Campana points, Height zeta functions, and log Manin's conjecture
aa r X i v : . [ m a t h . N T ] F e b CAMPANA POINTS, HEIGHT ZETA FUNCTIONS,AND LOG MANIN’S CONJECTURE
SHO TANIMOTO
Abstract.
This is a report of the author’s talk at RIMS workshop 2020 Problems andProspects in Analytic Number Theory held online on Zoom. We discuss a recent formulationof log Manin’s conjecture for klt Campana points and an approach to this conjecture usingthe height zeta function method. Introduction
One of fundamental tools in diophantine geometry is the notion of height functions andthis height function measures the geometric and arithmetic complexities of rational points onan algebraic variety. These are crucial to various finiteness results in diophantine geometrysuch as Mordell-Weil theorem, Siegel’s theorem, Mordell-Faltings’ theorem, and so on. Oneof basic properties of height functions is the Northcott property which claims that for aheight function associated to an ample divisor, the set of rational points whose height is lessthan T is finite. Thus one may consider the counting function of rational points of boundedheight, and one natural question is the asymptotic formula for such a counting function when T goes to infinity.Around the late 1980’s, Yuri Manin and his collaborators proposed a general frameworkto understand this asymptotic formula in terms of geometric and arithmetic invariants ofthe underlying projective variety, and this leads to Manin’s conjecture whose formulationis developed in a series of papers [FMT89], [BM90], [Pey95], [BT98a], [Pey03], [Pey17],and [LST18]. One of fertile testing grounds for this conjecture is a class of equivariantcompactifications of homogeneous spaces, and there are mainly two methods available, i.e.,the method of mixing and the height zeta function method.Mixing is a concept from ergodic theory, and this idea has been successfully used toprove equidistribution of rational points on homogeneous spaces acted by semi-simple groups([GMO08] and [GO11]). The height zeta function method can be applied to a variety of equi-variant compactifications of connected algebraic groups including, but not limited to, gener-alized flag varieties ([FMT89]), toric varieties ([BT96] and [BT98a]), equivariant compact-ifications of vector groups ([CLT02]), wonderful compactifications of semi-simple groups ofadjoint type ([STBT07]), and biequivariant compactifications of unipotent groups ([ST16]).The height zeta function method also has its advantage to studying the counting problemof integral points associated to a reduced boundary divisor, and this has been implementedfor equivariant compactifications of vector groups ([CLT12]), toric varieties ([CLT10b]), won-derful compactifications of semi-simple groups of adjoint type ([TBT13] and [Cho19]), andbiequivariant compactificaitons of the Heisenberg group ([Xia20]). These results suggestthat there should be an analogous formulation of log Manin’s conjecture for integral points, Date : February 26, 2021. owever certain subtleties of geometric and arithmetic nature prevent a general formulationof such a conjecture.Campana and subsequently Abramovich proposed the notion of Campana points in [Cam05]and [Abr09] and this notion interpolates between rational points and integral points. Thecounting problem of Campana points has been originally featured in [VV11], [BVV12], and[VV12]. Recently many mathematicians started to look at this problem and develop a se-ries of results, attested by [BY20], [PSTVA20], [PS20], [Xia20], and [Str20]. In [PSTVA20],Pieropan, Smeets, V´arilly-Alvarado, and the author initiated a systematic study of the count-ing problem for Campana points, and formulated a log Manin’s conjecture for klt Campanapoints. Then we confirmed this conjecture for equivariant compactifications of vector groupsusing the height zeta function method for vector groups which is developed by Chambert-Loirand Tschinkel in [CLT02] and [CLT12].In this survey paper, we discuss the formulation of log Manin’s conjecture for klt Campanapoints and applications of the height zeta function method to study this problem for variousequivariant compactifications of connected algebraic groups.Here is a plan of this paper: In Section 2, we review the notion of height functions. InSection 3, we introduce two definitions of (weak) Campana points. In Section 4, we discussa formulation of log Manin’s conjecture for klt Campana points. Finally in Section 5, wediscuss the height zeta function method and its applications to equivariant compactificationsof algebraic groups. Acknowledgements:
The author would like to thank Marta Pieropan, Arne Smeets, andTony V´arilly-Alvarado for collaborations helping to shape his perspective on log Manin’sconjecture for klt Campana points. The author would like to thank the organizers of RIMSworkshop Problems and Prospects in Analytic Number Theory for an opportunity to give atalk there. This work was supported by the Research Institute for Mathematical Sciences,an International Joint Usage/Research Center located in Kyoto University.Sho Tanimoto was partially supported by Inamori Foundation, by JSPS KAKENHI Early-Career Scientists Grant number 19K14512, and by MEXT Japan, Leading Initiative forExcellent Young Researchers (LEADER).2.
