Sphere Packings, Lattices and Groups

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.

Lectures on K3 Surfaces

In this chapter, we give the definition of Mordell–Weil lattice (in Sect. 6.5). First, we bring together the concepts from Chaps. 4 and 5 in order to gain a better understanding of the Neron–Severi lattice of an elliptic surface. This will lead to the announced notion of Mordell–Weil lattice which will be studied in detail in this chapter, but also throughout the remainder of this book.

Mordell-Weil Lattices

Historically the theory of quadratic forms has its origins in number-theoretic questions of the following type: Which integers can be written in the form x 2 + 2y 2 , which are sums of three squares, or more generally, which integers can be represented by an arbitrary quadratic form Σ a ij x i x j integral coefficients? This general question is exceptionally difficult and we are still quite far from a complete solution. It is natural and considerably simpler to first investigate these questions over the field of rational numbers, that is, to ask for rational instead of integral solutions to the equation Σ a ij x i x j = a. This leads to the problem of classification of quadratic forms over $ \mathbb{Q} $, which was first solved by Minkowski. His solution appears in this chapter basically unaltered, except for a few simplifications and the use of modern terminology. The Gaussian sums of Gauss and Dirich-let play a significant role in the more formal algebraic part of the theory.

Rational Quadratic Forms

We make a complete list of all possible ADE-types of singular fibers of complex elliptic K3 surfaces and the torsion parts of their MordellWeil groups.

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On elliptic K3 surfaces.

We present a complete list of extremal elliptic K3 surfaces (Theorem 1.1). As an application, we give a sufficient condition for the topological fundamental group of complement to an ADE-configuration of smooth rational curves on a K3 surface to be trivial (Proposition 4.1 and Theorems 4.3).

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Classification of extremal elliptic {$K3$} surfaces and fundamental groups of open {$K3$} surfaces

A K3 surface X defined over a field k of characteristic 0 is called singular if the Neron-Severi lattice NS(X) of X ⊗ k is of rank 20. Let X be a singular K3 surface defined over a number field F. For each embedding σ: F → C, we denote by T(X") the transcendental lattice of the complex K3 surface X σ obtained from X by σ. For each prime p of F at which X has a supersingular reduction Xp, we define L(X,p) to be the orthogonal complement of NS(X) in NS(Xp). We investigate the relation between these lattices T(X σ ) and L(X,p). As an application, we give a lower bound for the degree of a number field over which a singular K3 surface with a given transcendental lattice can be defined.

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Ichiro Shimada

Papers

On elliptic K3 surfaces.

Classification of extremal elliptic {$K3$} surfaces and fundamental groups of open {$K3$} surfaces

Transcendental lattices and supersingular reduction lattices of a singular K3 surface

Projective models of the supersingular K3 surface with Artin invariant 1 in characteristic 5

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