Related papers: The Unipotent Chabauty-Kim-Kantor Method for Relat…
In this thesis we develop a Chabauty-Kim theory for the relative completion of motivic fundamental groups, including Selmer stacks and moduli spaces of admissible torsors for the relative completion of the de Rham fundamental group. On one…
The Chabauty--Kim method was developed with the aim of approaching effective Faltings', the problem of explicitly determining the finite set of rational points on a hyperbolic curve. This method has seen success with the more particular…
In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian…
We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets…
The Chabauty--Coleman--Kim method, under favourable circumstances, describes the set of integral points of a hyperelliptic curve inside the $p$-adic zeroes of certain transcendental functions. For an elliptic curve of Mordell--Weil rank…
The Chabauty--Kim method is a tool for finding the integral or rational points on varieties over number fields via certain transcendental $p$-adic analytic functions arising from certain Selmer schemes associated to the unipotent…
The Chabauty--Kim method is a method for finding rational points on curves under certain technical conditions, generalising Chabauty's proof of the Mordell conjecture for curves with Mordell--Weil rank less than their genus. We show how the…
We give sufficient conditions for finiteness of linear and quadratic refined Chabauty-Kim loci of affine hyperbolic curves. We achieve this by constructing depth $\leq 2$ quotients of the fundamental group, following a construction of…
In this paper we prove the finiteness of the set of S-integral points of a punctured rational elliptic curve without complex multiplication using the Chabauty-Kim method. This extends previous results of Kim in the complex multiplication…
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…
The main point of the paper is to take the explicit motivic Chabauty-Kim method developed in papers of Dan-Cohen--Wewers and Dan-Cohen and the author and make it work for non-rational curves. In particular, we calculate the abstract form of…
Reducibility methods, aiming to simplify systems by conjugating them to those with constant coefficients, are crucial for studying the existence of quasiperiodic solutions. In KAM theory for PDEs, these methods help address the…
Faltings' theorem states that curves of genus $g \geq 2$ have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the number of rational points, but this bound is too large…
We give an alternative proof of Faltings's theorem (Mordell's conjecture): a curve of genus at least two over a number field has finitely many rational points. Our argument utilizes the set-up of Faltings's original proof, but is in spirit…
The Chabauty-Coleman method is a $p$-adic method for finding all rational points on curves of genus $g$ whose Jacobians have Mordell-Weil rank $r < g$. Recently, Edixhoven and Lido developed a geometric quadratic Chabauty method that was…
This thesis is devoted to the study of hyperbolic differential operators on globally hyperbolic manifolds, linear gauge theories and their quantisation. In the first part, we treat the Cauchy problem for symmetric hyperbolic systems and…
Let $X= \mathbb{P}^1 \setminus \{0,1,\infty\}$, and let $S$ denote a finite set of prime numbers. In an article of 2005, Minhyong Kim gave a new proof of Siegel's theorem for $X$: the set $X(\mathbb{Z}[S^{-1}])$ of $S$-integral points of…
Kitchloo and Morava give a strikingly simple picture of elliptic cohomology at the Tate curve by studying a completed version of $S^1$-equivariant $K$-theory for spaces. Several authors (cf [ABG],[KM],[L]) have suggested that an equivariant…
The Cauchy problem for a quasilinear system of hyperbolic-parabolic equations is addressed with the method of linearization and fixed point. Coupling between the hyperbolic and parabolic variables is allowed in the linearization and we do…
This paper presents a dissipativeness analysis of a quadrature method of moments (called HyQMOM) for the one-dimensional BGK equation. The method has exhibited its good performance in numerous applications. However, its mathematical…