Height functions
In this section we review the notion of height functions and their basic properties. Themain references are [HS00] and [CLT10a], and they include different treatments of heightfunctions. In [HS00] height functions are introduced using the machinery of Weil heightmachine and some basic properties of height functions such as the Northcott property areproved. In [CLT10a], adelic metrizations are used to define height functions, and this defi-nition is frequently used in the literature in Manin’s conjecture. It is well-known that twodefinitions are essentially equivalent. See [HS00] for more details. In this paper, we employthe definition of height functions using adelic metrizations described in [CLT10a].Let us fix our notation: let F be any number field and O F be its ring of integers. Wedenote the set of places of F by Ω F , the set of archimedean places by Ω ∞ F , and the set ofnon-archimedean places by Ω < ∞ F . For any finite set S ⊂ Ω F containing Ω ∞ F , O F,S denotes he ring of S -integers. For each v ∈ Ω F , we denote the completion of F with respect to v by F v . When v is non-archimedean, we denote the ring of integers for F v by O v with maximalideal m v and residue field k v of size q v . We denote the adele ring of F by A F .For each v ∈ Ω F , F v is a locally compact subgroup and it comes with a self-Haar measured x v = µ v which is normalized in a way Tate did in [Tat67]. We define the absolute value | · | v on F v by requiring µ v ( xB ) = | x | v · µ v ( B ) . This normalization satisfies the product formula, i.e., for any x ∈ F × , we have Y v ∈ Ω F | x | v = 1See [CLT10a] for more details.Let F be a number field and v ∈ Ω F be a place of F . Let U be an open set of F nv inthe analytic topology. A complex valued function on U is smooth if it is C ∞ when v isarchimedean and it is locally constant when v is non-archimedean. This notion is local andextends to any v -adic analytic manifold.Let X be a smooth variety defined over F v and L be a line bundle on X . For each localpoint x ∈ X ( F v ), we denote the fiber of L at x by L x . Definition 2.1.
A smooth metric on L is a collection of metrics k · k : L x ( F v ) → R ≥ for all x ∈ X ( F v ) such that • for ℓ ∈ L x ( F v ) \ { } , k ℓ k > • for any a ∈ F v , x ∈ X ( F v ), and ℓ ∈ L x ( F v ), k aℓ k = | a | v k ℓ k , and; • for any open subset U ⊂ X ( F v ) and any non-vanishing section f ∈ Γ( U, L ), thefunction x
7→ k f ( x ) k is smooth.An integral model of a projective variety can be used to define a metric on it: Example 2.2.
Let X be a smooth projective variety defined over F v and L be a line bundleon X where v is non-archimedean. Suppose that we have a flat projective O v -scheme X and a line bundle L on X extending X and L . Let x ∈ X ( F v ) = X ( O v ). Then we define asmooth metric on L by insisting that for any ℓ ∈ L x ( F v ) k ℓ k ≤ ⇐⇒ ℓ ∈ L x ( O v ) . This metric is called as the induced metric by an integral model ( X , L ).Next we define adelic metrizations on a smooth projective variety defined over a numberfield F . Definition 2.3.
