Algebraic Number Theory Research Articles

Arithmetic Geometry

Arithmetic geometry is at the interface between algebraic geometry and number theory, and studies schemes over the ring of integers of number fields, or their p -adic completions, and connects with representation theory, automorphic forms, Hodge theory, algebraic topology, and many other fields.

Integrality properties in the moduli space of elliptic curves: CM case

For each algebraic number [Formula: see text], a result of Habegger [P. Habegger, Singular moduli that are algebraic units, Algebra Number Theory 9(7) (2015) 1515–1524] shows that there are only finitely many singular moduli [Formula: see text] such that [Formula: see text] is an algebraic unit. His result uses Duke’s Equidistribution Theorem and is thus not effective. In this paper, we give an effective proof of Habegger’s result assuming that [Formula: see text] is not a singular modulus itself. We give an explicit bound, which depends only on [Formula: see text], on the discriminant [Formula: see text] associated with a singular modulus [Formula: see text] such that [Formula: see text] is a unit. This implies explicit bounds on the number of these singular moduli.

Neural Cryptography Based on Generalized Tree Parity Machine for Real-Life Systems

Traditional public key exchange protocols are based on algebraic number theory. In another perspective, neural cryptography, which is based on neural networks, has been emerging. It has been reported that two parties can exchange secret key pairs with the synchronization phenomenon in neural networks. Although there are various models of neural cryptography, called Tree Parity Machine (TPM), many of them are not suitable for practical use, considering efficiency and security. In this paper, we propose a Vector-Valued Tree Parity Machine (VVTPM), which is a generalized architecture of TPM models and can be more efficient and secure for real-life systems. In terms of efficiency and security, we show that the synchronization time of the VVTPM has the same order as the basic TPM model, and it can be more secure than previous results with the same synaptic depth.

On the generalization of Inoue manifolds

This paper is about a generalization of celebrated Inoue's surfaces. To each matrix M in SL(2n+1,ℤ) we associate a complex non-Kähler manifold TM of complex dimension n+1. This manifold fibers over S1 with the fiber T2n+1 and monodromy MT. Our construction is elementary and does not use algebraic number theory. We show that some of the Oeljeklaus-Toma manifolds are biholomorphic to the manifolds of type TM. We prove that if M is not diagonalizable, then TM does not admit a Kähler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds.

Improved divisor arithmetic on generic hyperelliptic curves

The divisor class group of a hyperelliptic curve defined over a finite field is a finite abelian group at the center of a number of important open questions in algebraic geometry, number theory and cryptography. Many of these problems lend themselves to numerical investigation, and as emphasized by Sutherland [14, 13], fast arithmetic in the divisor class group is crucial for their efficiency. Besides, implementations of these fundamental operations are at the core of the algebraic geometry packages of widely-used computer algebra systems such as Magma and Sage.

The Integral Points On A Class Of Elliptic Curve

It is very important to study the integer points of various elliptic curves in elementary number theory. On the one hand, as an important research object in elementary number theory, elliptic curve plays an indispensable role in the development of mathematics. On the other hand, as an important research object in elementary number theory, elliptic curve has practical application in many aspects. So far, there are some conclusions about elliptic curve y 2 = x 3 + a x + b, a, b ∈ Z. In 1987, D. Zagier proposed the integer points problem on y 2 = x 3 - 27x + 62, which is important for studying the arithmetic and geometric properties of elliptic curves. In 2009, Zhu H L and Chen J H used the methods of P-adic analysis and algebraic number theory to solve the problem of the integer points on y 2 = x 3 - 27x + 62. In 2010, Wu H M used the elementary methods to find all the integral points of elliptic curves y 2 = x 3 - 27x - 62. In 2015, Li Y Z and Cui B J used the elementary methods to solve the problem of the integer points on y 2 = x 3 - 21x - 90. In 2016, Guo J used the elementary methods to solve the problem of the integer points on y 2 = x 3 + 27x + 62. In 2017, Guo J used the elementary method proves that y 2 = x 3 - 21x + 90 has no integer points. Up to now, there is no relevant conclusions on the integral points of elliptic curves y 2 = x 3 + 19x - 46, which is the subject of this paper. By using congruence and Legendre Symbol, it can be proved that elliptic curve y 2 = x 3 + 19x - 46 has only one integer point: (x, y) = (2,0).

