Cyclic Extension Research Articles

Analytic ranks of elliptic curves over number fields

Let E E be an elliptic curve over Q \mathbb {Q} . Then, we show that the average analytic rank of E E over cyclic extensions of degree l l over Q \mathbb {Q} with l l a prime not equal to 2 2 , is at most 2 + r Q ( E ) 2+r_\mathbb {Q}(E) , where r Q ( E ) r_\mathbb {Q}(E) is the analytic rank of the elliptic curve E E over Q \mathbb {Q} . This bound is independent of the degree l l . Using a recent result of Bhargava, Taniguchi and Thorne [Improved error estimates for the Davenport–Heilbronn theorems, arxiv.org/abs/2107.12819, 2021], we obtain a non-trivial upper bound on the average analytic rank of E E over S 3 S_3 -fields.

Ramification filtration and differential forms

Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $\Gamma_{L}^{(v)}$ the ramification subgroups of $\Gamma_{L}=\operatorname{Gal}(L^{\mathrm{sep}}/L)$. We consider the category $\operatorname{M\Gamma}_{L}^{\mathrm{Lie}}$ of finite $\mathbb{Z}_p[\Gamma_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $\Gamma_L$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups $\Gamma_L^{(v)}$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$ can be explicitly extracted from some differential forms $\widetilde{\Omega} [N]$ on the Fontaine etale $\phi $-module $M(H)$ associated with $H$. The forms $\widetilde{\Omega}[N]$ are completely determined by a canonical connection $\nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic, which contain a primitive $p$th root of unity, we show that a similar problem for $\mathbb{F}_p[\Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding $\phi $-module together with the action of the Galois group of a cyclic extension $L_1$ of $L$ of degree $p$. Then our solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of $\operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the $p$-adic period of the formal group $\mathbb{G}_m$.

Ramification filtration and differential forms

Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $\Gamma_{L}^{(v)}$ the ramification subgroups of $\Gamma_{L}=\operatorname{Gal}(L^{\mathrm{sep}}/L)$. We consider the category $\operatorname{M\Gamma}_{L}^{\mathrm{Lie}}$ of finite $\mathbb{Z}_p[\Gamma_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $\Gamma_L$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups $\Gamma_L^{(v)}$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$ can be explicitly extracted from some differential forms $\widetilde{\Omega} [N]$ on the Fontaine etale $\phi $-module $M(H)$ associated with $H$. The forms $\widetilde{\Omega}[N]$ are completely determined by a canonical connection $\nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic, which contain a primitive $p$th root of unity, we show that a similar problem for $\mathbb{F}_p[\Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding $\phi $-module together with the action of the Galois group of a cyclic extension $L_1$ of $L$ of degree $p$. Then our solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of $\operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the $p$-adic period of the formal group $\mathbb{G}_m$. Bibliography: 21 titles.

Prediction of breakage in human hair caused by cyclical extension using infrared spectroscopy coupled with multivariate curve resolution.

We propose a method of prediction for hair breakage induced by change of molecular structure. The changes inside hair by cyclical extension were investigated using infrared (IR) spectroscopy coupled with multivariate curve resolution (MCR) and two-dimensional correlation spectroscopy (2D-COS). In bleached hair, cyclical extension stresses at 5% strain levels were seen to increase the signal of CH2 and that of component primarily derived from C=O at 55 cycles, SO3H signal at 75 cycles, ultimately leading to fiber breakage. The CH2 profile of bleached hair significantly increased at 55 cycles compared to before extension in almost agreement with component profile primarily derived from C=O. This agreement behavior was likely to surface translocation of the lipids such as wax esters and triacylglycerols inside hair caused by cyclical extension. Two-dimensional correlation spectra indicated that SO3H was produced by way of a cystine oxide by cyclical extension. In contrast, only CH2 signal gradually increased without breakage in untreated hair. Thus, the method proposed in this study monitoring three functional groups is expected to have potential application in prediction of hair breakage by cyclical extension such as everyday grooming actions.

