Higgs bundles over elliptic curves

Holomorphic Principal Bundles Over Elliptic Curves II: The Parabolic Construction

Let E be a smooth elliptic curve and let G be a simple complex algebraic group of rank r. We shall always assume that π1(G) is cyclic and that c is a generator. We shall freely identify c with the corresponding element in the center of the universal cover G̃ of G. A C∞ G-bundle ξ0 over E has a characteristic class c1(ξ0) ∈ H(E;π1(G)) ∼= π1(G) which determines ξ0 up to C ∞ isomorphism. The goal o...

Holomorphic Principal Bundles Over Elliptic Curves III: Singular Curves and Fibrations

Let G be a reductive complex linear algebraic group, which in this paper for simplicity we shall always assume to be simple and simply connected, and let Lie(G) = g. This paper is the third in a series concerned with the study of holomorphic principal G-bundles over elliptic fibrations. The first paper in the series studied the moduli space M of Gbundles on a smooth elliptic curve E by means of...

Vector bundles over algebraic curves

Elliptic Curves over C

that is both analytic (as a mapping of complex manifolds) and algebraic: addition of points in E(C) corresponds to addition in C modulo the lattice L. This correspondence between lattices and elliptic curves over C is known as the Uniformization Theorem; we will spend this lecture and part of the next proving it. To make the correspondence explicit, we need to specify the map Φ from C/L and an ...

Elliptic Curves over Q

All polynomials and rational functions in this essay are assumed to have coefficients in Q . Fix an integer n ≥ 1. An affine variety is a simultaneous irreducible system of polynomial equations in n variables. The Q -points, R -points and C points of the affine variety are all solutions of the polynomial system in Q , R n and C , respectively. Rational projective n-spaceg Q n is the set of line...