Related papers: Global discrepancy and small points on elliptic cu…

In this paper, we prove that for each number field $F$ there exists a uniform bound on the prime levels $p$ of elliptic curves $E/F$ for which $F(E[p])=F(\zeta_p)$. Under the Generalized Riemann Hypothesis, we also give uniform bounds on…

We study the asymptotic behaviour of the maximum interpoint distance of random points in a planar bounded set with an unique major axis and a boundary behaving like an ellipse at the endpoints. Our main result covers the case of uniformly…

We prove an equidistribution result for small subvarieties of an abelian variety which generalizes the Szpiro-Ullmo-Zhang theorem on equidistribution of small points.

Let $p \geq 5$ be a prime number. We find all the possible subgroups $G$ of ${\rm GL}_2 ( \mathbb{Z} / p \mathbb{Z} )$ such that there exists a number field $k$ and an elliptic curve ${\mathcal{E}}$ defined over $k$ such that the ${\rm Gal}…

We study the asymptotic distribution of CM points on the moduli space of elliptic curves over $\mathbb{C}_p$, as the discriminant of the underlying endomorphism ring varies. In contrast with the complex case, we show that there is no…

We initiate the theory of $\ell^p$-improving inequalities for arithmetic averages over hypersurfaces and their maximal functions. In particular, we prove $\ell^p$-improving estimates for the discrete spherical averages and some of their…

We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for the number of distinct group structures of all…

We prove that, when elliptic curves $E/\mathbb{Q}$ are ordered by height, the average number of integral points $\#|E(\mathbb{Z})|$ is bounded, and in fact is less than $66$ (and at most $\frac{8}{9}$ on the minimalist conjecture). By…

Let $p$ be a prime number and let $ k $ be a number field, which does not contain the field $\mathbb{Q} (\zeta_p + \bar{\zeta_p})$. Let $\mathcal{E}$ be an elliptic curve defined over $k$. We prove that if there are no $k$-rational torsion…

Let $C$ be an algebraic curve embedded transversally in a power $E^N$ of an elliptic curve $E$. In this article we produce a good explicit bound for the height of all the algebraic points on $C$ contained in the union of all proper…

We investigate some aspects of the $m$-division field $K({\mathcal{E}}[m])$, where $\mathcal{E}$ is an elliptic curve defined over a field $K$ with ${\textrm{char}}(K)\neq 2,3$ and $m$ is a positive integer. When $m=p^r$, with $p\geq 5$ a…

In this paper we give an upper bound for the number of integral points on an elliptic curve E over F_q[T] in terms of its conductor N and q. We proceed by applying the lower bounds for the canonical height that are analogous to those given…

In this paper we refine recent work due to A. Shankar, A. N. Shankar, and X. Wang on counting elliptic curves by conductor to the case of elliptic curves with a rational 2-torsion point. This family is a small family, as opposed to the…

We develop a general framework to study Szpiro's conjecture and the $abc$ conjecture by means of Shimura curves and their maps to elliptic curves, introducing new techniques that allow us to obtain several unconditional results for these…

We generalise the Siegel-Voloch theorem about S-integral points on elliptic curves as follows: let K/F denote a global function field over a finite field F of characteristic p>3, let S denote a finite set of places of K and let E/K denote a…

We consider the set of points in projective $n$-space that generate an extension of degree $e$ over given number field $k$, and deduce an asymptotic formula for the number of such points of absolute height at most $X$, as $X$ tends to…

We give an asymptotic lower bound on the number of field extensions generated by algebraic points on superelliptic curves over $\mathbb{Q}$ with fixed degree $n$ and discriminant bounded by $X$. For $C$ a fixed such curve given by an affine…

We show that for an elliptic curve E defined over a number field K, the group E(A) of points of E over the adele ring A of K is a topological group that can be analyzed in terms of the Galois representation associated to the torsion points…

Using the formalism of bar complexes and their relative versions, we give a new, purely algebraic, construction of the so-called universal elliptic KZB connection in arbitrary level. We compute explicit analytic formulae, and we compare our…

In this paper we investigate the Erd\"os/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been…