On the distribution of points of bounded height on equivariant compactifications of vector groups

We prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space. Oblatum 24-IV-2001 & 22-X-2001?Published online: 1 February 2002     

K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.     

Counting points of fixed degree and bounded height on linear varieties

We count points of fixed degree and bounded height on a linear projective variety defined over a number field k. If the dimension of the variety is large enough compared to the degree we derive asymptotic estimates as the height tends to infinity. This generalizes results of Thunder, Christensen and Gubler and special cases of results of Schmidt and Gao.     

Orbits of points on certain K3 surfaces

In this paper we show that, for a K3 surface within a certain class of surfaces and over a number field, the orbit of a point under the group of automorphisms is either finite or its exponent of growth is exactly the Hausdorff dimension of a fractal associated to the ample cone. In particular, the exponent depends on the geometry of the surface and not its arithmetic. For surfaces in this class, the exponent is 0.6527±0.0012.     

Torsion points on families of products of elliptic curves

In a recent paper we proved a special case of a variant of Pink's Conjecture for a variety inside a semiabelian scheme: namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes. Here we go ahead with the programme of settling the conjecture for general abelian surface schemes by completing the proof for all non-simple surfaces. This involves some entirely new and crucial issues.     

A nondensity property of preperiodic points on Chebyshev dynamical systems

Let k be a number field with algebraic closure , and let S be a finite set of primes of k, containing all the infinite ones. Consider a Chebyshev dynamical system on P². Fix the effective divisor D of P² that is equal to a line nondegenerate on²[−2,2]. Then we will prove that the set of preperiodic points on which are S-integral relative to D is not Zariski dense in P².     

The difference between the ordinary height and the canonical height on elliptic curves

We estimate the bounds for the difference between the ordinary height and the canonical height on elliptic curves over number fields. Our result is an improvement of the recent result of Cremona, Prickett, and Siksek [J.E. Cremona, M. Prickett, S. Siksek, Height difference bounds for elliptic curves over number fields, J. Number Theory 116 (2006) 42-68]. Our bounds are usually sharper than the other known bounds.     

Torsion points on the jacobian of a Fermat quotient

Let p≥5 be a prime, ζ a primitive pth root of unity and λ=1−ζ. For 1≤s≤p−2, the smooth projective model C_p,s of the affine curve v^p=u^s(1−u) is a curve of genus (p−1)/2 whose jacobian J_p,s has complex multiplication by the ring of integers of the cyclotomic field Q(ζ). In 1981, Greenberg determined the field of rationality of the p-torsion subgroup of J_p,s and moreover he proved that the λ³-torsion points of J_p,s are all rational over Q(ζ). In this paper we determine quite explicitly the λ³-torsion points of J_p,1 for p=5 and p=7, as well as some further p-torsion points which have interesting arithmetical applications, notably to the complementary laws of Kummer’s reciprocity for pth powers.     

Torsion points on some abelian surfaces of CM-type

Let A be a two-dimensional abelian variety of CM-type defined over Q, which is not simple over C. Let p be a prime number. We show that torsion points of A(Q) of prime order p are possible only for p≦7.     

Cubic surfaces violating the Hasse principle are Zariski dense in the moduli scheme

We construct new examples of cubic surfaces, for which the Hasse principle fails. Thereby we show that, over every number field, the counterexamples to the Hasse principle are Zariski dense in the moduli scheme of non-singular cubic surfaces.     

Rational points and cohomology of discriminant varieties

Let G be a finite unitary reflection group acting in a complex vector space . The discriminant varietyX_G of G is defined as the space of regular orbits of G on V. Classical examples include the varieties of complex polynomials of degree n with distinct (resp. non-zero distinct) roots. The normaliser of G in GL(V) acts on X_G; in this work we determine the action of on the cohomology of X_G. In the classical cases this amounts to computing the cohomology of X_G with certain local coefficient systems. Our methods are to compute equivariant weight polynomials by means of explicit counting of the rational points of certain varieties over finite fields, and then to exploit the weight purity of the relevant varieties. We obtain some power series identities as a byproduct.     

Rational points of bounded height on Del Pezzo surfaces of degree six

LetK be a number field. Denote byV ₃ a split Del Pezzo surface of degree six overK and by ω its canonical divisor. Denote byW ₃ the open complement of the exceptional lines inV ₃. LetN _{W s}(−ω, X) be the number ofK-rational points onW ₃ whose anticanonical heightH _−ω is bounded byX. Manin has conjectured that asymptoticallyN _{W 3}(−ω, X) tends tocX(logX)³, wherec is a constant depending only on the number field and on the normalization of the height. Our goal is to prove the following theorem: For each number fieldK there exists a constantc _K such thatN _{W 3}(−ω, X)≤c_KX(logX)^3+2r , wherer is the rank of the group of units ofO _K. The constantc _K is far from being optimal. However, ifK is a purely imaginary quadratic field, this proves an upper bound with a correct power of logX. The proof of Manin's conjecture for arbitrary number fields and a precise treatment of the constants would require a more sophisticated setting, like the one used by [Peyre] to prove Manin's conjecture and to compute the correct asymptotic constant (in some normalization) in the caseK=ℚ. Up to now the best result for arbitraryK goes back, as far as we know, to [Manin-Tschinkel], who gives an upper boundN _{W 3}(−ω,X)≤cX^l+ε. The author would like to express his gratitude to Daniel Coray and Per Salberger for their generous and indispensable support.     

Estimates for Tamagawa numbers of diagonal cubic surfaces

For diagonal cubic surfaces, we give an upper bound for E. Peyre's Tamagawa type number in terms of the coefficients of the defining equation. This bound shows that the reciprocal admits a fundamental finiteness property on the set of all diagonal cubic surfaces. As an application, we show that the infinite series of Tamagawa numbers related to the Fano cubic bundles considered by Batyrev and Tschinkel (1996) [BT] are indeed convergent.     

A lower bound for the canonical height on elliptic curves over abelian extensions

Let E/K be an elliptic curve defined over a number field, let ? be the canonical height on E, and let K^ab/K be the maximal abelian extension of K. Extending work of M. Baker (IMRN 29 (2003) 1571-1582), we prove that there is a constant C(E/K)>0 so that every nontorsion point P∈E(K^ab) satisfies .     

Separators of points on algebraic surfaces

For a finite set of points X⊆Pⁿ and for a given point P∈X, the notion of a separator of P in X (a hypersurface containing all the points in X except P) and of the degree of P in X, (the minimum degree of these separators) has been largely studied. In this paper we extend these notions to a set of points X on a projectively normal surface S⊆Pⁿ, considering as separators arithmetically Cohen-Macaulay curves and generalizing the case S=P² in a natural way. We denote the minimum degree of such curves as and we study its relation to . We prove that if S is a variety of minimal degree these two terms are explicitly related by a formula, whereas only an inequality holds for other kinds of surfaces.     

On the number of rational points of a plane algebraic curve

The number of F_q -rational points of a plane non-singular algebraic curve defined over a finite field F_q is computed, provided that the generic point of is not an inflexion and that is Frobenius non-classical with respect to conics. Received: 18 March 2003     

On primitive points of elliptic curves with complex multiplication

Let E be an elliptic curve defined over Q and P∈E(Q) a rational point of infinite order. Suppose that E has complex multiplication by an order in the imaginary quadratic field k. Denote by M_E,P the set of rational primes ? such that ? splits in k, E has good reduction at ?, and P is a primitive point modulo ?. Under the generalized Riemann hypothesis, we can determine the positivity of the density of the set M_E,P explicitly.