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

Let G be a sharply 3-transitive permutation set on PG(1,pm), p odd, pm > 9. Suppose G contains all the involutions in PGL(2,pm) and is such that if a permutation g lies in G then so does every power of g. We have that G necessarily coincides with PGL(2,pm).

A property of sharply 3–transitive finite permutation sets

Previous results on genera g of F_{q^2}-maximal curves are improved: (1) Either g\leq (q^2-q+4)/6, or g=\lfloor(q-1)^2/4\rfloor, or g=q(q-1)/2; (2) The hypothesis on the existence of a particular Weierstrass point in \cite{at} is proved; (3) For q\equiv 1\pmod{3}, q\ge 13, no F_{q^2}-maximal curve of genus (q-1)(q-2)/3 exists; (4) For q\equiv 2\pmod{3}, q\ge 11, the non-singular F_{q^2}-model of the plane curve of equation y^q+y=x^{(q+1)/3} is the unique F_{q^2}-maximal curve of genus g=(q-1)(q-2)/6; (5) Assume \dim(\cD_\cX)=5, and char(\fq)\geq 5. For q\equiv 1\pmod{4}, q\geq 17, the Fermat curve of equation x^{(q+1)/2}+y^{(q+1)/2}+1=0 is the unique F_{q^2}-maximal curve of genus g=(q-1)(q-3)/8. For q\equiv 3\pmod{4}, q\ge 19, there are exactly two F_{q^2}-maximal curves of genus g=(q-1)(q-3)/8, namely the above Fermat curve and the non-singular F_{q^2}-model of the plane curve of equation y^q+y=x^{(q+1)/4}. The above results provide some new evidences on maximal curves in connection with Castelnuovo's bound and Halphen's theorem, especially with extremal curves; see for instance the conjecture stated in Introduction.

On the genus of a maximal curve

Nakajimaʼs remark on Hennʼs proof

On cyclic semi-regular subgroups of certain 2-transitive permutation groups

Basic properties of polygons in Euclidean space and some related regularity questions were explored in the first part of the Nineteen century. Systematic investigations of polygons and their degree of regularity in a higher dimensional Euclidean space E r began in the 1970s, in the vein of Blumenthal’s fundamental work (Blumenthal et al. Theory and applications of distance geometry. Chelsea Publishing Co., New York, 1970) on distance preserving maps of E r . Such investigations were also stimulated by a practical question from organic chemistry posed to van der Waerden, see van der Waerden (Elem Math 25:73–78, 1970), and the subsequent discussion around it. An useful indicator of degree of regularity was introduced by Grunbaum (The geometry of metric and linear spaces. Springer, Berlin, 1975) who generalized the concept of an equilateral polygon in a higher dimensional Euclidean space E r : An n-gon ${P_1P_2\dots P_n}$ spanning E r is called at least k-equilateral, if

$${\overline{{P_1}{P_j}}=\overline{P_{1+h}P_{j+h}},\quad h=1,2,\ldots,n-1,\quad\quad\quad(0.1)}$$

holds for every ${1 < j \leq k+1}$ where the indices are taken mod n. If ${P_1P_2\dots P_n}$ is at least k-equilateral with ${k\geq [n/2]}$, then (0.1) holds for every ${1\le j\le n-1}$. In this case, ${P_1P_2\dots P_n}$ is called a totally equilateral n-gon since any two chords (or sides) P i P j and P k P l have equal length if the same holds for the chords (or sides) Q i Q j and Q k Q l of the planar regular n-gon ${Q_1Q_2\cdots Q_n}$. Grunbaum (The geometry of metric and linear spaces. Springer, Berlin, 1975) conjectured and Lawrence (k-equilateral (2k + 1)-gons span only even-dimensional spaces. Springer, Berlin, 1975) [independently van der Blij (Linear Algebra Appl 226/228: 345–352, 1995)] proved that totally equilateral polygons with odd number of vertices span always an even-dimensional space. In this paper, we prove that every at least (r + 1)-equilateral n-gon with ${n\geq r+1}$ is totally equilateral. This generalizes a previous result in three dimensional Euclidean space, see Korchmaros (Riv Mat Univ Parma (4) 1:45–50, 1975).

Gábor Korchmáros

Papers

A property of sharply 3–transitive finite permutation sets

On the genus of a maximal curve

Nakajimaʼs remark on Hennʼs proof

On cyclic semi-regular subgroups of certain 2-transitive permutation groups

Regular polygons in higher dimensional Euclidean spaces