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

This paper addresses the statistical problem of estimating the infinite-norm deviation from the empirical mean to the distribution mean for high-dimensional distributions on $\{0,1\}^d$, potentially with $d=\infty$. Unlike traditional…

We produce explicit elliptic curves over \Bbb F_p(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related…

Let E be an elliptic curve over Q, and let n=>1. The central object of study of this article is the division field Q(E[n]) that results by adjoining to Q the coordinates of all n-torsion points on E(Q). In particular, we classify all curves…

We establish asymptotic lower bounds for the number of elliptic curves over $\mathbb{Q}$ with prescribed entanglement of division fields, ordered by naive height. Such elliptic curves are obtained as $1$-parameter families arising from…

We prove an equidistribution result for totally geodesic submanifolds in a compact locally symmetric space. In the case of Hermitian locally symmetric spaces, this gives a convergence theorem for currents of integration along totally…

We formulate a multi-variable p-adic Birch and Swinnerton-Dyer conjecture for p-ordinary elliptic curves A over number fields K. It generalises the one-variable conjecture of Mazur-Tate-Teitelbaum, who studied the case K=Q and the…

Let $E$ be an elliptic curve with good reduction at a fixed odd prime $p$ and $K$ an imaginary quadratic field where $p$ splits. We give a growth estimate for the Mordell-Weil rank of $E$ over finite extensions inside the…

Let $ \lambda ^2 \in \mathbb N $, and in dimensions $ d\geq 5$, let $ A_{\lambda } f (x)$ denote the average of $ f \;:\; \mathbb Z ^{d} \to \mathbb R $ over the lattice points on the sphere of radius $\lambda$ centered at $x$. We prove $…

Given an elliptic curve $E$ over a finite field $\mathbb{F}_q$ we study the finite extensions $\mathbb{F}_{q^n}$ of $\mathbb{F}_q$ such that the number of $\mathbb{F}_{q^n}$-rational points on $E$ attains the Hasse upper bound. We obtain an…

In this paper we relate some classical normal forms for complex elliptic curves in terms of 4-point sets in the Riemann sphere. Our main result is an alternative proof that every elliptic curve is isomorphic as a Riemann surface to one in…

The Riesz $s$-energy of an $N$-point configuration in the Euclidean space $\mathbb{R}^{p}$ is defined as the sum of reciprocal $s$-powers of all mutual distances in this system. In the limit $s\to0$ the Riesz $s$-potential $1/r^s$ ($r$ the…

We establish new uniform height inequalities for rational points on higher-dimensional varieties, extending the classical Roth-Schmidt-Subspace paradigm to the Arakelov-theoretic setting. Our main result provides sharp bounds for heights…

We consider finite point subsets (distributions) in compact metric spaces. Non-trivial bounds for sums of distances between points of distributions and for discrepancies of distributions in metric balls are given in the case of general…

Let $P$ be a set of $n$ points in the real plane contained in an algebraic curve $C$ of degree $d$. We prove that the number of distinct distances determined by $P$ is at least $c_d n^{4/3}$, unless $C$ contains a line or a circle. We also…

We characterize the possible groups $E(\mathbb{Z}/N\mathbb{Z})$ arising from elliptic curves over $\mathbb{Z}/N\mathbb{Z}$ in terms of the groups $E(\mathbb{F}_p)$, with $p$ varying among the prime divisors of $N$. This classification is…

We consider the local-global principle for divisibility in the Mordell-Weil group of a CM elliptic curve defined over a number field. For each prime $p$ we give sharp lower bounds on the degree $d$ of a number field over which there exists…

In the first part of the paper, we fix a non-CM elliptic curve $E/\mathbb{Q}$ and an odd prime $\ell$ and investigate the distribution of invariants associated to the $\ell$-volcano containing the reduction $E_p$, as $p$ ranges over primes…

Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/Q and non-torsion point P in E(Q), there is at most one…

The well-known analogies between number fields and function fields have led to the transposition of many problems from one domain to the other. In this paper, we will discuss traffic of this sort, in both directions, in the theory of…

In this paper we develop a new explicit method to studying rational points near manifolds and obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve.…