Let X be a smooth projective variety defined over a number field F and L be a line bundle on X . An adelic metrization on L is a collection of v -adic smooth metrics {k · k v } v ∈ Ω F on X such that there exist a finite set S of places including Ω ∞ F , a flat O F,S -projective model X , and a line bundle L on X extending X and L such that for any v S ,the metric k · k v is induced by ( X , L ). Note that two integral models are isomorphic outsideof finitely many places so that two adelic metrizations differ only at finitely many places.Finally we define the notion of height functions: efinition 2.4. Let X be a smooth projective variety defined over a number field F and L = ( L, {k · k v } ) be an adelically metrized line bundle on X . For each rational point x ∈ X ( F ), choose ℓ ∈ L x ( F ) and we define the height function H L : X ( F ) → R ≥ by H L ( x ) = Y v ∈ Ω F k ℓ k − v . This is well-defined due to the product formula mentioned above.Here is an example of height functions for the projective space:
Example 2.5.
Let X = P n and L = O X (1). We consider the standard integral model X = P n O F . For each non-archimedean place v ∈ Ω F , we let k · k v be the metric at v inducedby X . For any archimedean place v , we define a smooth metric at v by insisting k ℓ ( x ) k v = | ℓ ( x ) | v pP ni =0 | x i | v , where x = ( x : · · · : x n ) ∈ X ( F ) and ℓ ∈ H ( X, O X (1)). Then it is an easy exercise toprove that the height function associated to L with this adelic metrization is given by H ( x ) = Y v ∈ Ω < ∞ F max {| x | v , · · · , | x n | v } Y v ∈ Ω ∞ F p | x | v + · · · + | x n | v . When F = Q , we may assume that x i ’s are integers and gcd( x , · · · , x n ) = 1. In thissituation, the above formula reduces to H ( x ) = p | x | ∞ + · · · + | x n | ∞ . Let us mention a few basic properties of height functions:
Proposition 2.6.
Let X be a smooth projective variety defined over a number field F and L = ( L, {k · k v } ) be an adelically metrized line bundle on X . Then the following statementsare true: • Let L ′ be another adelically metrized line bundle associated to L . Then there existpositive constants C ≤ C such that for any x ∈ X ( F ) , we have C H L ′ ( x ) ≤ H L ( x ) ≤ C H L ′ ( x ); • Let B be the base locus of the complete linear series | L | . Then there exists a positiveconstant C > such that for any x ∈ ( X \ B )( F ) , we have H L ( x ) ≥ C ; • When L is ample, for any real number T > the set { x ∈ X ( F ) | H L ( x ) ≤ T } is a finite set. The last property is called as the Northcott property which is fundamental in diophantinegeometry and it is also foudational for Manin’s conjecture. For more details, see [HS00]. . Campana points
In this section, we review the notion of Campana points. Campana points were originallyconsidered by Campana for curves in [Cam05], and its higher dimensional analogue wasexplored by Abramovich in [Abr09]. One may consider Campana points as integral pointson Campana orbifolds developed by again Campana himself:
Definition 3.1.
