Related papers: Computing p-adic heights on hyperelliptic curves

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 propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility…

The eigencurve is a powerful tool introduced by Coleman and Mazur to study $p$-adic families of overconvergent modular forms. In this article, we introduce an analogous set of tools for understanding families of "overconvergent" $p$-adic…

We use Arakelov theory to define a height on divisors of degree zero on a hyperelliptic curve over a global field, and show that this height has computably bounded difference from the N\'eron-Tate height of the corresponding point on the…

Let $p$ be an odd prime number. We propose an algorithm for computing rational representations of isogenies between Jacobians of hyperelliptic curves via-adic differential equations with a sharp analysis of the loss of precision.…

We establish a relation between intersection numbers of special cycles on a Shimura curve and special values of derivatives of metaplectic Eisenstein series at a place of bad reduction where p-adic uniformization in the sense of Cherednik…

In arithmetic and algebraic geometry, superspecial (s.sp.\ for short) curves are one of the most important objects to be studied, with applications to cryptography and coding theory. If $g \geq 4$, it is not even known whether there exists…

Special kinds of continued fractions have been proved to converge to transcendental real numbers by means of the celebrated Subspace Theorem. In this paper we study the analogous $p$--adic problem. More specifically, we deal with Browkin…

This paper deals with the determination of the S-curves in the theory of non-hermitian orthogonal polynomials with respect to exponential weights along suitable paths in the complex plane. It is known that the corresponding complex…

We propose a new algorithm for approximating the metric projection onto a superelliptic disk of order $p>1$, i.e., the convex hull of a superellipse (Lam\'e curve), and prove its convergence.

A new incremental algorithm for data compression is presented. For a sequence of input symbols algorithm incrementally constructs a p-adic integer number as an output. Decoding process starts with less significant part of a p-adic integer…

This note explores the consequences of Koenigsmann's model theoretic argument from the proof of the birational p-adic section conjecture for curves in the context of higher dimensional varieties over p-adic local fields.

In this paper, we are concerned with the computations of the $p$-rank of curves in two different setups. We first work with complete intersection varieties in $\mb{P}^n \text{ for}~n\ge 2$ and compute explicitly the action of Frobenius on…

We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more…

We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $\mathbb Q$ that is substantially faster than previous approaches. Our algorithm takes as input a…

Let $B$ be a simple CM abelian variety over a CM field $E$, $p$ a rational prime. Suppose that $B$ has potentially ordinary reduction above $p$ and is self-dual with root number $-1$. Under some further conditions, we prove the generic…

We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel…

Let $K$ be a local field of residue characteristic $p>0$. We explain how to compute the semistable reduction of $K$-curves $Y$ equipped with a degree-$p$ morphism from $Y$ to the projective line. This includes the reduction at $p$ of…

Relying on the classical connection between Backward Stochastic Differential Equations (BSDEs) and non-linear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional…

We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper…