Integral points of bounded height on partial equivariant compactifications of vector groups

We establish asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups.Comment: 34 pages; revised version; submitte

Concentration analysis and cocompactness

Loss of compactness that occurs in may significant PDE settings can be expressed in a well-structured form of profile decomposition for sequences. Profile decompositions are formulated in relation to a triplet $(X,Y,D)$, where $X$ and $Y$ are Banach spaces, $X\hookrightarrow Y$, and $D$ is, typically, a set of surjective isometries on both $X$ and $Y$. A profile decomposition is a representation of a bounded sequence in $X$ as a sum of elementary concentrations of the form $g_kw$, $g_k\in D$, $w\in X$, and a remainder that vanishes in $Y$. A necessary requirement for $Y$ is, therefore, that any sequence in $X$ that develops no $D$-concentrations has a subsequence convergent in the norm of $Y$. An imbedding $X\hookrightarrow Y$ with this property is called $D$-cocompact, a property weaker than, but related to, compactness. We survey known cocompact imbeddings and their role in profile decompositions

Tamagawa numbers of polarized algebraic varieties

Let ${\cal L} = (L, \| \cdot \|_v)$ be an ample metrized invertible sheaf on a smooth quasi-projective algebraic variety $V$ defined over a number field. Denote by $N(V,{\cal L},B)$ the number of rational points in $V$ having ${\cal L}$-height $\leq B$. We consider the problem of a geometric and arithmetic interpretation of the asymptotic for $N(V,{\cal L},B)$ as $B \to \infty$ in connection with recent conjectures of Fujita concerning the Minimal Model Program for polarized algebraic varieties. We introduce the notions of ${\cal L}$-primitive varieties and ${\cal L}$-primitive fibrations. For ${\cal L}$-primitive varieties $V$ over $F$ we propose a method to define an adelic Tamagawa number $\tau_{\cal L}(V)$ which is a generalization of the Tamagawa number $\tau(V)$ introduced by Peyre for smooth Fano varieties. Our method allows us to construct Tamagawa numbers for $Q$-Fano varieties with at worst canonical singularities. In a series of examples of smooth polarized varieties and singular Fano varieties we show that our Tamagawa numbers express the dependence of the asymptotic of $N(V,{\cal L},B)$ on the choice of $v$-adic metrics on ${\cal L}$.Comment: 54 pages, minor correction

Heights and measures on analytic spaces. A survey of recent results, and some remarks

This paper has two goals. The first is to present the construction, due to the author, of measures on non-archimedean analytic varieties associated to metrized line bundles and some of its applications. We take this opportunity to add remarks, examples and mention related results.Comment: 41 pages, final version. To appear in: Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry, edited by Raf Cluckers, Johannes Nicaise, Julien Seba

The dynamical Manin-Mumford problem for plane polynomial automorphisms

Let $f$ be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve $C$. We conjecture that this happens if and only if $f$ admits a time-reversal symmetry; in particular the Jacobian $\mathrm{Jac}(f)$ must be a root of unity. As a step towards this conjecture, we prove that the Jacobian of $f$ and all its Galois conjugates lie on the unit circle in the complex plane. Under mild additional assumptions we are able to conclude that indeed $\mathrm{Jac}(f)$ is a root of unity. We use these results to show in various cases that any two automorphisms sharing an infinite set of periodic points must have a common iterate, in the spirit of recent results by Baker-DeMarco and Yuan-Zhang.Comment: 45 pages. Theorems A and B are now extended to automorphisms defined over any field of characteristic zer