* 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

Groupes et algébres de lie

Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Arithmetic duality theorems

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

Basic Algebraic Geometry

has only trivial endomorphisms over an algebraic closure of the ground field K if the Galois group Gal(f) of the polynomial f ∈ K[x] is “very big”. More precisely, if f is a polynomial of degree n ≥ 5 and Gal(f) is either the symmetric group Sn or the alternating group An then End(J(C)) = Z. Notice that it easily follows that the ring of K-endomorphisms of J(C) coincides with Z and the real problem is how to prove that every endomorphism of J(C) is defined over K. There are some results of this type in the literature. Previously Mori [8], [9] has constructed explicit examples (in all characteristics) of hyperelliptic jacobians without nontrivial endomorphisms. In particular, he provided examples over Q with semistable Cf and big (doubly transitive) Gal(f) [9]. The semistability of Cf implies the semistability of J(Cf ) and, thanks to a theorem of Ribet [14], all endomorphisms of J(Cf ) are defined over Q. (Applying to Cf/Q the Shafarevich conjecture [17] (proven by Fontaine [3] and independently by Abrashkin [1], [2]) and using Lemma 4.4.3 and arguments on p. 42 of [16], one may prove that the Galois group Gal(f) of the polynomial f involved is S2g+1 where deg(f) = 2g + 1.) Andre ([7], pp. 294-295) observed that results of Katz ([5], [6]) give rise to examples of hyperelliptic jacobians J(Cf ) over the field of rational function C(z) with End(J(Cf )) = Z. Namely, one may take f(x) = h(x)−z where h(x) ∈ C[x] is a Morse function. In particular, this explains Mori’s example [8]

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2000/0007/0001/MRL-2000-0007-0001-a011.pdf

Hyperelliptic jacobians without complex multiplication

Let k be a field finitely generated over the field of rational numbers, and Br(k) the Brauer group of k. For an algebraic variety X over k we consider the cohomological Brauer–Grothendieck group Br(X). We prove that the quotient of Br(X) by the image of Br(k) is finite if X is a K3 surface. When X is an abelian variety over k, and X is the variety over an algebraic closure k of k obtained from X by the extension of the ground field, we prove that the image of Br(X) in Br(X) is finite.

/pdf/a-finiteness-theorem-for-the-brauer-group-of-abelian-56x3abjb8a.pdf

A finiteness theorem for the Brauer group of abelian varieties and 3 surfaces

Let $k$ be a field that is finitely generated over the field of rational numbers and $Br(k)$ the Brauer group of $k$. Let $X$ be an absolutely irreducible smooth projective variety over $k$, let $Br(X)$ be the cohomological Brauer-Grothendieck group of $X$ and $Br_0(X)$ the image of $Br(k)$ in $Br(X)$. We write $Br_1(X)$ for the subgroup of elements in $Br(X)$ that become trivial after replacing $k$ by its algebraic closure. 
We prove that $Br(X)/Br_0(X)$ is finite if $X$ is a $K3$ surface. When $X$ is (a principal homogeneous space of) an abelian variety over $k$ then we prove that $Br(X)/Br_1(X)$ is finite.

A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces

We prove that the jacobian of a hyperelliptic curve y^2=f(x) has no nontrivial endomorphisms over an algebraic closure of the ground field of characteristic zero if the Galois group of the polynomial f is ``very big''.

We obtain an easy sufficient condition for the Brauer group of a diagonal quartic surface D over Q to be algebraic. We also give an upper bound for the order of the quotient of the Brauer group of D by the image of the Brauer group of Q. The proof is based on the isomorphism of the Fermat quartic surface with a Kummer surface due to Mizukami.

/pdf/on-the-brauer-group-of-diagonal-quartic-surfaces-54ukb8gzyp.pdf

Yuri G. Zarhin

Papers

Hyperelliptic jacobians without complex multiplication

A finiteness theorem for the Brauer group of abelian varieties and 3 surfaces

A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces

Hyperelliptic jacobians without complex multiplication

On the Brauer group of diagonal quartic surfaces