Algebraic number fields generated by an infinite family of monogenic trinomials

For an infinite family of monogenic trinomials \(P(X)=X^3\pm 3rbX-b\in {\mathbb {Z}}[X]\), arithmetical invariants of the cubic number field \(L={\mathbb {Q}}(\theta )\), generated by a zero \(\theta\) of \(P(X)\), and of its Galois closure \(N=L(\sqrt{d_L})\) are determined. The conductor \(f\) of the cyclic cubic relative extension \(N/K\), where \(K={\mathbb {Q}}(\sqrt{d_L})\) denotes the unique quadratic subfield of \(N\), is proved to be of the form \(3^eb\) with \(e\in \lbrace 1,2\rbrace\), which admits statements concerning primitive ambiguous principal ideals, lattice minima, and independent units in \(L\). The number \(m\) of non-isomorphic cubic fields \(L_1,\ldots ,L_m\) sharing a common discriminant \(d_{L_i}=d_L\) with \(L\) is determined.

Hopf-Galois structures on cyclic extensions and skew braces with cyclic multiplicative group

Let $G$ and $N$ be two finite groups of the same order. It is well-known that the existences of the following are equivalent: (a) a Hopf-Galois structure of type $N$ on any Galois $G$-extension; (b) a skew brace with additive group $N$ and multiplicative group $G$; (c) a regular subgroup isomorphic to $G$ in the holomorph of $N$. We shall say that $(G,N)$ is realizable when any of the above exists. Fixing $N$ to be a cyclic group, W. Rump (2019) has determined the groups $G$ for which $(G,N)$ is realizable. In this paper, fixing $G$ to be a cyclic group instead, we shall give a complete characterization of the groups $N$ for which $(G,N)$ is realizable.

The growth of Tate–Shafarevich groups in cyclic extensions

Let $p$ be a prime number. Kęstutis Česnavičius proved that for an abelian variety $A$ over a global field $K$, the $p$-Selmer group $\mathrm {Sel}_{p}(A/L)$ grows unboundedly when $L$ ranges over the $(\mathbb {Z}/p\mathbb {Z})$-extensions of $K$. Moreover, he raised a further problem: is $\dim _{\mathbb {F}_{p}} \text{III} (A/L) [p]$ also unbounded under the above conditions? In this paper, we give a positive answer to this problem in the case $p \neq \mathrm {char}\,K$. As an application, this result enables us to generalize the work of Clark, Sharif and Creutz on the growth of potential $\text{III}$ in cyclic extensions. We also answer a problem proposed by Lim and Murty concerning the growth of the fine Tate–Shafarevich groups.

Shear rupture mechanism and dissipation phenomena in bias-extension test of pantographic sheets: Numerical modeling and experiments

In this paper, we show that a two-field model for pantographic sheets generalized to include a more refined description of the deformation of interconnecting pivots and structural dissipation is capable of describing the sheets mechanical behavior for (1) cyclic extension tests in elastic regimes and (2) damage initiation. To this aim, we performed two kinds of experimental tests. In the first ones, we tested, remaining in the elastic regime, pantographic sheets under different cyclic loads. Second, we extended the specimens reaching the first rupture. The numerical model was calibrated in order to predict (1) the observed amplitude and shape of hysteretic cycles and (2) the observed localization of the first rupture. In this framework, (1) a simultaneous rupture of several pivots is observed and theoretically justified, (2) the primary dissipation mechanism and rupture initiation are attributed to pivots’ shear deformation.

Cyclic polynomials arising from the functional equation for Dickson polynomials

In this paper, we study algebraic properties of a family of certain polynomials arising from the functional equation for Dickson polynomials. We see that the roots and discriminants of those polynomials have very simple expressions, and each polynomial is cyclic. Further, we provide an irreducibility criterion analogous to the well-known criterion of Vahlen-Capelli. We finish the paper by showing that any cyclic extension of a certain field comes from a member of the family.

The Tate-Shafarevich groups of multinorm-one tori

Let k be a global field and L be a product of cyclic extensions of k . Let T be the torus defined by the multinorm equation N L / k ( x ) = 1 and let T ˆ be its character group. The Tate-Shafarevich group and the algebraic Tate-Shafarevich group of T ˆ in degree 2 give obstructions to the Hasse principle and weak approximation for rational points on principal homogeneous spaces of T . We give concrete descriptions of these groups and provide several examples.

