, S ( ω )) to be m Res s =1 ζ K ( s )( b ( d, m ) − d Res s =1 ζ E ( s ) lim s → m (cid:18) s − m (cid:19) b ( d,m ) X χ ∈U b H ( φ m , χ ; − s ) . The result for S ⊃ S ( ω ) follows analogously upon redefining φ m,v to be identically1 for each v ∈ S \ S ( ω ) in Definition 4.9. (cid:3) Campana points
In this section we prove Theorem 1.4. We will be brief when the argument islargely similar to the case of weak Campana points, emphasising only the keydifferences. Fix a Galois extension
E/K of number fields with K -basis ω = { ω , . . . , ω d − } , let m ∈ Z ≥ and set T = T ω as in Section 3. AMPANA POINTS 29
Definition 6.1.
For each non-archimedean place v S ( ω ), let N ω ( x ) = Y w | v f w ( x )denote the v -adic decomposition of the norm form N ω associated to ω into irre-ducible polynomials f w ( x ) ∈ O v [ x ]. For each w | v , we define the functions e H w : T ( K v ) → R > , x max {| x i | deg f w v }| f w ( x ) | v ,ψ m,w : T ( K v ) → { , } , t v ( e H w ( t v ) = 1 or e H w ( t v ) ≥ q mv , . We then define the
Campana local indicator function ψ m,v : T ( K v ) → { , } , t v Y w | v ψ m,w ( t v ) . Setting ψ m,v = 1 for v ∈ S ( ω ), we then define the Campana indicator function ψ m : T ( A K ) → { , } , ( t v ) v Y v ∈ Val( K ) ψ m,v ( t v ) . If v S ′ ( ω ), then for each w | v , we also define the function σ m,w : T ( K v ) → { , } , t v ( e H w ( t v ) = 1 or e H w ( t v ) = q mv , σ m,v : T ( K v ) → { , } , t v Y w | v σ m,w ( t v ) . Letting σ m,v be the indicator function for K v for v ∈ S ′ ( ω ), we define the function σ m : T ( A K ) → { , } , ( t v ) v Y v ∈ Val( K ) σ m,v ( t v ) . Lemma 6.2.
The Campana O K,S ( ω ) -points of (cid:0) P d − K , ∆ ω m (cid:1) are precisely the rationalpoints t ∈ T ( K ) such that ψ m ( t ) = 1 .Proof. Taking coordinates t , . . . , t d − as in the proof of Lemma 4.11, we have e H w ( t ) = 1 | f w ( t , . . . , t d − ) | w = q v ( f w ( t ,...,t d − )) v = q n αw ( Z ( f w ) ,t ) v for all non-archimedean places v S ( ω ) and places w | v , where Z ( f w ) denotesthe Zariski closure of Z ( f w ) in P d − O K,S ( ω ) . (cid:3) Lemma 6.3.
For all v ∈ Val( K ) , the function ψ m,v is K v -invariant and on K v .Proof. For v ∈ S ( ω ) the result is trivial, so let v S ( ω ) and w | v . Since f w ( x · y ) = f w ( x ) f w ( y ) for x, y ∈ L , it follows as in the proof of Lemma 4.12 that e H w ( x · y ) ≤ e H w ( x ) e H w ( y ) for all x, y ∈ T ( K v ). Since H ′ v = Q w | v e H w , we have e H w ( K v ) = 1 for all w | v , hence it follows as in the proof of Corollary 4.14 that e H w and ψ m,w are K v -invariant. Since ψ m,v (1) = 1, we deduce that ψ m,v ( K v ) = 1. (cid:3) Proposition 6.4.
Given v S ′ ( ω ) and t v ∈ T ( K v ) , the image of t v in X ∗ ( T v ) is X w | v v (cid:16) e H w ( t v ) (cid:17) deg f w n w , with n w defined as in Proposition 3.4.Proof. Follows from Proposition 3.4. (cid:3)
Corollary 6.5.
For v S ′ ( ω ) non-archimedean with q v sufficiently large, χ anautomorphic character of T unramified at v and s ∈ C with Re s > , we have b H v ( ψ m,v , χ v ; − s ) = 1 + 1 q msv X w | v deg f w | m χ w ( π w ) m + O q ( m +1) sv ! . Proof.
