Weak approximation and the Hilbert property for Campana points

Streeter, Sam; Nakahara, Masahiro

doi:10.48550/arxiv.2010.12555

Cited by 4 publications

(10 citation statements)

References 31 publications

(29 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The hypothesis that the Campana points themselves are not thin is discussed by Nakahara and Streeter in [18]. The authors establish in [18,Theorem 1.1] a connection between thin sets of Campana points and weak approximation, in the spirit of Serre's arguments in [24,Theorem 3.5.7]. Combining this with [18,Corollary 1.4], it can be shown that this hypothesis holds for the orbifold we consider below.…”

Section: The Campana-manin Conjecturementioning

confidence: 83%

“…Then there is a thin set T of Campana O K,S -points such that Remark 2.8. The hypothesis that the Campana points themselves are not thin is discussed by Nakahara and Streeter in [18]. The authors establish in [18,Theorem 1.1] a connection between thin sets of Campana points and weak approximation, in the spirit of Serre's arguments in [24,Theorem 3.5.7].…”

Section: The Campana-manin Conjecturementioning

confidence: 99%

See 1 more Smart Citation

Sums of four squareful numbers

Shute

2021

Preprint

View full text Add to dashboard Cite

We find an asymptotic formula for the number of primitive vectors (z 1 , . . . , z 4 ) ∈ (Z =0 ) 4 such that z 1 , . . . , z 4 are all squareful and bounded by B, andOur result agrees in the power of B and log B with the Campana-Manin conjecture of Pieropan, Smeets, Tanimoto and Várilly-Alvarado. Contents 1. Introduction 1 2. The Campana-Manin conjecture 5 3. Dealing with the large coefficients 10 4. The circle method 14 5. Proof of Theorem 1.1 36 References 43

show abstract

Section: The Campana-manin Conjecturementioning

confidence: 83%

Section: The Campana-manin Conjecturementioning

confidence: 99%

Sums of four squareful numbers

Shute

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The authors establish in [10,Theorem 1.1] a connection between thin sets of Campana points and weak approximation, in the spirit of Serre's arguments in [15,Theorem 3.5.7]. Together with [10,Corollary 1.4], this implies that C is not itself thin. It remains to consider whether the second explanation above could hold.…”

Section: Introductionmentioning

confidence: 96%

“…(2) There is a thin set T ⊂ C of Campana points such that the removal of T from the count N 1 (B) reduces the leading constant to c PSTV-A . Recent work of Nakahara and Streeter in [10] tackles the question of when the set of Campana points corresponding to a log Fano orbifold (P n , D) can be a thin set. The authors establish in [10,Theorem 1.1] a connection between thin sets of Campana points and weak approximation, in the spirit of Serre's arguments in [15,Theorem 3.5.7].…”

Section: Introductionmentioning

confidence: 99%

“…Recent work of Nakahara and Streeter in [10] tackles the question of when the set of Campana points corresponding to a log Fano orbifold (P n , D) can be a thin set. The authors establish in [10,Theorem 1.1] a connection between thin sets of Campana points and weak approximation, in the spirit of Serre's arguments in [15,Theorem 3.5.7]. Together with [10,Corollary 1.4], this implies that C is not itself thin.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the leading constant in the Manin-type conjecture for Campana points

Shute¹

2021

Preprint

View full text Add to dashboard Cite

We compare the Manin-type conjecture for Campana points recently formulated by Pieropan, Smeets, Tanimoto and Várilly-Alvarado with an alternative prediction of Browning and Van Valckenborgh in the special case of the orbifold (P 1 , D), where. We find that the two predicted leading constants do not agree, and we discuss whether thin sets could explain this discrepancy. Motivated by this, we provide a counterexample to the Manin-type conjecture for Campana points, by considering orbifolds corresponding to squareful values of binary quadratic forms.

show abstract

On the leading constant in the Manin-type conjecture for Campana points

Shute

2022

Acta Arith.

View full text Add to dashboard Cite

On the leading constant in the Manin-type conjecture for Campana points by Alec Shute (Klosterneuburg)1. Introduction. The study of Campana points is an emerging area of interest in arithmetic geometry as a way to interpolate between rational and integral points. Campana orbifolds, first introduced in [4] and [5], consist of a variety X and a weighted boundary divisor D of X. The Campana points associated to the orbifold (X, D) can be viewed as rational points of X that are integral with respect to D. In the recent paper [16], Pieropan, Smeets, Tanimoto and Várilly-Alvarado formulate a Manin-type conjecture for the quantitative study of Campana points on Fano Campana orbifolds, which henceforth we shall refer to as the PSTV-A conjecture. The authors establish their conjecture in the special case of vector group compactifications, using the height zeta function method developed by Chambert-Loir and Tschinkel [7], [8].The arithmetic study of Campana points is still in its early stages. Initial results in [2], [22] and [3], which predate the formulation of the PSTV-A conjecture, concern squareful and m-full values of hyperplanes of P n+1 . (We recall that a non-zero integer z is m-full if for any prime p dividing z, we have p m | z, and squareful if it is 2-full.) Following discussions in the Spring 2006 MSRI program on rational and integral points on higherdimensional varieties, Poonen [17] posed the problem of finding the number of coprime integers z 0 , z 1 such that z 0 , z 1 and z 0 + z 1 are all squareful and bounded by B. In the language of the PSTV-A conjecture, this corresponds to counting Campana points on the orbifold (P 1 , D), where D is the divisorUpper and lower bounds for this problem were obtained by Browning and Van Valckenborgh [2], but finding an asymptotic formula remains wide open. Van Valckenborgh [22] considers a higher-dimensional

show abstract

Weak approximation and the Hilbert property for Campana points

Cited by 4 publications

References 31 publications

Sums of four squareful numbers

Sums of four squareful numbers

On the leading constant in the Manin-type conjecture for Campana points

On the leading constant in the Manin-type conjecture for Campana points

Contact Info

Product

Resources

About