42 citations · 119 across the 37 of their papers we have counts for
28 papers · 1 filter
Nonabelian Chabauty for the Thrice-punctured Line over Cyclotomic Fields
Minhyong Kim, Xiang Li, Martin Lüdtke
In this paper we study the motivic Chabauty--Kim method, which aims to determine the set of -integral points of , over cyclotomic fiel…
Path Integrals and p-adic L-functions
Magnus Carlson, Hee-Joong Chung, Dohyeong Kim +3
We prove an arithmetic path integral formula for the inverse p-adic absolute values of the Kubota-Leopoldt p-adic L-functions at roots of unity.
A note on abelian arithmetic BF-theory
Magnus Carlson, Minhyong Kim
We compute some arithmetic path integrals for BF-theory over the ring of integers of a totally imaginary field, which evaluate to natural arithmetic invariants associated to $\math…
Arithmetic Gauge Theory: A Brief Introduction
Minhyong Kim
Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Di…
Abelian arithmetic Chern-Simons theory and arithmetic linking numbers
Hee-Joong Chung, Dohyeong Kim, Minhyong Kim +3
Following the method of Seifert surfaces in knot theory, we define arithmetic linking numbers and height pairings of ideals using arithmetic duality theorems, and compute them in t…
Arithmetic Chern-Simons Theory II
Hee-Joong Chung, Dohyeong Kim, Minhyong Kim +2
We apply ideas of Dijkgraaf and Witten on three-dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fie…
Arithmetic Chern-Simons Theory I
Minhyong Kim
In this paper, we apply ideas of Dijkgraaf and Witten on 2+1 dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebra…
Diophantine geometry and non-abelian reciprocity laws I
Minhyong Kim
We use non-abelian fundamental groups to define a sequence of higher reciprocity maps on the adelic points of a variety over a number field satisfying certain conditions in Galois…
On the 2-part of the Birch-Swinnerton-Dyer conjecture for elliptic curves with complex multiplication
John Coates, Minhyong Kim, Zhibin Liang +1
Given an elliptic curve E over Q with complex multiplication having good reduction at 2, we investigate the 2-adic valuation of the algebraic part of the L-value at 1 for a family…
A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves
Jennifer Balakrishnan, Ishai Dan-Cohen, Minhyong Kim +1
We state a conjectural criterion for identifying global integral points on a hyperbolic curve over in terms of Selmer schemes inside non-abelian cohomology functors wi…
Irreducible modular representations of the Borel subgroup of GL_2(Q_p)
Laurent Berger, Mathieu Vienney
Let E be a finite extension of Fp. Using Fontaine's theory of (phi,Gamma)-modules, Colmez has shown how to attach to any irreducible E-linear representation of Gal(Qpbar/Qp) an inf…
Non-abelian fundamental groups in arithmetic geometry
Minhyong Kim
This is a brief exposition of the mathematical themes that motivate the special programme at the Newton Institute in 2009. It is mostly intended for the general public having mathe…
Galois theory and Diophantine geometry
Minhyong Kim
This is an essay to accompany the author's lecture at the introductory workshop on `Nonabelian fundamental groups in arithmetic geometry' at the Newton Institute, Cambridge in July…
Massey products for elliptic curves of rank 1
Minhyong Kim
For an elliptic curve over Q of analytic rank 1, we use the level-two Selmer variety and secondary cohomology products to find explicit analytic defining equations for global integ…
Selmer varieties for curves with CM Jacobians
John Coates, Minhyong Kim
We study the Selmer variety associated to a canonical quotient of the $\Q_p$-pro-unipotent fundamental group of a smooth projective curve of genus at least two defined over $\Q$ wh…
p-adic L-functions and Selmer varieties associated to elliptic curves with complex multiplication
Minhyong Kim
We show how non-vanishing of p-adic L functions controls the dimensions of Selmer varieties associated to the complement of the origin in an elliptic curve with CM. As a corollary,…
Diophantine Geometry as Galois Theory in the Mathematics of Serge Lang
Minhyong Kim
A remark about the role of Galois theory in Diophantine geometry as reflected in the work of Serge Lang. An entry in `The mathematical contributions of Serge Lang.'
Remark on fundamental groups and effective Diophantine methods for hyperbolic curves
Minhyong Kim
We discuss how non-commutative fundamental groups could eventually contribute to algorithms for finding rational points on hyperbolic curves.
The l-component of the unipotent Albanese map
Minhyong Kim, Akio Tamagawa
We study the \Q_p-unipotent Albanese map for curves over local fields with residue characteristic different from p and show that it has finite image. As a corollary, we deduce a si…
The unipotent Albanese map and Selmer varieties for curves
Minhyong Kim
We discuss -adic unipotent Albanese maps for curves of positive genus, extending the theory of -adic multiple polylogarithms. This construction is then used to relate linear…
On relative computability for curves
Minhyong Kim
We discuss a rational version of a conjecture of Matiyasevich, Davis, and Putnam on the relative decidability of the finiteness problem for Diophantine equations with respect to th…
The motivic fundamental group of the projective line minus three points and the theorem of Siegel
Minhyong Kim
In this paper, we establish a link between the structure theory of the pro-unipotent motivic fundamental group of the projective line minus three points and Diophantine geometry. I…
Torsion points on modular curves and Galois theory
Kenneth A. Ribet, Minhyong Kim
We outline the proof of a theorem of M. Baker and A. Tamagawa that gives a complete description of the torsion points on a modular curve embedded in its Jacobian using the notion o…
A note on Szpiro's inequality for curves of higher genus
Minhyong Kim
This is a short addendum to a note of Beauville on the subject of the title. We prove an inequality that takes into account the constant part of the Jacobian.