**Preface**

This book grew out of the NATO Advanced Study Institute held in Ohrid, Macedonia in 2014. The focus of the Advanced Study Institute was on the arithmetic of the superelliptic curves and their applications. From the mathematical point of view the conference was organized in three parts.

In the first part the focus is on superelliptic curves over the complex numbers, the automorphisms groups, and the corresponding Hurwitz loci. In this volume such topics correspond to papers [1–4]. In the second part, the book focuses on the arithmetic part of the subject. It covers some of the fundamental concepts in the area such as the method of descent, Chabauty and Mordell-Weill sieve, rational points on curves, heights on curves, and p-adic methods. It consists of papers [5–11]. The third part of this volume focuses in some applications of such curves. It consists of papers [12–15].

We have tried to keep the same structure for this book as that of the conference. Since the chapters are based mostly on the lectures of the ASI, this volume is more similar to a textbook than a volume proceedings. We have gone to great lengths to coordinate the lectures among the authors such that they follow some logical line and the material is not repeated from chapter to chapter.

We have included some papers which provide some background for the readers and were not part of the lectures in the conference. Hopefully, that will make this volume locically more complete and easier to read. Having said that we assume that the reader is familiar with the basic facts of algebraic geometry, commutative algebra, and number theory.

In [1] we make the case for superelliptic curves. The article is intended to be a motivation for the study of superelliptic curves which were the main focus of the NATO ASI. While the scope of interesting mathematical problems related to superelliptic curves is very broad and the applications include different areas of sciences, in this paper the authors focus on few of the arithmetic problems related to the moduli space of curves, automorphisms groups, minimal models of curves, and rational points on curves with the intention to emphasize the methods of extending the knowledge of hyperelliptic curves to all superelliptic curves.

A superelliptic curve, simply stated, is an algebraic curve with equation y^{n} = f(x) defined over some field k. The well known hyperelliptic curves, y^{2} = f(x), are the simplest case of superelliptic curves. There is a natural question to ask whether the rich mathematical theory of the hyperelliptic curves can be extended to all superelliptic curves. Moreover, one wonders if all of the applications of hyperelliptic curves such as cryptography, mathematical physics, quantum computation, diophantine geometry, etc. can carry over to the superelliptic curves.

In [2] the authors focus on the Weierstrass points of superelliptic curves. The Weierstrass points are an important concept of the theory of Riemann surfaces and algebraic curves and serve as a prerequisite for studying the automorphisms of the curves. In this lecture is given a detailed account of holomorphic and meromorphic functions of Riemann surfaces, the proofs of the Weierstrass and Noether gap theorems, the Riemann-Hurwitz theorem, and the basic definitions and properties of higher-order Weierstrass points.

In Lecture 1 the authors cover some of the basic material on Riemann surfaces and algebraic curves and their Weierstrass points. They describe some facts on the fixed points of automorphisms and prove the Hurwitz theorem on the bound of the automorphism group.

In Lecture 2, they describe linear systems and inflection points with an eye toward the Wronskian form, which is useful in computing these inflection points. Then, using a special linear system, they define Weierstrass points. They give the basic definitions of Weierstrass points using Riemann-Roch spaces as well as with spaces of holomorphic differentials and generalize this definition to discuss higher-order Weierstrass points (which are called q-Weierstrass points). Properties of these points, along with proofs and references, are given. This part concludes with Hurwitz’s theorem, which gives an upper bound for the number of automorphisms of a curve of genus g.

In Lecture 3, they examine Weierstrass points in a few contexts. First, they see some results pertaining to Weierstrass points on superelliptic curves, which can be thought of as generalizations of hyperelliptic curves. After that, they investigate group actions on non-hyperelliptic curves of genus g = 3.

In [3] is given a description of the theta functions of algebraic curves, half-integer theta-nulls, and the fundamental theta functions. The authors describe how to determine such fundamental theta functions and components of the moduli space in terms of such functions. Several open problems are suggested.

