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

Transcendence degree one function fields over a finite field with many automorphisms

In a previous paper 1, all point sets of minimum size in PG(2,q), blocking all external lines to a given irreducible conic , have been determined for every odd q. Here we obtain a similar classification for those point sets of minimum size, which meet every external and tangent line to . © 2004 Wiley Periodicals, Inc.

Blocking sets of nonsecant lines to a conic in PG(2,q), q odd

The sporadic complete 12-arc in PG(2, 13) contains eight points from a conic. In PG(2,q) with q>13 odd, all known complete k-arcs sharing exactly ½(q+3) points with a conic have size at most ½(q+3)+2, with only two exceptions, both due to Pellegrino, which are complete (½(q+3)+3) arcs, one in PG(2, 19) and another in PG(2, 43). Here, three further exceptions are exhibited, namely a complete (½(q+3)+4)-arc in PG(2, 17), and two complete (½(q+3)+3)-arcs, one in PG(2, 27) and another in PG(2, 59). The main result is Theorem 6.1 which shows the existence of a (½(qr+3)+3)-arc in PG(2,qr) with r odd and q≡3 (mod 4) sharing ½(qr+3) points with a conic, whenever PG(2,q) has a (½(qr+3)+3)-arc sharing ½(qr+3) points with a conic. A survey of results for smaller q obtained with the use of the MAGMA package is also presented. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 25–47, 2010

On arcs sharing the maximum number of points with ovals in cyclic affine planes of odd order

Some finite translation planes arising from A6-1nvariant ovoids of the Klein quadric

We investigate the structure of a collineation group G leaving invariant a unital U in a finite projective plane ∏ of even order n=m2. When G is transitive on the points of U and the socle of G has even order, then ∏ must be a Desarguesian plane, U a classical unital and PSU(3,m2)≤G≤PГU(3,m2) — for m>2. The primitive case follows as an easy corollary.

Gábor Korchmáros

Papers

Transcendence degree one function fields over a finite field with many automorphisms

Blocking sets of nonsecant lines to a conic in PG(2,q), q odd

On arcs sharing the maximum number of points with ovals in cyclic affine planes of odd order

Some finite translation planes arising from A6-1nvariant ovoids of the Klein quadric

Collineation groups preserving a unital in a projective plane of even order