Let F be an arbitrary field and X be a smooth projective variety definedover F . Let D ǫ = P α ∈A ǫ α D α be an effective Q -divisor on X with D α ’s irreducible anddistinct. We say ( X, D ǫ ) is a Campana orbifold if the following statements are true: • For any α ∈ A , a non-negative rational number ǫ α takes the form of1 − m α , where m α is a positive integer or + ∞ ; • the reduced divisor D = P α ∈A D α is a strict normal crossings divisor.We say a Campana orbifold ( X, D ǫ ) is Fano if − ( K X + D ǫ ) is ample.Let ( X, D ǫ ) be a Campana orbifold. Then ( X, D ǫ ) is a divisorial log terminal (dlt for short)pair in the sense of birational geometry. When ǫ α < α , ( X, D ǫ ) is a kawamata logterminal (klt for short) pair. See [KM98] for the definitions and their basic properties. Wesay a Campana orbifold ( X, D ǫ ) is klt if ǫ α < α ∈ A .To define the notion of Campana points, one needs to fix an integral model of a Campanaorbifold. Let ( X, D ǫ ) be a Campana orbifold defined over a number field F with D ǫ = P α ∈A ǫ α D α . Let S be a finite set of places including all archimedean places. A good integralmodel of ( X, D ǫ ) away from S is a flat projective O F,S -scheme X such that X is extending X and X is regular. Let D α be the Zariski closure of D α in X and let D ǫ = P α ∈A ǫ α D α .Let us fix a good integral model of a Campana orbifold ( X, D ǫ ) as above. Let A ǫ = { α ∈A | ǫ α = 0 } . We set X ◦ = X \ ∪ α ∈A ǫ D α . Let P ∈ X ◦ ( F ) be a rational point and v S be a non-archimedean place of F . Then we may consider P as an O v -point P v ∈ X ( O v ) byvaluative criterion for properness. Since P v
6⊂ D α for any α ∈ A ǫ , the pullback of D α via P v defines an ideal in O v . We denote its colength by n v ( D α , P ). When P ∈ D α for some α ∈ A ǫ , we formally set n v ( D α , P ) = + ∞ . The total intersection number is given by n v ( D ǫ , P ) = X α ∈A ǫ ǫ α n v ( D α , P ) . Now we are ready to define two notions of Campana points:
Definition 3.2.
We say P ∈ X ( F ) is a weak Campana O F,S -point on ( X , D ǫ ) if the followingstatements are true: • we have P ∈ ( X \ ∪ ǫ α =1 D α )( O F,S ), and; • for v S , if n v ( D ǫ , P ) >
0, then n v ( D ǫ , P ) ≤ X α ∈A ǫ n v ( D α , P ) ! − . We denote the set of weak Campana O F,S -points by ( X , D ǫ ) w ( O F,S ). efinition 3.3. We say P ∈ X ( F ) is a Campana O F,S -point on ( X , D ǫ ) if the followingstatements are true: • we have P ∈ ( X \ ∪ ǫ α =1 D α )( O F,S ), and; • for v S and for all α ∈ A ǫ with ǫ α < n v ( D α , P ) >
0, we have n v ( D α , P ) ≥ m α , where ǫ α = 1 − m α .A Campana O F,S -point is klt when the underlying Campana orbifold is a klt pair.We denote the set of Campana O F,S -points by ( X , D ǫ )( O F,S ). Then we have the followinginclusions: X ◦ ( O F,S ) ⊂ ( X , D ǫ )( O F,S ) ⊂ ( X , D ǫ ) w ( O F,S ) ⊂ X ( F ) , where X ◦ = X \ ( ∪ α ∈A ǫ D α ). When ǫ α = 0 for all α ∈ A , the rightmost two inclusions areequalities. When ǫ α = 1 for all α ∈ A ǫ , the leftmost two inclusions are equalities.Here is an example of klt Campana points: Example 3.4.
For simplicity, let us assume that F = Q and S = {∞} . Let X = P n and H = V ( x ) be a hyperplane. Let m be a positive integer and ǫ = 1 − /m . We define D ǫ = ǫH. We consider the standard integral model of X . Then a rational point x = ( x : · · · : x n ) ∈ X ( Q ) with x i ∈ Z and gcd( x , · · · , x n ) = 1 is a Campana Z -point if x = 0 or x = 0 andthe following statement is true: for any prime number p we have p | x = ⇒ p m | x . Any non-zero integer with this property is said to be m -full. When m = 2, it is said to besquarefull. 4. Log Manin’s conjecture
Let X be a smooth projective variety defined over a number field F and L = ( L, {k · k v } )be an adelically metrized line bundle on X . We consider the associated height function H L : X ( F ) → R > . When L is ample, this height function satisfies the Northcott property so that for any subset Q ⊂ X ( F ) and any positive real number T > N ( Q, L , T ) = { P ∈ Q | H L ( P ) ≤ T } . Manin’s conjecture predicts the asymptotic formula of the above function for an appropriate Q , and a natural question is to extend this conjecture to integral points and Campanapoints. In [PSTVA20], Pieropan, Smeets, V´arilly-Alvarado and the author formulated thislog version of Manin’s conjecture when the underlying Campana orbifold is a klt log Fanopair. In this section, we review a general formulation of this log Manin’s conjecture. .1. Two birational invariants.