In [4] are studied the automorphism groups of real curves admitting a regular meromorphic function f of degree p, so called real cyclic p-gonal curves.

In [5] is given an introduction to reduction theory of binary forms. Since Gauss, the reduction theory of integral binary quadratic forms is quite completely understood. For binary Hermitian forms, this was studied starting with Hermite, Bianchi, and much developed by Elstrodt, Grunewland, and Mennicke see [5].

In 1848, Hermite introduced a reduction theory for binary forms of degree n which was developed more fully in 1917 by Julia in his thesis. For reducing binary forms of degree n, Julia introduced an irrational SL_{2}(${\mathbb{Z}}$)-invariant of binary forms which is known in the literature as Julia’s invariant. The author uses reduction theory of binary forms to study the following problems.

1) For any binary form f defined over a number field K, find an GL_{2}(K)-equivalent one f′ such that f′ has minimal height as defined in [7].

2) Given a fixed value of the discriminant Δ, or a given set of invariants, enumerate up to an SL_{2}([Dscr ]K)-equivalence, all forms f with the given discriminant or this set of invariants.

In [6] the author describes the analogue of the Sato-Tate conjecture for an abelian variety over a number field; this predicts that the zeta functions of the reductions over various finite fields, when properly normalized, have a limiting distribution predicted by a certain group-theoretic construction related to Hodge theory, Galois images, and endomorphisms. After making precise the definition of the Sato-Tate group appearing in this conjecture, we describe the classification of Sato-Tate groups of abelian surfaces due to Fité–Kedlaya–Rotger–Sutherland.

In [7] the authors cover some of the basics of the theory of heights. They define the minimal height of binary forms and moduli height for algebraic curves and prove that the moduli height of superelliptic curves [Hscr ](f) ≤ c_{0}$\tilde{H}$(f) where c_{0} is a constant and $\tilde{H}$ the minimal height of the corresponding binary form.

In Lecture 1 are defined affine and projective heights on the projective space, multiplicative and logarithmic heights, and absolute heights. The authors describe Northcott’s theorem, Kronecker’s theorem, and Segre embedding.

In Lecture 2, the authors focus on the heights of polynomials, Gauss lemma on heights, Gelfand’s inequality, and bounds on heights of homogenous polynomials acting on them by linear transformations on variables. The main focus of this lecture is on the heights of binary forms. For any binary form f they provide bounds for f^{M} when M ∊ SL_{2}(K). This leads to the definition of the minimal height $\tilde{H}$(f) and moduli height [Hscr ](f) for binary forms. They prove that [Hscr ](f) ≤ c_{0}$\tilde{H}$(f)^{n0} for any binary form f, where c_{0} and n_{0} are constants depending only on the degree of the binary form f.

Lecture 3 focuses on heights of algebraic curves. The main focus is in providing equations for the algebraic curves with “small” coefficients as continuation of our previous work. Hence, the concept of height is the natural concept to be used. For a genus g ≥ 2 algebraic curve χ_{g} defined over an algebraic number field K they define the height H_{K}(χ_{g}) and show that this is well-defined. This is basically the minimum height among all curves which are isomorphic to χ_{g} over K. ${\macr{H}_K (\chi_g)}$ is the height over the algebraic closure $\macr{K}$.

The main theme of [8] is how one can try to prove that a given curve over [Qscr ] does not possess rational points, with a focus on hyperelliptic curves. After introducing hyperelliptic curves and summarizing their most important properties, the author studies the notion of ‘everywhere local solubility’. A curve (or more generally, any variety) over [Qscr ] is everywhere locally soluble or ELS, if it has real points and points over [Qscr ]_{p} for all primes p. The author explains that it can be effectively tested whether a curve is ELS or not. Obviously, any curve with rational points is also ELS, so this provides a necessary condition. In most cases of interest (curves of genus zero being the exception), the necessary condition is not sufficient, and indeed a large proportion of curves are ELS, but do not possess rational points. We therefore need more sophisticated ways of proving the non-existence of rational points. One general approach that can be used is descent. In its most general version, it uses an unramified and geometrically Galois covering π : D → C of the given curve C. Associated to this covering is its Selmer set Sel(π). The Selmer set is finite and computable (at least in principle) and receives a map from the set of rational points C([Qscr ]). So Sel(π) = φ implies that the curve has no rational points. In the last part of these notes, we explain how a certain Selmer set, the 2-Selmer set, of a hyperelliptic curve can be computed in practice.

