Quasi-fonctions et Hauteurs sur les Varietes Abeliennes

Neron, André

doi:10.2307/1970644

Cited by 137 publications

(64 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…This follows immediately from Theorem 2 in [34] once we have shown that for −1 − a ≤ σ ≤ 1 + b we have (29) |f λ,χ (σ + it, g)| ≤ c 1 e…”

Section: Lemma 85mentioning

confidence: 71%

“…This is less trivial than the example given above because one has to determine a fundamental domain modulo the action of the units (the proofs appeared in [39]). In 1965, Neron [29] gave asymptotics for the number of rational points of bounded height on abelian varieties. The classical circle method, now a whole field in itself, proves asymptotics on complete intersections of small degree.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Height zeta functions of toric bundles over flag varieties

Strauch¹,

Tschinkel

1999

Sel. math., New ser.

View full text Add to dashboard Cite

“…This follows immediately from Theorem 2 in [34] once we have shown that for −1 − a ≤ σ ≤ 1 + b we have (29) |f λ,χ (σ + it, g)| ≤ c 1 e…”

Section: Lemma 85mentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Height zeta functions of toric bundles over flag varieties

Strauch¹,

Tschinkel

1999

Sel. math., New ser.

View full text Add to dashboard Cite

“…A Néron-Tate height function [18] is the unique representative of the class of height functions attached to a line bundle that is in reality a function of degree at most two, namely, a quadratic form plus a linear functional on A(Q). The quadratic part of a Néron-Tate height function associated with an ample line bundle N on A gives a non-degenerate non-negative quadratic form on the real iv vector space R ⊗ A(Q).…”

Section: ∈ Pic V We Have H(l ⊗ M P ) = H(l P ) + H(m P ) + O(1mentioning

confidence: 99%

Integral and rational points on algebraic curves of certain types and their Jacobian varieties over number fields

Fujimori¹

1997

Tohoku Math. Publ.

View full text Add to dashboard Cite

The author does not know how to thank Professor Yasuo Morita for his patient and warm encouragement. Without it, the author could not be able to write up the thesis. He thanks Professor Atsushi Sato for having made some comments on the drafts of my paper and preprint.Thanks also go to Professor Tadao Oda and other staffs of the institute for many things and a comfortable environment.1991 Mathematics Subject Classification. Primary 11G30; Secondary 11D41, 14G05, 14H25. Author addresses:Mathematical Institute, Tohoku University, Sendai 980-77, Japan IntroductionThe logarithmic absolute height function is a fundamental tool when we investigate the distribution of rational points on a projective variety V over the rational number field Q.LetQ be an algebraic closure of Q. A (logarithmic absolute) height function is a real-valued function on the set V (Q) = Hom(SpecQ, V ) of Q-valued points on the variety V . It is defined up to a bounded function for the pair V and a line bundle L on V . We denote one of the representatives by h V (L, ·), or simply, by h(L, ·). The set of rational points, orwhere v runs through the set of rational prime numbers and the infinite prime, and | · | v is the standard absolute value on Q defined by v. This is a height function attached to the hyperplane section sheaf O(1) on Taking an embedding ι : V → P N , we obtainA height function is additive with respect to the tensor operation on the group Pic V of line bundles on the projective variety V . For L and of rational points on the abelian variety A is finitely generated, which was substantially proved in [16] and [25]. Let R be the rank of A(Q). The quadratic form attached to the ample line bundle N on A determines an R-dimensional Euclidean space structure on R ⊗ A(Q) and we seeIf we start from polynomial equations with coefficients in the rational integer ring Z, which define a quasi-projective variety over Q, we naturally where r(T, a) is the rank of the group of rational points on the Jacobian variety of the plane curve overMumford where O C (−(2g − 2)P ) is the line bundle on C one of whose rational sections has −(2g − 2)P as the corresponding divisor, and Pic • is the functor which associates the group of isomorphism classes of line bun-height function attached to the invertible sheaf Ω. We call it the canonical height function on the curve C. is naturally regarded as a subset of the set of rational points on the planeWe denote by J a the Jacobian variety of C a . Corollary 0.5. If a is n-th power-free and |a| > a 1 , then where g is the genus of C and phrase "in a neighborhood" means that the function of v on the left side of the equation becomes bounded on the image of C(Q).For P ∈ C(Q), we see by the theoremThis leads to a new proof of a fact which is usually an application of the Riemann-Hurwitz formula.Corollary 0.8. Notation and Terminology 0.11. For the objects X, Y, Z and the We denote respectively by Z, Q, and R the ring of rational integers, the field of rational numbers, and the field of real numbers. A finite extensio...

show abstract

“…This is applied to the distribution of integral points on abelian varieties in Section 5.4. where h (w, z) is a hermitian quadratic form on C^ and K (k) is real valued. Then for all zeC^ one has (see [33]) <D, (^(z))-(9)>,= -log |9(z)|+7r/2^(z, z). The subscript v added to one of the above symbols denotes that it is relative to the valuation v, if more than one valuation is being considered.…”

Section: Local Neron Heights and Integral Pointsmentioning

confidence: 99%

The density of rational and integral points on algebraic varieties

Brown¹

1992

Ann. Sci. École Norm. Sup.

View full text Add to dashboard Cite

Quasi-fonctions et Hauteurs sur les Varietes Abeliennes

Cited by 137 publications

References 4 publications

Height zeta functions of toric bundles over flag varieties

Height zeta functions of toric bundles over flag varieties

Integral and rational points on algebraic curves of certain types and their Jacobian varieties over number fields

The density of rational and integral points on algebraic varieties

Contact Info

Product

Resources

About