Rational points and non-anticanonical height functions

Frei, Christopher; Loughran, Daniel

doi:10.1090/proc/14248

Cited by 4 publications

(1 citation statement)

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“…For such a height function, [FL17] establishes Manin's conjecture when the base is the projective space using conic bundle structures. Our theorem is flexible in the sense that S can be other Fano manifold other than the projective space.…”

Section: Local Tamagawa Measures Of Conics In Familiesmentioning

confidence: 98%

On upper bounds of Manin type

Tanimoto

2018

Preprint

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We introduce a certain birational invariant of a polarized algebraic variety and use that to obtain upper bounds for the counting functions of rational points on algebraic varieties. Using our theorem, we obtain new upper bounds of Manin type for 28 deformation types of smooth Fano 3-folds of Picard rank ≥ 2 following Mori-Mukai's classification. We also find new upper bounds for polarized K3 surfaces S of Picard rank 1 using Bayer-Macrì's result on the nef cone of the Hilbert scheme of two points on S.where Eff 1 (X) is the cone of pseudo-effective divisors on X. Here is the statement of weak Manin's conjecture:Conjecture 1.1 (Weak Manin's conjecture/Linear growth conjecture). Let X be a geometrically uniruled smooth projective variety defined over a number field k and let L be a big and nef divisor on X. Then there exists a non-empty Zariski open subset U ⊂ X such that for any ǫ > 0 N(U, L, T ) = O ǫ (T a(X,L)+ǫ ).Note that for L = −K X , we have a(X, L) = 1 so that it backs up the word "linear growth".Remark 1.2. There are counterexamples to a version of this conjecture where one assumes L to be only big but not nef. See [LST18, Section 5.1]

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Section: Local Tamagawa Measures Of Conics In Familiesmentioning

confidence: 98%

On upper bounds of Manin type

Tanimoto

2018

Preprint

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Manin's conjecture for certain spherical threefolds

Derenthal

Gagliardi

2018

Advances in Mathematics

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We prove Manin's conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for a spherical threefold with canonical singularities and two infinite families of spherical threefolds with log terminal singularities. Moreover, we show that one of these families does not satisfy a conjecture of Batyrev and Tschinkel on the leading constant in the asymptotic formula. Our proofs are based on the universal torsor method, using Brion's description of Cox rings of spherical varieties.

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On upper bounds of Manin type

Tanimoto

2020

Alg. Number Th.

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Rational points and non-anticanonical height functions

Cited by 4 publications

References 20 publications

On upper bounds of Manin type

On upper bounds of Manin type

Manin's conjecture for certain spherical threefolds

On upper bounds of Manin type

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