Related papers: Quadratic Chabauty: p-adic height pairings and int…

Given a smooth, proper, geometrically integral curve $X$ of genus $g$ with Jacobian $J$ over a number field $K$, Chabauty's method is a $p$-adic technique to bound $\# X(K)$ when $\mathrm{rank}\ J(K) < g$. We study limitations of a variant…

We compute the Mordell-Weil groups of the modular Jacobian varieties of hyperelliptic modular curves $X_1(M, MN)$ over every number field which is the composition of quadratic fields. Also we prove criteria for the existence of elliptic…

In this paper, we present efficient algorithms for computing the number of points and the order of the Jacobian group of a superelliptic curve over finite fields of prime order p. Our method employs the Hasse-Weil bounds in conjunction with…

For a superelliptic curve $\mathcal X$, defined over $\mathbb Q$, let $\mathfrak p$ denote the corresponding moduli point in the weighted moduli space. We describe a method how to determine a minimal integral model of $\mathcal X$ such…

In this paper, we extend a uniformity result of Dimitrov-Gao-Habegger to dimension two and use it to get a uniform bound on the set of all quadratic points for non-hyperelliptic non-bielliptic curves in terms of the Mordell-Weil rank.

We study an infinite family of Mordell curves (i.e. the elliptic curves in the form y^2=x^3+n, n \in Z) over Q with three explicit integral points. We show that the points are independent in certain cases. We describe how to compute bounds…

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 explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in…

We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus four, five, and six. We…

We develop a strategy for bounding from above the height of rational points of modular curves with values in number fields, by functions which are polynomial in the curve's level. Our main technical tools come from effective Arakelov…

We study abelian varieties defined over function fields of curves in positive characteristic $p$, focusing on their arithmetic within the system of Artin-Schreier extensions. First, we prove that the $L$-function of such an abelian variety…

Quadratic points of a surface in the projective 3-space are the points which can be exceptionally well approximated by a quadric. They are also singularities of a 3-web in the elliptic part and of a line field in the hyperbolic part of the…

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…

We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usually called the method of Chabauty--Kim or non-abelian…

Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a…

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a…

We prove a general formula for the $p$-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above $p$. The formula is in terms of the cyclotomic derivative of a…

Consider a smooth, geometrically irreducible, projective curve of genus $g \ge 2$ defined over a number field of degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that…

Consider a hyperelliptic integral $I=\int P/(Q\sqrt{S}) dx$, $P,Q,S\in\mathbb{K}[x]$, with $[\mathbb{K}:\mathbb{Q}]<\infty$. When $S$ is of degree $\leq 4$, such integral can be calculated in terms of elementary functions and elliptic…

In this paper, we prove a combination theorem for a relatively acylindrical graph of relatively hyperbolic groups (Theorem 1.1). Here, we are extending the technique of [Tom21] and constructing Bowditch boundary of the fundamental group of…