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

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)

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Character Theory of Finite Groups

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

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 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

It has been known for a long time that the Deligne–Lusztig curves associated to the algebraic groups of type ${^2A_2,\,^2B_2}$ and ${^2G_2}$ defined over the finite field ${\mathbb {F}_n}$ all have the maximum number of ${\mathbb {F}_n}$-rational points allowed by the Weil “explicit formulas”, and that these curves are ${\mathbb {F}_{q^2}}$-maximal curves over infinitely many algebraic extensions ${\mathbb {F}_{q^2}}$ of ${\mathbb {F}_n}$. Serre showed that an ${\mathbb {F}_{q^2}}$-rational curve which is ${\mathbb {F}_{q^2}}$-covered by an ${\mathbb {F}_{q^2}}$-maximal curve is also ${\mathbb {F}_{q^2}}$-maximal. This has posed the problem of the existence of ${\mathbb {F}_{q^2}}$-maximal curves other than the Deligne–Lusztig curves and their ${\mathbb {F}_{q^2}}$-subcovers, see for instance Garcia (On curves with many rational points over finite fields. In: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pp. 152–163. Springer, Berlin, 2002) and Garcia and Stichtenoth (A maximal curve which is not a Galois subcover of the Hermitan curve. Bull. Braz. Math. Soc. (N.S.) 37, 139–152, 2006). In this paper, a positive answer to this problem is obtained. For every q = n3 with n = pr > 2, p ≥ 2 prime, we give a simple, explicit construction of an ${\mathbb {F}_{q^2}}$-maximal curve ${\mathcal {X}}$ that is not ${\mathbb {F}_{q^2}}$-covered by any ${\mathbb {F}_{q^2}}$-maximal Deligne–Lusztig curve. Furthermore, the ${\mathbb {F}_{q^2}}$-automorphism group Aut${(\mathcal {X})}$ has size n3(n3 + 1)(n2 − 1)(n2 − n + 1). Interestingly, ${\mathcal {X}}$ has a very large ${\mathbb {F}_{q^2}}$-automorphism group with respect to its genus ${g = \frac{1}{2}\,(n^3 + 1)(n^2 - 2) + 1}$.

A new family of maximal curves over a finite field

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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Curves of large genus covered by the hermitian curve

On Curves Covered by the Hermitian Curve

This paper is concerned with certain point-sets T in a projective plane PG (2, q) over GF (q) which have only three characters with respect to the lines. We assume throughout this paper that for any line l of πwhere It is easily seen that if t = 1 then T is a (q + 1)-arc, i.e. an oval; otherwise T is a (q+t, t)-arc of type (0, 2, t). Therefore (q+t, t)-arcs of type (0, 2, t) appear to be a generalization of ovals and there are interesting connections between ovals and (q + t, t)-arcs of type (0, 2, t) from various points of view. Our purpose is to investigate such particular (k, t)-arcs using some ideas of B. Segre developed for ovals in three fundamental papers [16, 17, 18]. For these papers and more recent results in this direction the reader is referred to [6], chapter 10 and [9]. General results concerning (k, n)-arcs may be found in [6], chapter 12; see also [4, 7, 20, 23].

Gábor Korchmáros

Papers

Algebraic Curves Over a Finite Field

A new family of maximal curves over a finite field

Curves of large genus covered by the hermitian curve

On Curves Covered by the Hermitian Curve

On (q + t)-arcs of type (0, 2, t) in a desarguesian plane of order q