Let X be a smooth projective variety defined over a field F . Let D , D are Q -divisors on X . We say D and D are numerically equivalent if for anycurve C ⊂ X , we have D .C = D .C . In this case we write D ≡ D . We define the spaceof Q -divisors up to numerical equivalence as N ( X ) Q = { D : Q -divisors } / ≡ . We set N ( X ) := N ( X ) Q ⊗ Q R . Then we define the cone of pseudo-effective divisors byEff ( X ) := the cone of effective Q -divisors ⊂ N ( X ) . Now we are ready to introduce two birational invariants which play central roles in Manin’sconjecture:
Definition 4.1.
Let (
X, D ǫ ) be a klt Campana orbifold defined over a field F and L be anample Q -divisor on X . We define the Fujita invariant or a -invariant by a ( X, D ǫ , L ) := inf { t ∈ R | tL + K X + D ǫ ∈ Eff ( X ) } . Next assume that a ( X, D ǫ , L ) >
0. Then we define the b -invariant by b ( F, X, D ǫ , L ) := codimension of the minimal face of Eff ( X )containing a ( X, D ǫ , L ) L + K X + D ǫ .It is explained in [PSTVA20, Section 3.6.2] that these invariants are birational invariants. Example 4.2.
Let (
X, D ǫ ) be a klt Fano orbifold defined over a field F and L = − ( K X + D ǫ ).Then we have a ( X, D ǫ , L ) = 1 , b ( F, X, D ǫ , L ) = ρ ( X ) = dim N ( X ) . Thin exceptional sets.
The notion of thin sets has been explored by Serre to studyGalois inverse problem, and it is also fundamental to Manin’s conjecture. Let us give thedefinition of thin sets for Campana points:
Definition 4.3.
Let (
X, D ǫ ) be a klt Campana orbifold defined over a number field F . Let S be a finite set of places of F including Ω ∞ F and we fix a good integral model away from S X →
Spec O F,S .A type I thin set is a set of the form V ( F ) ∩ ( X , D ǫ )( O F,S ) , where V ⊂ X is a proper closed subset of X .A type II thin set is a set of the form f ( Y ( F )) ∩ ( X , D ǫ )( O F,S ) , where f : Y → X is a dominant generically finite morphism of degree ≥ F with Y integral.A thin set is any subset of a finite union of type I and type II thin sets.Here is an example of thin sets: Example 4.4.
Let X = P with D ǫ = 0 defined over a number field F . We consider themorphism f : P → P , ( x : x ) ( x d : x d )with d ≥
2. Then f ( X ( F )) ⊂ X ( F ) is a thin set. .3. Log Manin’s conjecture for klt Campana points.
Finally we state log Manin’sconjecture for klt Campana points:
Conjecture 4.5 (Log Manin’s conjecture for klt Campana points) . Let (
X, D ǫ ) be a kltFano orbifold defined over a number field F and L = ( L, {k · k v } ) be an adelically metrizedample line bundle. Assume that ( X , D ǫ )( O F,S ) is not thin. Then there exists a thin set Z ⊂ ( X , D ǫ )( O F,S ) such that N (( X , D ǫ )( O F,S ) \ Z, L , T ) ∼ c ( F, X , D ǫ , L , Z ) T a ( X,D ǫ ,L ) (log T ) b ( F,X,D ǫ ,L ) − , as T → ∞ . Here the leading constant c ( F, X , D ǫ , L , Z ) is analogous to Peyre’s constantdeveloped in [Pey95] and [BT98a] and its definition is given in [PSTVA20, Section 3.3]. Remark 4.6.
