Campana points and powerful values of norm forms

We give an asymptotic formula for the number of weak Campana points of bounded height on a family of orbifolds associated to norm forms for Galois extensions of number fields. From this formula we derive an asymptotic for the number of elements with m-full norm over a given Galois extension of $$\mathbb {Q}$$ Q . We also provide an asymptotic for Campana points on these orbifolds which illustrates the vast difference between the two notions, and we compare this to the Manin-type conjecture of Pieropan, Smeets, Tanimoto and Várilly-Alvarado.

Gauge groups and bialgebroids

We study the Ehresmann–Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the gauge groupoid of a classical principal bundle. We show that the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid, and we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include: Galois objects of Taft algebras, a monopole bundle over a quantum sphere and a not faithfully flat Hopf–Galois extension of commutative algebras. For each of the latter two examples, there is in fact a suitable invertible antipode for the bialgebroid making it a Hopf algebroid.

Designing a Block Cipher in Galois Extension Fields for IoT Security

This paper focuses on a block cipher adaptation of the Galois Extension Fields (GEF) combination technique for PRNGs and targets application in the Internet of Things (IoT) space, an area where the combination technique was concluded as a quality stream cipher. Electronic Codebook (ECB) and Cipher Feedback (CFB) variations of the cryptographic algorithm are discussed. Both modes offer computationally efficient, scalable cryptographic algorithms for use over a simple combination technique like XOR. The cryptographic algorithm relies on the use of quality PRNGs, but adds an additional layer of security while preserving maximal entropy and near-uniform distributions. The use of matrices with entries drawn from a Galois field extends this technique to block size chunks of plaintext, increasing diffusion, while only requiring linear operations that are quick to perform. The process of calculating the inverse differs only in using the modular inverse of the determinant, but this can be expedited by a look-up table. We validate this GEF block cipher with the NIST test suite. Additional statistical tests indicate the condensed plaintext results in a near-uniform distributed ciphertext across the entire field. The block cipher implemented on an MSP430 offers a faster, more power-efficient alternative to the Advanced Encryption Standard (AES) system. This cryptosystem is a secure, scalable option for IoT devices that must be mindful of time and power consumption.

On the p-adic Beilinson conjecture and the equivariant Tamagawa number conjecture

Let E/K be a finite Galois extension of totally real number fields with Galois group G. Let p be an odd prime and let $$r>1$$ r > 1 be an odd integer. The p-adic Beilinson conjecture relates the values at $$s=r$$ s = r of p-adic Artin L-functions attached to the irreducible characters of G to those of corresponding complex Artin L-functions. We show that this conjecture, the equivariant Iwasawa main conjecture and a conjecture of Schneider imply the ‘p-part’ of the equivariant Tamagawa number conjecture for the pair $$(h^0(\mathrm {Spec}(E))(r), \mathbb {Z}[G])$$ ( h 0 ( Spec ( E ) ) ( r ) , Z [ G ] ) . If $$r>1$$ r > 1 is even we obtain a similar result for Galois CM-extensions after restriction to ‘minus parts’.

Constructing totally p-adic numbers of small height

Bombieri and Zannier gave an effective construction of algebraic numbers of small height inside the maximal Galois extension of the rationals which is totally split at a given finite set of prime numbers. They proved, in particular, an explicit upper bound for the lim inf of the height of elements in such fields. We generalize their result in an effective way to maximal Galois extensions of number fields with given local behavior at finitely many places.