Power Line Communication with Robust Timing and Carrier Recovery against Narrowband Interference for Smart Grid.

Power line communication (PLC) is an important interconnection technology for the smart grid, but the robustness of PLC transmission is faced with a great challenge due to strong non-Gaussian noise and interference. In this paper, a narrowband interference (NBI) resistant preamble is designed, and an effective timing and frequency synchronization method is proposed for OFDM-based PLC systems in the smart grid, which is capable of simultaneously conveying some bits of transmission parameter signaling (TPS) as well. In the time domain, the cyclic extension of the training OFDM symbol is scrambled, which makes it feasible to combat against NBI contamination. More accurate timing detection and sharper correlation peak can be implemented under the power line channel and the AWGN channel in the presence of NBI, compared with the conventional Schmidl’s and Minn’s methods with the same preamble length. Furthermore, the TPS transmitted using the proposed method is also immune from the NBI. The proposed method is capable of improving the synchronization performance of the PLC transmission significantly, which is verified by theoretical analysis and computer simulations.

On cyclic algebraic-geometry codes

In this paper we initiate the study of cyclic algebraic geometry codes. We give conditions to construct cyclic algebraic geometry codes in the context of algebraic function fields over a finite field by using their group of automorphisms. We prove that cyclic algebraic geometry codes constructed in this way are closely related to cyclic extensions. We also give a detailed study of the monomial equivalence of cyclic algebraic geometry codes constructed with our method in the case of a rational function field.

English

For any number field K with non-elementary 3-class group Cl(3,K) = C(3^e) x C(3), e >= 2, the punctured capitulation type kappa(K) of K in its unramified cyclic cubic extensions Li, 1 <= i <= 4, is an orbit under the action of S3 x S3. By means of Artin's reciprocity law, the arithmetical invariant kappa(K) is translated to the punctured transfer kernel type kappa(G2) of the automorphism group G2 = Gal(F(3,2,K)/K) of the second Hilbert 3-class field of K. A classification of finite 3-groups G with low order and bicyclic commutator quotient G/G' = C(3^e) x C(3), 2 <= e <= 6, according to the algebraic invariant kappa(G), admits conclusions concerning the length of the Hilbert 3-class field tower F(3,infty,K) of imaginary quadratic number fields K.

Biquaternion division algebras and fourth power-central elements

Let A be a biquaternion division algebra over a field F ( char F ≠ 2 ). We prove that an element e ∈ F ⁎ such that A F ( e 4 ) = 0 exists if and only if there is a decomposition A ≃ ( a , b ) ⊗ ( c , d ) with 《 − a , b , c , d 》 = 0 . This statement strengthens the result of Rost, Serre and Tignol ( [7] ), where the additional condition − 1 ∈ F ⁎ holds. Another result is a construction of a field F with − 1 ∈ F ⁎ , a quartic cyclic field extension L / F , and quaternion algebras ( a , b ) and ( c , d ) over F such that ( a , b ) L = ( c , d ) L = 0 , but the quadratic form 〈 a , b , a b 〉 ⊥ 《 c , d 》 is anisotropic. As a consequence, for any algebraically closed field k 0 , we give an example of an anisotropic 7-dimensional quadratic form Ψ over k 0 ( z , x , y ) with ind ( C 0 ( Ψ ) ) = 4 such that for any discrete k 0 ( z ) -valuation v on k 0 ( z , x , y ) the form Ψ becomes isotropic over the completion k 0 ( z , x , y ) v . In other words, the form Ψ provides a counterexample to the strong Hasse principle for 7-dimensional quadratic forms with respect to the rational extension K ( x , y ) / K , where K = k 0 ( z ) .

Cyclic cubic extensions of ℚ

We determine the irreducible trinomials [Formula: see text] for integers [Formula: see text] which generate precisely all possible Galois extensions of degree [Formula: see text] over [Formula: see text]. The proof, although involved, is elementary and one can parametrize all these polynomials explicitly. As an accidental by-product of the results, we prove that infinitely many primes congruent to [Formula: see text] or [Formula: see text] mod [Formula: see text] are sums of two rational cubes - thereby, giving the first unconditional result on a classical open problem.

