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We explore log Manin's conjecture for integral points and its connections to $\mathbb A^1$-connectedness. We prove log Manin's conjecture for Campana rational curves and for $\mathbb A^1$-curves on split toric varieties. Our arguments…

We classify elliptic curves over the rationals whose N\'eron model over the integers is semi-abelian, with good reduction at p=2, and whose Mordell--Weil group contains an element of order two that stays non-trivial at p=2. Furthermore, we…

We show that the Shimura varieties of level one parametrizing QM-abelian varieties have rarely rational points.

The Poincar\'e torsor of a Shimura family of abelian varieties can be viewed both as a family of semi-abelian varieties and as a mixed Shimura variety. We show that the special subvarieties of the latter cannot all be described in terms of…

In this note we study numerically the combinatorics of curves and geodesics on the torus with one boundary component. A potential computational difficulty is avoided by counting inside specific orbits of the mapping class group up to a…

Let $A$ be an abelian variety defined over a number field $K$. We say that a point $P \in A(\overline{\mathbb{Q}})$ is primitive if there is no $Q \in A(\overline{\mathbb{Q}})$ defined on the field of definition of $P$ over $K$ such that…

For rank-two $A$-motives defined over local fields with odd characteristic, we give an analogue of a theorem of Imai stating that abelian varieties with good reduction over $p$-adic fields have only finitely many torsion points values in…

In this paper, we complete the long-standing challenge to establish a Khintchine-type theorem for arbitrary nondegenerate manifolds in $\mathbb{R}^n$. In particular, our main result finally removes the analyticity assumption from the…

Let $A$ be an abelian variety over a $p$-adic field $K$ and $L$ an algebraic infinite extension over $K$. We consider the finiteness of the torsion part of the group of rational points $A(L)$ under some assumptions. In 1975, Hideo Imai…

Inspired by the work of Boris Vertman on refined analytic torsion for manifolds with boundary, in this paper we extend the construction of the Cappell-Miller analytic torsion to manifolds with boundary. We also compare it with the refined…

Let A be an abelian variety of dimension g defined over a number field K. We study the size of the torsion group A(F)_{tors} where F/K is a finite extension and more precisely we study the possible exponent \gamma in the inequality…

We prove a $p$-adic analogue of the Andr\'{e}-Oort conjecture for subvarieties of the universal abelian varieties containing a dense set of special points. Let $g$ and $n$ be integers with $n \geq 3$ and $p$ a prime number not dividing $n$.…

We study the Mumford--Tate conjecture for hyperk\"{a}hler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O'Grady's ten dimensional…

Manin's conjecture for the asymptotic behavior of the number of rational points of bounded height on del Pezzo surfaces can be approached through universal torsors. We prove several auxiliary results for the estimation of the number of…

We shall consider sections of an elliptic scheme $\mathcal{E}$ over a(n affine) base curve $B$, and study the points of $B$ where the section takes a torsion value. In particular, we shall relate the distribution in $B$ of these points with…

We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the N\'eron-Tate height of generators of its Mordell-Weil group. The…

We study the rationality properties of the moduli space $\mathcal{A}_g$ of principally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the results to arithmetic questions. In particular we show that any principally polarised abelian…

The Serre conjecture II predicts that every torsor under a semisimple, simply connected, algebraic group over a field of cohomological dimension at most 2 and of degree of imperfection at most 1 has a rational point. We generalize this…

We give an alternative proof of Faltings's theorem (Mordell's conjecture): a curve of genus at least two over a number field has finitely many rational points. Our argument utilizes the set-up of Faltings's original proof, but is in spirit…

We associate to an analytic subvariety of a torus a tropical variety. In the first part, we generalize the results from tropical algebraic geometry to this non-archimedean analytic situation. The periodic case is applied to a totally…