A representative of RΓ(N,T) for higher dimensional twists of ℤpr(1)

Let [Formula: see text] be a Galois extension of [Formula: see text]-adic number fields and let [Formula: see text] be a de Rham representation of the absolute Galois group [Formula: see text] of [Formula: see text]. In the case [Formula: see text], the equivariant local [Formula: see text]-constant conjecture describes the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin [Formula: see text]-functions and it can be formulated as the vanishing of a certain element [Formula: see text] in [Formula: see text]; a similar approach can be followed also in the case of unramified twists [Formula: see text] of [Formula: see text] (see [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383; D. Izychev and O. Venjakob, Equivariant epsilon conjecture for 1-dimensional Lubin–Tate groups, J. Théor. Nr. Bordx. 28(2) (2016) 485–521]). One of the main technical difficulties in the computation of [Formula: see text] arises from the so-called cohomological term [Formula: see text], which requires the construction of a bounded complex [Formula: see text] of cohomologically trivial modules which represents [Formula: see text] for a full [Formula: see text]-stable [Formula: see text]-sublattice [Formula: see text] of [Formula: see text]. In this paper, we generalize the construction of [Formula: see text] in Theorem 2 of [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383] to the case of a higher dimensional [Formula: see text].

Galois corings and groupoids acting partially on algebras

Bagio and Paques [Partial groupoid actions: globalization, Morita theory and Galois theory, Comm. Algebra 40 (2012) 3658–3678] developed a Galois theory for unital partial actions by finite groupoids. The aim of this note is to show that this is actually a special case of the Galois theory for corings, as introduced by Brzeziński [The structure of corings, Induction functors, Maschke-type theorem, and Frobenius and Galois properties, Algebr. Represent. Theory 5 (2002) 389–410]. To this end, we associate a coring to a unital partial action of a finite groupoid on an algebra [Formula: see text], and show that this coring is Galois if and only if [Formula: see text] is an [Formula: see text]-partial Galois extension of its coinvariants.

Galois and Class Field Theory for Quantum Chemists

Quantum mechanics could be studied through polynomial algebra, as has been demonstrated by a work (“An approach to first-principles electronic structure calculation by symbolic-numeric computation” by A. Kikuchi). We carry forward the algebraic method of quantum mechanics through algebraic number theory; the basic equations are represented by the multivariate polynomial ideals; the symbolic computations process the ideal and disentangle the eigenstates as the algebraic variety; upon which one canbuild the Galois extension of the number field, in analogy with the univariate polynomial case, to investigate the hierarchy of solutions; the Galois extension is accompanied with the group operations, which permute the eigenstates from one to another, and furnish the quantum system with a non-apparent symmetry. Besides, this sort of algebraic quantum mechanics is an embodiment of the class field theory; some of the important consequences of the latter emerge in quantum mechanics. We shall demonstrate these points through simple models; we will see the use of computational algebra facilitates such sort of analysis, which might often be complicated if we try to solve them manually.

Artin L-functions of almost monomial Galois groups

If {K/\mathbb{Q}} is a finite Galois extension with an almost monomial Galois group and if {s_{0}\in\mathbb{C}\setminus\{1\}} is not a common zero for any two Artin L-functions associated to distinct complex irreducible characters of the Galois group, then all Artin L-functions of {K/\mathbb{Q}} are holomorphic at {s_{0}}. We present examples and basic properties of almost monomial groups.

Galois and Class Field Theory for Quantum Chemists

Quantum mechanics could be studied through polynomial algebra, as has been demonstrated by a work (``An approach to first-principles electronic structure calculation by symbolic-numeric computation'' by A. Kikuchi). We carry forward the algebraic method of quantum mechanics through algebraic number theory; the basic equations are represented by the multi-variate polynomial ideals; the symbolic computations process the ideal and disentangle the eigenstates as the algebraic variety; upon which one can build the Galois extension of the number field, in analogy with the univariate polynomial case, to investigate the hierarchy of solutions; the Galois extension is accompanied with the group operations, which permute the eigenstates from one to another, and furnish the quantum system with a non-apparent symmetry. Besides, this sort of algebraic quantum mechanics is an embodiment of the class field theory; some of the important consequences of the latter emerge in quantum mechanics. We shall demonstrate these points through simple models; we will see the use of computational algebra facilitates such sort of analysis, which might often be complicated if we try to solve them manually.