Related papers: Number fields generated by points in linear system…
We present new constructions of complex and p-adic Darmon points on elliptic curves over base fields of arbitrary signature. We conjecture that these points are global and present numerical evidence to support our conjecture.
We show that the geometric classification of smooth projective curves admitting infinitely many points of degree $d\leq 5$ extends from number fields to function fields of characteristic 0. Over number fields, this classification was…
This article is devoted to the investigation of structure of wrap groups of fiber bundles over ultra-normed infinite fields and more generally over Cayley-Dickson algebras. Iterated wrap groups are studied as well. Their smashed products…
We study the existence problem and the enumeration problem for sections of Serre fibrations over compact orientable surfaces. When the fundamental group of the fiber is finite, a complete solution is given in terms of 2-dimensional…
Graphs on integer points of polytopes whose edges come from a set of allowed differences are studied. It is shown that any simple graph can be embedded in that way. The minimal dimension of such a representation is the fiber dimension of…
We prove a first principle of preservation of multiplicity in difference geometry, paving the way for the development of a more general intersection theory. In particular, the fibres of a \sigma-finite morphism between difference curves are…
Motivated by the intermediate Lang conjectures on hyperbolicity and rational points, we prove new finiteness results for non-constant morphisms from a fixed variety to a fixed variety defined over a number field by applying Faltings's…
In this paper, we classify the possible group structures on the set of $R$-valued points of an abelian variety, where $R$ is any real closed field. We make use of a family of abelian varieties that, in effect, allows one to quantify over…
A theorem of B. Green states that if A is a Dedekind ring whose fraction field is a local or global field, every normal projective curve over Spec(A) has a finite morphism to P^1_A. We give a different proof of a variant of this result…
Given a birational parameterization $\phi: \mathbb{P}_k^2 - rightarrow \mathbb{P}_k^3$ of an algebraic surface $\mathscr S\subset \mathbb{P}_k^3$, we bound the number of 1-dimensional fibers of the canonical projection of the graph of…
In this article we prove some interesting results on field generated by division points of several formal groups of same height, already implicit in the treatment in appendix-A of my M.Sc thesis (Points of Small Height in Certain Nonabelian…
We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curves defined over a number field. This method is used to solve some arithmetic problems that remained open.
We give a combinatorial description of closed curves on oriented surfaces in terms of certain permutations, called charts. We describe automorphisms of curves in terms of charts and compute the total number of curves counted with…
In this paper, we study square functions for extension operators over finite-type, planar curves endowed with the Euclidean arclength measure. We prove new results for curves of the form $(T,\phi(T))$ where $\phi(T)$ is a polynomial of…
We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in one variable over a finite field is finiteley generated.
I prove three classification results about harmonic morphisms whose fibers have dimension one. All are valid when the domain is at least of dimension 4. (The character of this overdetermined problem is very different when the dimension of…
We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic…
A result of Belyi can be stated as follows. Every curve defined over a number field can be expressed as a cover of the projective line with branch locus contained in a rigid divisor. We define the notion of geometrically rigid divisors in…
We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length…
We study automorphism groups of fibered surfaces for finite cyclic covering fibrations of an elliptic surface. We estimate the order of a finite subgroup of automorphism groups in terms of the genus of the fiber, the genus of the base…