The commuting graph of the cyclic extension of an abelian group

In this article, using the method of local analysis of abelian groups, we classify the commuting graph of the prime cyclic extension of an abelian group completely and obtain certain prominent parameters like independent number, clique number and second clique number of the graphs.

Ultrasensitive Graphene-Based Nanobiosensor for Rapid Detection of Hemoglobin in Undiluted Biofluids.

Detection of hemoglobin (Hb), a critical part of the biological system that is responsible for oxygen transportation, is of great significance on clinical diagnosis of various diseases. Particularly, time-efficient Hb detection under nanomole levels has drawn much attention in recent years. Herein, we present a graphene field effect transistor (GFET)-based aptameric nanobiosensor for rapid detection of Hb in undiluted biofluids including serum and urine and for the first time use polyethylenimine (PEI), a kind of comparatively low-cost polymer consisting of numerous amino groups, which can be directly linked with the anchor molecule without any pretreatment as the graphene surface passivation agent. Experimental results indicate the PEI-modified graphene aptameric nanobiosensor can respond to the Hb concentration change in a few minutes (6-8 min) with estimated detection limits of 10.6 fM in 1× PBS, 14.2 fM in undiluted serum, and 11.9 fM in undiluted urine, respectively. Further, considering the potential use of our sensor for implantable and wearable applications, we also examine the sensing performance of the sensor fabricated on an ultrathin flexible polyethylene terephthalate (PET) substrate. The Hb detection results are almost invariable even after 100 cycles of cyclic extension by 120% or 100 cycles of bending with a radius of 1 mm. Hence, our sensor holds great potential for accurate monitoring of nanomole levels of Hb in clinical applications.

Galois module structure of the units modulo $$p^m$$ of cyclic extensions of degree $$p^n$$

Let p be prime, and $$n,m \in \mathbb {N}$$ . When K/F is a cyclic extension of degree $$p^n$$ , we determine the $$\mathbb {Z}/p^m\mathbb {Z}[\text {Gal}(K/F)]$$ -module structure of $$K^\times /K^{\times p^m}$$ . With at most one exception, each indecomposable summand is cyclic and free over some quotient group of $$\text {Gal}(K/F)$$ . For fixed values of m and n, there are only finitely many possible isomorphism classes for the non-free indecomposable summand. These Galois modules act as parameterizing spaces for solutions to certain inverse Galois problems, and therefore this module computation provides insight into the structure of absolute Galois groups. More immediately, however, these results show that Galois cohomology is a context in which seemingly difficult module decompositions can practically be achieved: when $$m,n>1$$ the modular representation theory allows for an infinite number of indecomposable summands (with no known classification of indecomposable types), and yet the main result of this paper provides a complete decomposition over an infinite family of modules.

On the prime graph of a finite group with unique nonabelian composition factor

We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group G isospectral to a finite simple group has a unique nonabelian composition factor, that is, the quotient of G by the solvable radical of G is an almost simple group. The main goal of this paper is to prove that this almost simple group is a cyclic extension of its socle. To this end, we consider a general situation when G is an arbitrary group with unique nonabelian composition factor, not necessarily isospectral to a simple group, and study the prime graph of G, where the prime graph of G is the graph whose vertices are the prime numbers dividing the order of G and two such numbers r and s are adjacent if and only if and G has an element of order rs. Namely, we establish some sufficient conditions for the prime graph of such a group to have a vertex adjacent to all other vertices. Besides proving the main result, this allows us to refine a recent result by Cameron and Maslova concerning finite groups almost recognizable by prime graph.

Galois scaffolds and Galois module structure for totally ramified [formula omitted]-extensions in characteristic 0

Recently, much work has been done to investigate Galois module structure of local field extensions, particularly through the use of Galois scaffolds. Given a totally ramified p-extension of local fields L/K, a Galois scaffold gives us a K-basis for K[G] whose effect on the valuation of elements of L is easy to determine. In 2013, N.P. Byott and G.G. Elder gave sufficient conditions for the existence of Galois scaffolds for cyclic extensions of degree p2 in characteristic p. We take their work and adapt it to cyclic extensions of degree p2 in characteristic 0.