Related papers: The Chabauty--Coleman method and p-adic linear for…
We formulate a non-abelian generalisation of a conjecture of Stoll, which conjecturally describes the structure of the loci cut out by Kim's method of non-abelian Chabauty. We prove the rank 0 quadratic case of this conjecture, which in…
We establish a new perturbation theory for orthogonal polynomials using a Riemann--Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with…
We investigate several interrelated foundational questions pertaining to the study of motivic dga's of Dan-Cohen--Schlank [8] and Iwanari [13]. In particular, we note that morphisms of motivic dga's can reasonably be thought of as a…
We present several upper bounds for the height of global residues of rational forms on an affine variety. As a consequence, we deduce upper bounds for the height of the coefficients in the Bergman-Weil trace formula. We also present upper…
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 goal of this paper is to study a $p$-adic analog of the joint of the conjectures of Andr\'e--Oort and Andr\'e--Pink. More precisely, on a product of ordinary Siegel formal moduli schemes, we study the distribution of points whose…
We prove a special case of the following conjecture of Zilber-Pink generalising the Manin-Mumford conjecture : let $X$ be a curve inside an Abelian variety $A$ over $\bar{\Q}$, provided $X$ is not contained in a torsion subvariety, the…
$p$-adic continued fractions, as an extension of the classical concept of classical continued fractions to the realm of $p$-adic numbers, offering a novel perspective on number representation and approximation. While numerous $p$-adic…
The mathematical basis of p-adic Higgs mechanism discussed in papers [email protected] 9410058-62 is considered in this paper. The basic properties of p-adic numbers, of their algebraic extensions and the so called canonical…
This is a sequel of our paper [arXiv:1809.08425] on the Quot-scheme limit and variational properties of Donaldson's functional, which established its coercivity for slope stable holomorphic vector bundles over smooth projective varieties.…
We study the inverse Jacobian problem for the case of Picard curves over $\mathbb{C}$. More precisely, we elaborate on an algorithm that, given a small period matrix $\Omega\in \mathbb{C}^{3\times 3}$ corresponding to a principally…
We generalize the work of Dem'janenko and Silverman for the Fermat quartics, effectively determining the rational points on the curves $x^{2m}+ax^m+ay^m+y^{2m}=b$ whenever the ranks of some companion hyperelliptic Jacobians are at most one.…
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 establish a new transcendence criterion of $p$-adic continued fractions which are called Ruban continued fractions. By this result, we give explicit transcendental Ruban continued fractions with bounded $p$-adic absolute value of partial…
Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the…
For any family of elliptic curves over the rational numbers with fixed $j$-invariant, we prove that the existence of a long sequence of rational points whose $x$-coordinates form a non-trivial arithmetic progression implies that the…
Jacobi's results on the computation of the order and of the normal forms of a differential system are translated in the formalism of differential algebra. In the quasi-regular case, we give complete proofs according to Jacobi's arguments.…
Let $k$ be a number field. We investigate the Mordell-Weil ranks of Jacobian varieties $J_C$ associated with algebraic curves $C$ of genus $g \geq 1$ defined by affine equations of the form $y^s=x(ax^r+b)$, where $a, b \in k$ ($ab \neq 0$),…
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…
Variational inequalities represent a broad class of problems, including minimization and min-max problems, commonly found in machine learning. Existing second-order and high-order methods for variational inequalities require precise…