Since χ v , H v and ψ m,v are T ( O v )-invariant and v S ′ ( ω ), we have b H v ( ψ m,v , χ v ; − s ) = Z T ( K v ) ψ m,v ( t v ) χ v ( t v ) H v ( t v ) s dµ v = X t v ∈ T ( K v ) /T ( O v ) ψ m,v (cid:0) t v (cid:1) χ v (cid:0) t v (cid:1) H v (cid:0) t v (cid:1) s = X n v ∈ X ∗ ( T v ) ψ m,v ( n v ) χ v ( n v ) e ϕ Σ ( n v ) s log q v = ∞ X r =0 γ χ,v,r q rsv , where γ χ,v,r = X n v ∈ X ∗ ( T v ) H v ( n v )= q rv ψ m,v ( n v ) χ v ( n v ) . Put d w = deg f w and let n v = P w | v α w n w ∈ X ∗ ( T v ) with min w { α w } = 0. ByProposition 3.4 and Note 3.1, we havelog q v H v ( n v ) = X w | v d w α w , χ v ( n v ) = Y w | v χ w ( π w ) d w α w ,ψ m,v ( n v ) = ( α w = 0 or α w ≥ md w for all w | v, ψ m,v ( n v ) = 0 whenever q v ≤ H v ( n v ) ≤ q m − v , hence γ χ,v,r = 0 for1 ≤ r ≤ m −
1. Further, we see that ψ m,v ( n v ) = 1 and H v ( n v ) = q mv if and only ifthere is exactly one place w | v such that α w = md w and α w = 0 for w = w , so γ χ,v,m = X w | v deg f w | m χ w ( π w ) m . Since | ψ m,v ( n v ) χ v ( n v ) | ≤
1, we deduce that | γ χ,v,r | ≤ { β , . . . , β d ∈ Z ≥ : d X i =1 β i = r } ≤ d r . Analogously to the proof of Corollary 5.14, we deduce for q v sufficiently large that X r = m +1 γ χ,v,r q rsv = O q ( m +1) sv ! . (cid:3) AMPANA POINTS 31
Proposition 6.6.
For all places v S ′ ( ω ) with q v sufficiently large, we have b H v ( ψ m,v , χ v ; − s ) = L v ( χ m , ms ) O q ( m +1) sv !! , Re s > . Proof.
Let v S ′ ( ω ) with q v sufficiently large as in Corollary 6.5. If v is totallysplit in E/K , then deg f w = 1 for all w | v , so Corollary 6.5 gives b H v ( ψ m,v , χ v ; − s ) = 1 + 1 q msv X w | v χ mw ( π w ) + O q ( m +1) sv ! , and so we deduce the equality, since L v ( χ m , ms ) = Y w | v (cid:18) − χ mw ( π w ) q msv (cid:19) − = 1 + 1 q msv X w | v χ mw ( π w ) + O q ( m +1) sv ! . Now let v have inertia degree d v > E/K . Then deg f w = d v | d for all w | v .If d and m are coprime, then d v ∤ m , hence γ χ,v,m = 0 and the result follows from L v ( χ m , ms ) = Y w | v (cid:18) − χ mw ( π w ) q mv s (cid:19) − = 1 + O (cid:18) q d v msv (cid:19) . If d is prime, then v is inert, so T ( O v ) = T ( K v ), b H v ( ψ m,v , χ v ; − s ) = 1, and L v ( χ m , ms ) = 1 − q dmsv = 1 + O q ( m +1) sv ! . (cid:3) Proposition 6.7.
For any χ ∈ U , we have b H ( ψ m , χ ; − s ) = Y v |∞ b H v (1 , χ v ; − s ) L ( χ m , ms ) e G m,χ ( s ) , where e G m,χ ( s ) is a function which is holomorphic on Re s ≥ m , e G m, (cid:0) m (cid:1) = 0 and b H ( ψ m , − s ) has a simple pole at s = m .Proof. Defining e G m,χ,v ( s ) = b H v ( ψ m,v ,χ v ; − s ) L v ( χ m ,ms ) for each place v ∤ ∞ , it follows as in theproof of Corollary 5.14 that e G m,χ,v ( s ) is holomorphic and bounded uniformly interms ǫ and v on Re s ≥ ǫ for all ǫ >
0. Since Proposition 6.6 gives e G m,χ,v ( s ) = 1 + O q ( m +1) sv ! , it follows as in the proof of Corollary 5.14 that e G m,χ ( s ) is holomorphic and uni-formly bounded with respect to χ for Re s ≥ m with e G m, (cid:0) m (cid:1) = 0. Then, since L (1 , ms ) = ζ E ( ms ) , we conclude from Theorem 2.20 that b H ( ψ m , − s ) has a simple pole at s = m . (cid:3) Definition 6.8.
For Re s ≫
0, define the functions e Z m : C → C , s X x ∈ P d − ( K ) ψ m ( x ) H ( x ) s , e Ω m = e Z m ( s ) (cid:18) s − m (cid:19) . The proofs of the following two results are analogous to those of their weakCampana counterparts, namely Theorem 5.17 and Lemma 5.20 respectively.
Theorem 6.9.
The function e Ω m ( s ) admits a holomorphic extension to Re s ≥ m . Lemma 6.10.
The characters χ ∈ U contributing to the simple pole of e Z m ( s ) at s = m are precisely the characters χ ∈ U [ m ] such that e G m,χ (cid:0) m (cid:1) = 0 . Proposition 6.11.
The limit lim s → m (cid:18) s − m (cid:19) X χ ∈U [ m ] b H ( ψ m , χ ; − s ) is non-zero.Proof. By the same reasoning as in the proof of Proposition 5.21, we have X χ ∈U [ m ] b H ( ψ m , χ ; − s ) = X χ ∈U [ m ] Z T ( A K ) ψ m ( t ) χ ( t ) H ( t ) s dµ = |U [ m ] | Z T ( A K ) U [ m ] ,ψm H ( t ) s dµ, where T ( A K ) U [ m ] ,ψ m = { t ∈ T ( A K ) : ψ m ( t ) = χ ( t ) = 1 for all χ ∈ U [ m ] } . Now, take χ ∈ U [ m ], v S ′ ( ω ) non-archimedean. If σ v ( t v ) = 1 for some t v ∈ T ( K v ), then the image of t v in X ∗ ( T v ) is of the form P w | v − α w n w , where each α w is either 0 or md v for d v the common inertia degree of the places of E over v , so χ v ( t v ) = Y w | v χ w ( π w ) d v α w = 1 , since each d v α w is 0 or m and χ = χ m = 1. Then χ v ( t v ) = 1 for all χ ∈ U [ m ]. Inparticular, we deduce that T ( A K ) σ m ⊂ T ( A K ) U [ m ] ,ψ m , where T ( A K ) σ m = { t ∈ T ( A K ) : σ m ( t ) = 1 } . Then it suffices to prove thatlim s → m (cid:18) s − m (cid:19) b H ( σ m , − s ) = 0 . Analogously to the proof of Proposition 6.7, we may deduce that b H ( σ m , − s ) = ζ E ( ms ) e G m ( s )for e G m ( s ) a function holomorphic on Re s ≥ m with e G m (cid:0) m (cid:1) = 0, so the resultfollows. (cid:3) Proof of Theorem 1.4.
Since e Ω m (cid:0) m (cid:1) = 0 by Proposition 6.11, the result for S = S ( ω ) now follows from [3, Thm. 3.3.2, p. 624] and Theorem 6.9, taking e c ( ω , m, S ( ω )) = m Res s =1 ζ K ( s ) d Res s =1 ζ E ( s ) lim s → m (cid:18) s − m (cid:19) X χ ∈U [ m ] b H ( ψ m , χ ; − s ) . The result for S ⊃ S ( ω ) follows analogously upon redefining ψ m,v to be identically1 for each v ∈ S \ S ( ω ). (cid:3) AMPANA POINTS 33 Comparison to Manin-type conjecture
In this section we compare the leading constant in Theorem 1.4 with the Manin–Peyre constant in the conjecture of Pieropan, Smeets, Tanimoto and V´arilly-Alvarado.7.1.
Statement of the conjecture.
Let (
X, D ǫ ) be a smooth Campana orbifoldover a number field K which is klt (i.e. ǫ α < α ∈ A ) and Fano (i.e. − ( K X + D ǫ ) is ample). Let ( X , D ǫ ) be a regular O K,S -model of (
X, D ǫ ) for somefinite set S ⊂ Val( K ) containing S ∞ (i.e. X regular over O K,S ). Let L = ( L, k · k )be an adelically metrised big line bundle with associated height function H L : X ( K ) → R > (see [3, § U ⊂ X ( K ) and any B ∈ R > , we define N ( U, L , B ) = { P ∈ U : H L ( P ) ≤ B } . Definition 7.1.
