Chapter V: Beyond Heights: Slopes and Distribution of Rational Points

Abstract

The distribution of rational points of bounded height on algebraic varieties is far from uniform. Indeed the points tend to accumulate on thin subsets which are images of non-trivial finite morphisms. The problem is to find a way to characterise the points in these thin subsets. The slopes introduced by Jean-Benoît Bost are a useful tool for this problem. These notes will present several cases in which this approach is fruitful. We shall also describe the notion of locally accumulating subvarieties which arises when one considers rational points of bounded height near a fixed rational point.Résumé. -La distribution des points rationnels de hauteur bornée sur les variétés algébriques est loin d'être uniforme les points peuvent s'accumuler sur l'image de variétés formant un ensemble mince. La difficulté est de pouvoir caractériser les points de ces ensembles accumulateurs. Les pentes de la géométrie d'Arakelov forment un outil utile pour attaquer cette problématique. Ces notes présenteront différents exemples où cette approche est efficace. On évoquera également la question des sousvariétés localement accumulatrices qui apparaissent lorsqu'on considère les points de hauteur bornée au voisinage d'un point rationnel.w {0} → P n (K w ) be the natural projection. Than µ w is defined byfor any borelian subset U in P n (K w ), where B • w (1) denotes the ball of radius 1 for • w .

Equidistribution of rational points and the geometric sieve for toric varieties

“…13 For all non-archimedean ν ∈ Val(Q), the adelic measures ω V ν , ω T UT ν "patch together", in the following sense (cf. [38,Corollary 2.23], [34,Theorem 4.33]). Write…”

“…(See [34, §3.24].) Definition 2.1 (See [34] Hypotheses 3.27). We call a nice Q-projective variety V almost Fano if it satisfies the following hypotheses.…”

“…(See e.g [38,. Conjecture 7.12], and also[34, Remark 4.34 (i)].) Recall that M ⊂ V(Q) denotes the fixed thin subset.…”

“…[43, §2.7] and[12, §1] for background materials of this property 2. Note that our definition of counting measures differs slightly from[34, Definition 3.8] 3. i.e., for every continuous function with compact support f : V(A k ) → R, we have as B → ∞,(B(log B) r−1 ) −1 V(A k ) f d δ (V(k)\M) B → α(V)β (V) V(A k ) f d ω V .Since V(A k ) is compact, this is equivalent to the weak convergence.…”

Geometric Batyrev-Manin-Peyre for equivariant compactifications of additive groups

“…Let M k be the localisation of the Grothendieck ring of varieties over k at the class L of the affine line, that is M k = KVar k [L −1 ]. Our focus will be on studying the asymptotic behaviour of the class [M n ] in M k endowed with the weight topology of [1], following a question raised by Peyre [19,Question 5.4].…”