In [9] it is offered a brief sketch of various methods for computing the set of rational points on a curve, focusing on Chabauty and the Mordell–Weil sieve to give the reader with only modest knowledge of algebraic geometry a feel for some methods for computing the set of rational points on a curve of genus ≥ 2, especially Chabauty and the Mordell–Weil sieve.

In [10] the author describes how to prove the Mordell-Weil theorem for Jacobians of hyperelliptic curves over [Qscr ] and how to compute the rank and generators for the Mordell-Weil group.

In [11] the author gives an overview of some p-adic algorithms for computing with elliptic and hyperelliptic curves, starting with Kedlaya’s algorithm. While the original purpose of Kedlaya’s algorithm was to compute the zeta function of a hyperelliptic curve over a finite field, it has since been used in a number of other applications. In particular, the author describes how to use Kedlaya’s algorithm to compute Coleman integrals and p-adic heights on elliptic and hyperelliptic curves. Throughout, the author gives several numerical examples, and concludes by showing how to use Coleman integrals to explicitly find integral points on hyperelliptic curves whose Jacobians have Mordell-Weil rank equal to their dimension.

The goal of [12] this chapter is twofold. First, to provide a self contained, detailed and rigorous mathematical introduction to some aspects of the quantum error-correcting codes and especially quantum stabilizer codes and their connection to self-orthogonal linear codes. This has been done without venturing that much, if at all, into the world of physics. While most of the results presented are not new, it is not easy to extract a precise mathematical formulation of results and to provide their rigorous proofs by reading the vast number of papers in the field, quite a few of which are written by computer scientists or physicists. It is this formulation and proofs, some of which are new, that we present here. Techniques from algebra of finite fields as well as representations of finite abelian groups have been employed. The second goal is the construction of some stabilizer codes via self-orthogonal linear codes associated to algebraic curves.

In [13] is discussed encryption based on Walks in Algebraic Graphs (EWAGRA) which is used for protection of authors’ rights, access to electronic books or documents located at certain knowledge base (Information Quality Assurance Support Systems of a university, digital library supporting distance education, various digital archives and etc). The method allows to generate nonlinear stream ciphers, which have some similarity with a one-time pad: different keys produce distinct ciphertexts from the same plaintext.

In [14] the authors focus on Riemann zeta function and Riemann hypothesis over algebraic geometry codes. The paper provides a short and gentle introduction to zeta functions and the Riemann hypothesis for linear codes. This paper is organized as follows: section two covers the basics of linear codes, weight enumerators, and an elementary proof of MacWilliams’ identity.

In section three, the authors provide some historical background and motivation, introduce zeta functions for linear curves, and show various functional identities for zeta polynomials and zeta functions. Next, Riemann hypothesis is introduced for general virtual weight enumerators - a straight forward generalization of (code) weight enumerators. Last, a discussion on (virtual) codes that satisfy this hypothesis follows.

The paper [15] discusses a tight connection between Galois geometry, linear codes, and incidence matrices, that leads to new invariants for incidence structures. These new invariants admit both an algebraic and a geometric description, and are motivated by the longstanding Hamada conjecture about the minimum rank of the incidence matrices of the classical geometric designs and their use for the construction of majority-logic decodable error-correcting codes.

We hope that this volume will be useful to graduate students and new researchers in the field. We have urged the authors to keep the level of exposition as elementary as possible and in the same time include open research problems when possible.

We want first to thank NATO, for providing the funds of the Institute. Without such support this institute would have not been possible. Finally, we want to thank all the participants in the conference. Particular thanks to all the authors who contributed to this volume.