Let V be a variety over a field k of characteristic zero, and let A ⊂ V ( k ). We say that A is of type I if there is a proper Zariski closed subset W ⊂ V with A ⊂ W ( k ). We say that A is of type II if there is a normalgeometrically irreducible variety V ′ with dim V ′ = dim V and a finite surjectivemorphism φ : V ′ → V of degree ≥ A ⊂ φ ( V ′ ( k )). We say that A is thin ifit is contained in a finite union of subsets of V ( k ) of types I and II.We are now ready to give the statement of the conjecture. Conjecture 7.2. [27, Conj. 1.1, p. 3]
Suppose that L is nef and ( X , D ǫ )( O K,S ) is not thin. Then there exists a thin set Z ⊂ ( X , D ǫ )( O K,S ) and explicit positiveconstants a = a (( X, D ǫ ) , L ) , b = b ( K, ( X, D ǫ ) , L ) and c = c ( K, S, ( X , D ǫ ) , L , Z ) such that, as B → ∞ , we have N (( X , D ǫ )( O K,S ) \ Z, L , B ) ∼ cB a (log B ) b − . Interpretation for norm orbifolds.
The orbifold (cid:0) P d − K , ∆ ω m (cid:1) in Theorem1.4 is klt and Fano. It is smooth precisely when d = 2. The O K,S ( ω ) -model (cid:16) P d − O K,S ( ω ) , D ω m (cid:17) is regular. The Batyrev–Tschinkel height arises from an adelicmetrisation L of − K P d − = O ( d ). According to [27, § c (cid:16) K, S, (cid:16) P d − O K,S ( ω ) , D ω m (cid:17) , L , Z (cid:17) = 1 d τ (cid:0) K, S ( ω ) , (cid:0) P d − K , ∆ ω m (cid:1) , L , Z (cid:1) , where τ (cid:0) K, S ( ω ) , (cid:0) P d − K , ∆ ω m (cid:1) , L , Z (cid:1) = Z P d − ( K ) ǫ H ( t ) − m dτ P d − . Here, τ P d − is the Tamagawa measure defined in [20, Def. 2.8, p. 372], and P d − ( K ) ǫ denotes the topological closure of the Campana points inside P d − ( A K ). If oneassumes that weak approximation for Campana points holds for this orbifold (see[27, Question 3.8, p. 15]), it follows from the definition of τ P d − that τ = m Res s =1 ζ K ( s )Res s =1 ζ E ( s ) lim s → m (cid:18) s − m (cid:19) b H ( ψ m , − s ) . Given the assumption on weak approximation, the conjectural leading constant is c (cid:16) K, S, (cid:16) P d − O K,S ( ω ) , D ω m (cid:17) , L , Z (cid:17) = m Res s =1 ζ K ( s ) d Res s =1 ζ E ( s ) lim s → m (cid:18) s − m (cid:19) b H ( ψ m , − s ) . On the other hand, the leading constant given by Theorem 1.4 in this case is e c ( ω , m, S ( ω )) = m Res s =1 ζ K ( s ) d Res s =1 ζ E ( s ) lim s → m (cid:18) s − m (cid:19) X χ ∈U [ m ] b H ( ψ m , χ ; − s ) . We observe that our constant differs from the conjectural one in the potentialinclusion of non-trivial characters in the limit.7.3.
The quadratic case.
We now consider the case d = 2, in which the orbifoldin Theorem 1.4 is smooth. Here, forthcoming work of Nakahara and the author[22] shows that weak approximation for Campana points holds for any m ∈ Z ≥ .In Theorem 1.4, it is not clear that there are non-trivial characters contributingto the leading constant and whether their contribution is positive if so. However,we now exhibit an extension for which a non-trivial character contributes positivelyto the leading constant, and all contributing characters do so positively. Proposition 7.3.
