Related papers: Subfields of ample fields I. Rational maps and def…

Let $f:\mathbb{K}^n\rightarrow\mathbb{K}^m$ be a generically finite polynomial map of degree $d$ between affine spaces. In arXiv:1411.5011 we proved that if $\mathbb{K}$ is the field of complex or real numbers, then the set $S_f$ of points…

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a…

Among geometrically rational surfaces, del Pezzo surfaces of degree two over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we…

We prove that an infinite field interpretable in a $p$-adically closed field $K$ is definably isomorphic to a finite extension of $K$. The result remains true in any $P$-minimal field where definable functions are generically…

For an imaginary quadratic field $k$ of class number $>1$, we prove that there are only finitely many isomorphism classes of rational indefinite quaternion division algebras $B$ such that the associated Shimura curve $M^B$ has $k$-rational…

Given a number field $\mathbb{K} \subset \mathbb{C}$ that is not contained in $\mathbb{R}$, we prove the existence of a dense set of entire maps $f \colon \mathbb{C} \rightarrow \mathbb{C}$ whose preperiodic points and multipliers all lie…

We prove that any non-isotrivial elliptic K3 surface over an algebraically closed field $k$ of arbitrary characteristic contains infinitely many rational curves. In the case when $\mathrm{char}(k)\neq 2,3$, we prove this result for any…

Let K be an algebraically closed field of characteristic zero. Given a polynomial f(x,y) in K[x,y] with one place at infinity, we prove that either f is equivalent to a coordinate, or the family (f+c) has at most two rational elements. When…

Let $\mathcal M=\langle K;O\rangle$ be a real closed valued field and let $k$ be its residue field. We prove that every interpretable field in $\mathcal M$ is definably isomorphic to either $K$, $K(\sqrt{-1})$, $k$, or $k(\sqrt{-1})$. The…

For any affine hypersurface defined by a complete symmetric polynomial in $k\geq 3$ variables of degree $m$ over the finite field $\mathbb{F}_{q}$ of $q$ elements, a special case of our theorem says that this hypersurface has at least…

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…

We improve a bound due to the second author on number of rational points on smooth surfaces in $\mathbb{P}^3$ over finite fields and look at families of surfaces that achieve or nearly achieve this bound, for which we compute their exact…

Building on recent work of Bhargava--Elkies--Schnidman and Kriz--Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.

We show that any smooth projective cubic hypersurface of dimension at least $29$ over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous…

Let $K$ be a complete discrete valued field of characteristic $p$ with residue $k$ which is not necessarily perfect. We prove the Conjecture in \cite{cs} that a $p$-algebra over $K$ contains a totally ramified cyclic maximal subfield if it…

We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets…

We show that projective K3 surfaces with odd Picard rank contain infinitely many rational curves. Our proof extends the Bogomolov-Hassett-Tschinkel approach, i.e., uses moduli spaces of stable maps and reduction to positive characteristic.

If the $\ell$-adic cohomology of a projective smooth variety, defined over a local field $K$ with finite residue field $k$, is supported in codimension $\ge 1$, then every model over the ring of integers of $K$ has a $k$-rational point. For…

Consider a finite l-group acting on the affine space of dimension n over a field k, whose characteristic differs from l. We prove the existence of a fixed point, rational over k, in the following cases: --- The field k is p-special for some…

Let X be a smooth quartic surface not containing lines, defined over a number field K. We prove that there are only finitely many bitangents to X which are defined over K. This result can be interpreted as saying that a certain surface,…