Most hyperelliptic curves over Q have no rational points

By a hyperelliptic curve over Q, we mean a smooth, geometrically irreducible, complete curve C over Q equipped with a fixed map of degree 2 to P^1 defined over Q. Thus any hyperelliptic curve C over Q of genus g can be embedded in weighted projective space P(1,1,g+1) via an equation of the form C : z^2 = f(x,y) = f_0 x^n + f_1 x^{n-1} y + ... + f_n y^n where n=2g+2, the coefficients f_i lie in Z, and f factors into distinct linear factors over Q-bar. Define the height H(C) of C by H(C):=max{|f_i|}, and order all hyperelliptic curves over Q of genus g by height. Then we prove that, as g tends to infinity: 1) a density approaching 100% of hyperelliptic curves of genus g have no rational points; 2) a density approaching 100% of those hyperelliptic curves of genus g that have points everywhere locally fail the Hasse principle; and 3) a density approaching 100% of hyperelliptic curves of genus g have empty Brauer set, i.e., have a Brauer-Manin obstruction to having a rational point. We also prove positive proportion results of this type for individual genera, including g = 1.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1208.100

Nonexistence of Solutions to Certain Families of Diophantine Equations

In this work, I examine specific families of Diophantine equations and prove that they have no solutions in positive integers. The proofs use a combination of classical elementary arguments and powerful tools such as Diophantine approximations, Lehmer numbers, the modular approach, and earlier results proved using linear forms in logarithms. In particular, I prove the following three theorems. Main Theorem I. Let a, b, c, k ∈ Z+ with k ≥ 7. Then the equation (a^2cX^k − 1)(b^2cY^k − 1) = (abcZ^k − 1)^2 has no solutions in integers X, Y , Z \u3e 1 with a^2X^k ̸= b^2Y^k. Main Theorem II. Let L, M, N ∈ Z+ with N \u3e 1. Then the equation NX^2 + 2^L3^M = Y^N has no solutions with X, Y ∈ Z+ and gcd(NX,Y) = 1. Main Theorem III. Let p be an odd rational prime and let N, α, β, γ ∈ Z with N \u3e 1, α ≥ 1, and β, γ ≥ 0. Then the equation X^{2N} +2^{2α}5^{2β}p^{2γ} =Z^5 has no solutions with X, Z ∈ Z+ and gcd(X, Z) = 1

Nonexistence of Solutions to Certain Families of Diophantine Equations

In this work, I examine specific families of Diophantine equations and prove that they have no solutions in positive integers. The proofs use a combination of classical elementary arguments and powerful tools such as Diophantine approximations, Lehmer numbers, the modular approach, and earlier results proved using linear forms in logarithms. In particular, I prove the following three theorems. Main Theorem I. Let a, b, c, k ∈ Z+ with k ≥ 7. Then the equation (a^2cX^k − 1)(b^2cY^k − 1) = (abcZ^k − 1)^2 has no solutions in integers X, Y , Z \u3e 1 with a^2X^k ̸= b^2Y^k. Main Theorem II. Let L, M, N ∈ Z+ with N \u3e 1. Then the equation NX^2 + 2^L3^M = Y^N has no solutions with X, Y ∈ Z+ and gcd(NX,Y) = 1. Main Theorem III. Let p be an odd rational prime and let N, α, β, γ ∈ Z with N \u3e 1, α ≥ 1, and β, γ ≥ 0. Then the equation X^{2N} +2^{2α}5^{2β}p^{2γ} =Z^5 has no solutions with X, Z ∈ Z+ and gcd(X, Z) = 1

Deterministic polynomial factoring over finite fields: A uniform approach via P-schemes

We introduce a family of combinatorial objects called P-schemes, where P is a collection of subgroups of a finite group G. A P-scheme is a collection of partitions of right coset spaces H\G, indexed by H ∈ P, that satisfies a list of axioms. These objects generalize the classical notion of association schemes as well as m-schemes (Ivanyos et al., 2009). We apply the theory of P-schemes to deterministic polynomial factoring over finite fields: suppose f(X) ∈ Z[X] and a prime number pare given, such that f(X) :=f(X) modpfactorizes into n =deg(f)distinct linear factors over the finite field F_p. We show that, assuming the generalized Riemann hypothesis (GRH), f(X)can be completely factorized in deterministic polynomial time if the Galois group G of f(X)is an almost simple primitive permutation group on the set of roots of f(X), and the socle of Gis a subgroup of Sym(k)for kup to 2^O(√log n). This is the first deterministic polynomial-time factoring algorithm for primitive Galois groups of superpolynomial order. We prove our result by developing a generic factoring algorithm and analyzing it using P-schemes. We also show that the main results achieved by known GRH-based deterministic polynomial factoring algorithms can be derived from our generic algorithm in a uniform way. Finally, we investigate the schemes conjecturein Ivanyos et al. (2009), and formulate analogous conjectures associated with various families of permutation groups. We show that these conjectures form a hierarchy of relaxations of the original schemes conjecture, and their positive resolutions would imply deterministic polynomial-time factoring algorithms for various families of Galois groups under GRH

Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page

HILBERT BASES, DESCENT STATISTICS, AND COMBINATORIAL SEMIGROUP ALGEBRAS

The broad topic of this dissertation is the study of algebraic structure arising from polyhedral geometric objects. There are three distinct topics covered over three main chapters. However, each of these topics are further linked by a connection to the Eulerian polynomials. Chapter 2 studies Euler-Mahonian identities arising from both the symmetric group and generalized permutation groups. Specifically, we study the algebraic structure of unit cube semigroup algebra using Gröbner basis methods to acquire these identities. Moreover, this serves as a bridge between previous methods involving polyhedral geometry and triangulations with descent bases methods arising in representation theory. In Chapter 3, the aim is to characterize Hilbert basis elements of certain -lecture hall cones. In particular, the main focus is the classification of the Hilbert bases for the 1 mod cones and the -sequence cones, both of which generalize a previous known result. Additionally, there is much broader characterization of Hilbert bases in dimension ≤ 4 for -generated Gorenstein lecture hall cones. Finally, Chapter 4 focuses on certain algebraic and geometric properties of -lecture hall polytopes. This consists of partial classification results for the Gorenstein property, the integer-decomposition property, and the existence of regular, unimodular triangulations