For a smooth geometrically rationally connected projective variety X definedover a number field F , it is expected that X ( F ) is not thin as soon as there is a rationalpoint. Indeed, Colliot-Th´etl`ene’s conjecture predicts that the set of rational points is densein the Brauer-Manin set, and this implies that X satisfies weak weak approximation. It isknown that weak weak approximation property implies non-thinness of the set of rationalpoints. The corresponding statement for klt Campana points, i.e., weak weak approximationfor klt Campana sets implies non-thiness of the set of klt Campana points is established in[NS20]. So it is natural to expect that the assumption of Conjecture 4.5 is true as long asthere is a klt Campana point. Remark 4.7.
It is well-documented in the case of rational points that in Conjecture 4.5 itis important to remove the contribution of a thin set Z from the counting function. Thereis a series of papers ([LT17], [Sen21], and [LST18]) studying birational geometry of thinexceptional subsets for rational points. In [LST18], Lehmann, Sengupta, and the authorproposed a conjectural description of thin exceptional subsets and proved that it is indeed athin set using the minimal model program and the boundedness of singular Fano varieties.It would be interesting to perform a similar study for klt Campana points.Conjecture 4.5 is known in the following cases: • projective space with a boundary being the union of hyperplanes ([VV11], [VV12],[BVV12], and [BY20]); • equivariant compactifications of vector groups ([PSTVA20]); • toric varieties defined over Q ([PS20]) and; • biequivaraint compactifications of the Heisenberg group ([Xia20]).One can also consider a similar counting problem for weak Campana points, however thisproblem is much harder than Conjecture 4.5. At the moment of writing this paper, we donot know how one should formulate a log Manin’s conjecture for weak Campana points, but[Str20] takes the first step towards to this problem.5. Height zeta functions
Let F be a number field and G be a connected linear algebraic group defined over F . Let X be a smooth projective equivariant compacitification of G , i.e., X contains G as a Zariskiopen subset, and the right action of G extends to X . In this situation, the boundary D = X \ G = [ α ∈A D α s a divisor where each D α is an irreducible component. After applying an equivariantresolution, we may assume that D = P α ∈A D α is a divisor with strict normal crossings. Wealso fix an adelic metriazation for O ( D α ) for every α ∈ A .For each α ∈ A , we choose m α which is a positive integer or + ∞ , and set ǫ α = 1 − m α .We consider D ǫ = X α ∈A ǫ α D α , and ( X, D ǫ ) is a Campana orbifold. Let us fix a finite set S of places including Ω ∞ F and agood integral model X away from S extending X . When − ( K X + D ǫ ) is ample (or moregenerally big), it is natural to consider the counting problem of O F,S -Campana points on( X , D ǫ ). There is a general approach to this problem which is called as the height zetafunction method.Let Pic( X ) G be the Picard group of G -linearlized line bundles on X up to isomorphisms.(If the reader is not familiar with G -linearlizations, she/he may ignore this term for now.)After tensoring by Q , boundary components D α form a basis for Pic( X ) G Q . We choose asection f α ∈ H ( X, O ( D α )) , corresponding to D α . Then we define a local height pairing: for any place v ∈ Ω F , H v : G ( F v ) × Pic( X ) G C → C × , g v , X α ∈A s α D α ! Y α ∈A k f α ( g v ) k − s α v . Using this local height pairing, we define the global height pairing as the Euler product: H := Y v ∈ Ω F H v : G ( A F ) × Pic( X ) G C → C × . Applying the definition of Campana points to local points, for each v S , one can definethe Campana set ( X , D ǫ )( O v ) ⊂ X ( F v ) . We set G ( F v ) ǫ = G ( F v ) ∩ ( X , D ǫ )( O v ) , and let δ ǫ,v ( g v ) be the characteristic function of G ( F v ) ǫ on G ( F v ). When v ∈ S , we