Let K = Q (cid:0) √− (cid:1) , E = Q (cid:0) √− , √ (cid:1) and m = 2 . Choosethe K -basis ω = { , √− } of E . Then S ( ω ) = S ∞ , U [2] > , and for every χ ∈ U [2] , we have lim s → m b H ( ψ m , χ ; − s ) > .Proof. Writing a · b · √− as ( a, b ) and G = Gal( E, K ) as { G , g } , we have(1 , = (1 , , (1 , · (0 ,
1) = (0 , , (0 , = ( − , , G ((1 , , , G ((0 , , , g ((1 , , , g ((0 , , − , and clearly 1 = (1 , S ( ω ) = S ∞ . Note that N ω ( x, y ) = x − xy + y .Since Cl( E ) ∼ = Z / Z , the Hilbert class field M of E is quadratic over E . Weobtain the unramified Hecke character χ M of E , which is defined for all split w ∈ Val( E ) by χ M,w ( π w ) = − χ M istrivial on A ∗ K , it may be viewed inside U [2], hence U [2] > χ ∈ U [2]. To show that lim s → b H ( ψ , χ ; − s ) >
0, it suffices to show that b H v (cid:0) ψ ,v , χ v ; − (cid:1) > v ∈ Val( K ). If v | ∞ , then b H v (cid:0) ψ ,v , χ v ; − (cid:1) = b H v (cid:0) , − (cid:1) >
0, as χ v gives a continuous homomorphism from T ( K v ) /T ( O v ) ∼ = R > to µ , and R > has no proper open subgroups. If v is inert, then we have b H v (cid:0) ψ ,v , χ v ; − (cid:1) = 1. If v is split, then N ω ( x, y ) = ( x − θ y )( x − θ y ) for θ , θ ∈O v roots of z − z +1. By Proposition 6.4, we have H v = H ′ v if and only if there areno ( a, b ) ∈ ( K ∗ v ) with min { v ( a ) , v ( b ) } = 0 such that v ( a − θ y ) , v ( a − θ y ) ≥ v ( a − θ y ) , v ( a − θ y ) ≥
1, then we deduce from the equalities θ ( a − θ y ) − θ ( a − θ y ) = ( θ − θ ) a, ( a − θ y ) − ( a − θ y ) = ( θ − θ ) b, that v ( a ) , v ( b ) ≥ − v ( θ − θ ). Since min { v ( a ) , v ( b ) } = 0, we have H ′ v = H v if and only if v ( θ − θ ) ≥
1. Since ( θ − θ ) = −
3, the only such place is theunique place v of K above 3, and v ( θ − θ ) = 1.For any split place v = v , we have K v = T ( O v ) and ψ ,v = φ ,v , so b H v (cid:18) ψ ,v , χ v ; − (cid:19) = 1 + ∞ X n =2 c χ,v,n − c χ,v,n − q n v AMPANA POINTS 35 by Proposition 5.7. In fact, for w and w the places of E over v , we have χ w ( π w ) = χ w ( π w ) − ∈ { , − } , hence c χ,v,n − c χ,v,n − = 2 χ w ( π w ) n , so b H v (cid:18) ψ ,v , χ v ; − (cid:19) = 1 + ∞ X n =2 χ w ( π w ) n q n v = 1 + 2 q v − q − v χ w ( π w ) ! > . It only remains to check that b H v (cid:0) ψ ,v , χ v ; − (cid:1) >
0. We will make use of thefollowing property of valuations: v ( α + β ) ≥ min { v ( α ) , v ( β ) } , with equality if v ( α ) = v ( β ). (7.1)Assume that, for a, b ∈ (cid:0) K ∗ v (cid:1) as above, we have v ( a − θ b ) ≥
2. We claim that v ( a − θ b ) = 1. First, we deduce from (7.1) that2 = v ( a − θ b ) = v (( a − θ b ) + ( θ − θ ) b ) ≥ min { v ( a − θ b ) , v (( θ − θ ) b ) } , with equality if v ( a − θ b ) = v (( θ − θ ) b ). Since v (cid:0) (2 θ i − (cid:1) = v ( −
3) =2, we have v (2 θ i −
1) = 1, hence v ( θ i ) = v (2 θ i ) = v (1) = 0 by (7.1), somin { v ( a ) , v ( θ b ) } = min { v ( a ) , v ( b ) } = 0. Then, since v ( a − θ b ) ≥
2, itfollows that v ( a ) = v ( θ b ). We deduce that v ( a ) = v ( θ b ) = v ( b ), so v ( a ) = v ( b ) = min { v ( a ) , v ( b ) } = 0. Since v ( θ − θ ) = 1, we have v (( θ − θ ) b ) = 1.It follows that v ( a − θ b ) = 1 by (7.1).We deduce that ψ ,v ( t v ) = 1 if and only if t v ∈ K v , hence b H v (cid:18) ψ ,v , χ v ; − (cid:19) = Z K v dµ v > K v ⊂ T ( O v ) is of finite index for all v ∈ Val( K ). (cid:3) Possible thin sets.
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Department of Mathematical Sciences, University of Bath, Claverton Down,Bath, BA2 7AY, UK
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