Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

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Analytic Number Theory

For an algebraic function fieldF having a finite constant field, letg(F) (resp.N(F)) denote the genus ofF (resp. the number of places ofF of degree one). We construct a tower of function fields$F_1 \subseteq F_2 \subseteq F_3 \subseteq \ldots $ over$\mathbb{F}_{q^2 } $ such that the ratioN(Fi)/g(Fi) tends to the Drinfeld-Vladut boundq−1.

A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound

Journal of Algebra

Poised to become the leading reference in the field, the Handbook of Finite Fields is exclusively devoted to the theory and applications of finite fields. More than 80 international contributors compile state-of-the-art research in this definitive handbook. Edited by two renowned researchers, the book uses a uniform style and format throughout and each chapter is self contained and peer reviewed. The first part of the book traces the history of finite fields through the eighteenth and nineteenth centuries. The second part presents theoretical properties of finite fields, covering polynomials, special functions, sequences, algorithms, curves, and related computational aspects. The final part describes various mathematical and practical applications of finite fields in combinatorics, algebraic coding theory, cryptographic systems, biology, quantum information theory, engineering, and other areas. The book provides a comprehensive index and easy access to over 3,000 references, enabling you to quickly locate up-to-date facts and results regarding finite fields.

Handbook of Finite Fields

The book is devoted to the theory of algebraic geometric codes, a subject formed on the border of several domains of mathematics On one side there are such classical areas as algebraic geometry and number theory; on the other, information transmission theory, combinatorics, finite geometries, dense packings, etc The authors give a unique perspective on the subject Whereas most books on coding theory build up coding theory from within, starting from elementary concepts and almost always finishing without reaching a certain depth, this book constantly looks for interpretations that connect coding theory to algebraic geometry and number theory There are no prerequisites other than a standard algebra graduate course The first two chapters of the book can serve as an introduction to coding theory and algebraic geometry respectively Special attention is given to the geometry of curves over finite fields in the third chapter Finally, in the last chapter the authors explain relations between all of these: the theory of algebraic geometric codes

Algebraic Geometric Codes: Basic Notions

... The summary above gives but a glimpse of the richness of the contents of the book, which is an outstanding contribution to the literature on algebraic curves, a true vade mecum for researchers and students in the field. Mathematical Reviews

Algebraic Curves Over a Finite Field

https://www.ime.unicamp.br/~ftorres/RESEARCH/ARTS_PDF/FGT_JNT.pdf

On Maximal Curves

We generalize results of Homma and Kim [J. Pure Appl. Algebra Vol. 162, (2001), pp. 273--290] concerning an improvement on the Goppa bound on the minimum distance of certain Goppa codes.

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On Goppa Codes and Weierstrass Gaps at Several Points

We prove the following result which was conjectured by Stichtenoth and Xing: letg be the genus of a projective, irreducible non-singular algebraic curve over the finite field$\mathbb{F}_{q^2 } $ and whose number of$\mathbb{F}_{q^2 } $-rational points attains the Hasse-Weil bound; then either 4g≤(q−1)2 or 2g=(q−1)q.

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The genus of curves over finite fields with many rational points

For the Hermitian curve H defined over the finite field , we give a complete classification of Galois coverings of H of prime degree. The corresponding quotient curves turn out to be special cases of wider families of curves -covered by H arising from subgroups of the special linear group SL(2,F q ) or from subgroups in the normaliser of the Singer group of the projective unitary group . Since curves -covered by H are maximal over , our results provide some classification and existence theorems for maximal curves having large genus, as well as several values for the spectrum of the genera of maximal curves. For every q 2, both the upper limit and the second largest genus in the spectrum are known, but the determination of the third largest value is still in progress. A discussion on the “third largest genus problem“ including some new results and a detailed account of current work is given.

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Fernando Torres

Papers

Algebraic Curves Over a Finite Field

On Maximal Curves

On Goppa Codes and Weierstrass Gaps at Several Points

The genus of curves over finite fields with many rational points

Curves of large genus covered by the hermitian curve