set δ ǫ,v ≡ δ ǫ as the Euler product: δ ǫ = Y v ∈ Ω F δ ǫ,v : G ( A F ) → R ≥ . For g ∈ G ( A F ) and s ∈ Pic( X ) G C , we define the height zeta function by Z ( g, s ) := X γ ∈ G ( F ) H ( γg, s ) − δ ǫ ( γg )When ℜ ( s ) is sufficiently large, this function converges to a continuous function in g ∈ G ( F ) \ G ( A F ) and a holomorphic function in s ∈ Pic( X ) G C .A relation of this height zeta function to log Manin’s conjecture is given by Tauberiantheorem. Indeed, if one can prove that for an ample (or big) line bundle L , Z (id , sL ) admits meromorphic continuation to a half plane ℜ ( s ) ≥ a with a unique pole at s = a of order b with a > N ( G ( F ) ǫ , L , T ) ∼ cT a (log T ) b − , where c is a positive constant related to the leading constant of Z (id , sL ) at s = a . Thus ourgoal is reduced to obtain a meromorphic continuation of Z (id , s ).To this end, for s ≫
0, one can prove that Z ( g, s ) ∈ L ( G ( F ) \ G ( A F )) , thus one may apply spectral decomposition of this Hilbert space to Z ( g, s ) and use thisspectral decomposition to obtain a meromorphic continuation.This program has been pioneered mainly by Tschinkel and his collaborators, and has beencarried out in the following cases: • rational points on toric varieties ([BT96], [BT98b]); • rational points on equivariant compacitifications of vector groups ([CLT02]); • rational points on wonderful compactifications of semi-simple groups of adjoint type([STBT07]); • rational points on biequivariant compactifications of unipotent groups ([ST16]); • integral points on equivariant compacitificaitons of vector groups ([CLT12]); • integral points on toric varieties ([CLT10b]); • integral points on wonderful compactificaitons of semi-simple groups of adjoint type([TBT13] and [Cho19]); • Campana points on equivariant compacitifications of vector groups ([PSTVA20]); • Campana points on biequivariant compacitifications of the Heisenberg group ([Xia20]),and; • weak Campana points on certain toric varieties ([Str20]).It would be interesting to explore Campana points on other algebraic groups. In particular,the treatment of integral points on toric varieties ([CLT10b]) is known to be incomplete, andthere is some technical issue on this paper. It would be interesting to apply the height zetafunction method to klt Campana points on toric varieties and see whether we have a similarissue.Finally for the readers who are interested in working examples of this program, we rec-ommend them to consult [PSTVA20, Interlude I]. References [Abr09] Dan Abramovich. Birational geometry for number theorists. In
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Duke Math. J. , 161(15):2799–2836, 2012.[FMT89] Jens Franke, Yuri I. Manin, and Yuri Tschinkel. Rational points of bounded height on Fanovarieties.
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Ann. Sci. ´Ec. Norm. Sup´er. (4) , 41(3):383–435, 2008.[GO11] Alex Gorodnik and Hee Oh. Rational points on homogeneous varieties and equidistribution ofadelic periods.
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Proc. Lond. Math. Soc. , 2020. online publication.[Sen21] A. K. Sengupta. Manin’s conjecture and the Fujita invariant of finite covers.
Algebra NumberTheory , 2021. to appear.[ST16] Joseph Shalika and Yuri Tschinkel. Height zeta functions of equivariant compactifications ofunipotent groups.
Comm. Pure Appl. Math. , 69(4):693–733, 2016.[STBT07] Joseph Shalika, Ramin Takloo-Bighash, and Yuri Tschinkel. Rational points on compactifica-tions of semi-simple groups.
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Algebra Number Theory , 6(5):1019–1041, 2012.[Xia20] Huan Xiao. Campana points on biequivariant compactifications of the Heisenberg group. sub-mitted, 2020.
Department of Mathematics, Faculty of Science, Kumamoto University, Kurokami 2-39-1Kumamoto 860-8555 JapanPriority Organization for Innovation and Excellence, Kumamoto University
Email address : [email protected]@kumamoto-u.ac.jp