Optimal sup norm bounds for newforms on GL2 with maximally ramified central character

Abstract Recently, the problem of bounding the sup norms of L 2 {L^{2}} -normalized cuspidal automorphic newforms ϕ on GL 2 {\mathrm{GL}_{2}} in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character χ of ϕ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general χ. If the level N is a square, our result reduces to ∥ ϕ ∥ ∞ ≪ N 1 4 + ϵ , \|\phi\|_{\infty}\ll N^{\frac{1}{4}+\epsilon}, at least under the Ramanujan Conjecture. In particular, when χ has conductor N, this improves upon the previous best known bound ∥ ϕ ∥ ∞ ≪ N 1 2 + ϵ {\|\phi\|_{\infty}\ll N^{\frac{1}{2}+\epsilon}} in this setup (due to [A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 2017, 1009–1045]) and matches a lower bound due to [N. Templier, Large values of modular forms, Camb. J. Math. 2 2014, 1, 91–116], thus our result is essentially optimal in this case.

Hilbert modular forms and codes over [formula omitted

Let p be an odd prime and consider the finite field Fp2. Given a linear code C⊂Fp2n, we use algebraic number theory to construct an associated lattice ΛC⊂OLn for L an algebraic number field and OL the ring of integers of L. We attach a theta series θΛC to the lattice ΛC and prove a relation between θΛC and the complete weight enumerator evaluated on weight one theta series.

Dihedral universal deformations

This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1) representation theory, (2) algebraic number theory, (3) modularity. As to (1), we prove that the universal deformation is dihedral if all infinitesimal deformations are dihedral. Concerning (2) in the setting of Galois representations of number fields, we give sufficient conditions to ensure that the universal deformation relatively unramified outside a finite set of primes is dihedral, and discuss in how far these conditions are necessary. As side-results, we obtain cases of the unramified Fontaine–Mazur conjecture, and in many cases positively answer a question of Greenberg and Coleman on the splitting behaviour at p of p-adic Galois representations attached to newforms. As to (3), we prove a modularity theorem of the form ‘ $$R=\mathbb {T}$$ ’ for parallel weight one Hilbert modular forms for cases when the minimal universal deformation is dihedral.

The finite subgroups of $${\text {SL}}(3,\overline{F})$$

We attempt to give a complete exposition of the classification of the finite subgroups of $${\text {SL}}(3,\overline{F})$$, where $$\overline{F}$$ is a separably closed field of characteristic not dividing the order of the finite group, in contemporary language, completing some of the proofs of Blichfeldt from the early twentieth century. The classification—originating from Jordan, Klein, Valentiner—has applications in algebraic geometry, number theory and mathematical physics.

Review of Applied Number Theory

Number theory (or the theory of numbers [1]) starts from elementary (not necessarily simple and easy, and in fact, it is often very hard) number theory and grows up with analytic number theory (including additive and multiplicative number theory), algebraic number theory, geometric number theory (including arithmetic algebraic geometry), combinatorial number theory, computational number theory (including algorithmic number theory), to name just a few. Applied number theory, on the other hand, is involved in the application of various branches of number theory to a wide range of areas including, e.g., physics, chemistry, biology, graphics, arts, music, and particularly computing and digital communications [2]. Number theory was once viewed as the purest of the pure mathematics, with little application to other areas. However, with the advent of modern computers and digital communications, number theory becomes increasingly important and applicable to many areas ranging from natural sciences, engineering to social sciences.

Multifractality of light in photonic arrays based on algebraic number theory

Many natural patterns and shapes, such as meandering coastlines, clouds, or turbulent flows, exhibit a characteristic complexity that is mathematically described by fractal geometry. Here, we extend the reach of fractal concepts in photonics by experimentally demonstrating multifractality of light in arrays of dielectric nanoparticles that are based on fundamental structures of algebraic number theory. Specifically, we engineered novel deterministic photonic platforms based on the aperiodic distributions of primes and irreducible elements in complex quadratic and quaternions rings. Our findings stimulate fundamental questions on the nature of transport and localization of wave excitations in deterministic media with multi-scale fluctuations beyond what is possible in traditional fractal systems. Moreover, our approach establishes structure–property relationships that can readily be transferred to planar semiconductor electronics and to artificial atomic lattices, enabling the exploration of novel quantum phases and many-body effects.

Bazı Hardy Toplamları ve Ramanujan Toplamının Ortalama Değeri Hakkında Yeni Bağıntılar

Dedekind sum first occured naturally in Dedekind’s transformation law of his eta-function. In analogy, Hardy sums are encountered in the transformation formula of the theta function. Up to now, they have many of remarkable applications in analytic number theory (Dedekind's η-function), algebraic number theory (class numbers), lattice point problems, topology and algebraic geometry. Miscellaneous arithmetical properties of these sums have been analyzed by many scholars. Recently, considering the characteristics of Hardy sums and other celebrated sums such as Ramanujan sum and Kloosterman sum, interesting and meaningful identities have been obtained. In this paper, we continue to focus on arithmetical aspects of Hardy sums and Ramanujan sum. More precisely, we consider a mean value problem of these sums and Ramanujan sum with the help of the features of Dirichlet L-functions and present some computational formulas for them.

