Publication

$K3$ surfaces with maximal complex multiplication

2025

Abstract

Let X be a complex projective K3 surface having complex multiplication by a CM field E , and let T_{X} be its transcendental lattice. We say that X has maximal complex multiplication if \mathrm{End}{\mathrm{Hdg}}(T{X}) is the ring of integers of E .For which CM fields E does such a K3 surface exist? What are the possibilities for the transcendental lattices, Picard lattices of these surfaces? The aim of this paper is to study these questions and give some examples.

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2025

Abstract

Official source

Related concepts (32)

K3 surface

In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface in complex projective 3-space.

Complex multiplication

In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points.

List of complex and algebraic surfaces

This is a list of named algebraic surfaces, compact complex surfaces, and families thereof, sorted according to their Kodaira dimension following Enriques–Kodaira classification. Projective plane Cone (geometry) Cylinder Ellipsoid Hyperboloid Paraboloid Sphere Spheroid Cayley nodal cubic surface, a certain cubic surface with 4 nodes Cayley's ruled cubic surface Clebsch surface or Klein icosahedral surface Fermat cubic Monkey saddle Parabolic conoid Plücker's conoid Whitney umbrella Châtelet surfaces Dupin

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