Approximation diophantienne et distribution locale sur une surface torique II

Abstract

Nous proposons une formule empirique pour le problème de distribution locale des points rationnels de hauteur bornée. Il s'agit d'une version locale du principe de Batyrev-Manin-Peyre. Nous la vérifions pour une surface torique, sur laquelle des courbes rationnelles cuspidales et des courbes rationnelles nodales toutes les deux contribuent aux meilleures approximations en dehors d'un fermé de Zariski. Nous démontrons qu'en enlevant une partie mince, il existe une mesure limite et une formule asymptotique pour le grossissement critique.

Rational approximations on toric varieties

“…For X a variety defined over a number field K and for every fixed Q ∈ X( K), choose some distance function d ν (•, Q) with respect to some fixed place ν of K. Choose a height function H L associated to some fixed line bundle L. Then the constant α L,ν (Q, X) is defined as the infimum of positive real numbers γ such that the inequality (2) d ν (P, Q) γ H L (P ) 1, has infinitely solutions P i ∈ X(K) such that d ν (P i , Q) → 0. It measures the local behaviour via the complexity increasing of rational points when approaching the fixed point on the variety and it also plays a central role in the author's recent investigation [Hua17b] [Hua17a] [Hua18] on local distribution of rational points since it helps detect the locally accumulating subvarieties (Definition 4.3). These subvarieties contain rational points that are "closer" to the given point so that when γ is sufficiently close to α L,ν (Q, X), almost all solutions of the inequality (2) are located there.…”

“…It turns out that the local behaviour of rational points is very complicated even for simplest varieties other than the projective spaces. See [McK07], [MR15], [MR16], [Hua17b], [Hua17a], [Hua18]. All of them essentially consider (rational) surfaces, more precisely, (weak) del Pezzo surfaces of degree 3 and they all satisfy Conjecture 1.1.…”

“…Note that (the desingularisation of) this toric variety does not verify Hypothesis (*). See also the surfaces Y 3 , Y 4 in [Hua17a] and [Hua18].The actual phenomenon is that certain singular curves, whose approximation constant equal their degree divided by some factor coming from K. Roth's theorem (1) or their singular multiplicity, give better approximation than the smooth ones. Moreover the nodal type and the cuspical type singularities have different contributions.…”