Computing unit groups of curves

The group of units modulo constants of an affine variety over an algebraically closed field is free abelian of finite rank. Computing this group is difficult but of fundamental importance in tropical geometry, where it is necessary in order to realize intrinsic tropicalizations. We present practical algorithms for computing unit groups of smooth curves of low genus. Our approach is rooted in divisor theory, based on interpolation in the case of rational curves and on methods from algebraic number theory in the case of elliptic curves.

Computational method of algebraic number theory in particular area

Computational method of algebraic number theory in particular area

Proving quadratic reciprocity: explanation, disagreement, transparency and depth

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late eighteenth century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation.

Rooting out letters: octagonal symbol alphabets and algebraic number theory

It is widely expected that NMHV amplitudes in planar, maximally supersymmetric Yang-Mills theory require symbol letters that are not rationally expressible in terms of momentum-twistor (or cluster) variables starting at two loops for eight particles. Re- cent advances in loop integration technology have made this an ‘experimentally testable’ hypothesis: compute the amplitude at some kinematic point, and see if algebraic symbol letters arise. We demonstrate the feasibility of such a test by directly integrating the most difficult of the two-loop topologies required. This integral, together with its rotated image, suffices to determine the simplest NMHV component amplitude: the unique component finite at this order. Although each of these integrals involve algebraic symbol alphabets, the combination contributing to this amplitude is — surprisingly — rational. We describe the steps involved in this analysis, which requires several novel tricks of loop integration and also a considerable degree of algebraic number theory. We find dramatic and unusual simplifications, in which the two symbols initially expressed as almost ten million terms in over two thousand letters combine in a form that can be written in five thousand terms and twenty-five letters.

Complex group algebras of almost simple unitary groups

The aim of this article is to contribute to a question of R. Brauer that “when do non-isomorphic groups have isomorphic complex group algebras?” Let H and G be finite groups where and let denote the first column of the complex character table of H. In this article, we show that if then provided that q + 1 divides neither n nor n – 1. Consequently, it is shown that G is uniquely determined by the structure of its complex group algebra. This in particular extends a recent result of Bessenrodt et al. [Algebra Number Theory 9 (2015), 601–628] to the almost simple groups of arbitrary rank.

From Periods to Anabelian Geometry and Quantum Amplitudes

Periods are algebraic integrals, extending the class of algebraic numbers, and playing a central, dual role in modern Mathematical-Physics: scattering amplitudes and coefficients of de Rham isomorphism. The Theory of Periods in Mathematics, with their appearance as scattering amplitudes in Physics, is discussed in connection with the Theory of Motives, which in turn is related to Conformal Field Theory (CFT) and Topological Quantum Field Theory (TQFT), on the physics side. There are three main contributions. First, building a bridge between the Theory of Algebraic Numbers and Theory of Periods, will help guide the developments of the later. This suggests a relation between the Betti-de Rham theory of periods and Grothendieck’s Anabelian Geometry, towards perhaps an algebraic analog of Hurwitz Theorem, relating the algebraic de Rham cohomology and algebraic fundamental group, both pioneered by A. Grothendieck. Second, a homotopy-homology refinement of the Theory of Periods will help explain the connections with quantum amplitudes. The novel approach of Yves Andre to Motives via representations of categories of diagrams, relates from a physical point of view to generalized TQFTs. Finally, the known “universality” of Galois Theory, as how symmetries “grow”, controlling the structure of the objects of study, is discussed, in relation to the above several areas of research, together with ensuing further insight into the Mathematical-Physics symbiosis. To better understand and investigate Kontsevich-Zagier conjecture on abstract periods, the article ponders on the case of algebraic Riemann Surfaces representable by Belyi maps. Reformulation of cohomology of cyclic groups as a discrete analog of de Rham cohomology and the Arithmetic Galois Theory will provide a purely algebraic toy-model of the said algebraic homology/homotopy group theory of Grothendieck as part of Anabelian Geometry. The corresponding Platonic Trinity 5,7,11/TOI/E678 leads to connections with ADE-correspondence, and beyond, e.g. Theory of Everything (TOE) and ADEX-Theory. In perspective of the “Ultimate Physics Theory”, quantizing “everything”, i.e. cyclotomic quantum phase and finite Platonic-Hurwitz geometry of qubit frames as baryons, could perhaps be “The Eightfold (Petrie polygon) Way” to finally understand what quark flavors and fermion generations really are.

Аспекты применения алгебраической теории чисел в финансовой математике

Аспекты применения алгебраической теории чисел в финансовой математике