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Henry H Kim - One of the best experts on this subject based on the ideXlab platform.
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the average of the smallest prime in a conjugacy class
arXiv: Number Theory, 2016Co-Authors: Peter J Cho, Henry H KimAbstract:Let $C$ be a conjugacy class of $S_n$ and $K$ an $S_n$-field. Let $n_{K,C}$ be the smallest prime which is ramified or whose Frobenius Automorphism Frob$_p$ does not belong to $C$. Under some technical conjectures, we compute the average of $n_{K,C}$. For $S_3$ and $S_4$-fields, our result is unconditional. For $S_n$-fields, $n=3,4,5$, we give a different proof which depends on the strong Artin conjecture. Let $N_{K,C}$ be the smallest prime for which Frob$_p$ belongs to $C$. For $S_3$-fields, we obtain an unconditional result for the average of $N_{K,C}$ for $C=[(12)]$.
Ivan Fesenko - One of the best experts on this subject based on the ideXlab platform.
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9. Local reciprocity cycles
2015Co-Authors: Ivan FesenkoAbstract:In this section we introduce a description of totally ramified Galois extensions of a local field with finite residue field (extensions have to satisfy certain arithmetical restrictions if they are infinite) in terms of subquotients of formal power series Fsepp [[X]]. This description can be viewed as a non-commutative local reciprocity map (which is not in general a homomorphism but a cocycle) which directly describes the Galois group in terms of certain objects related to the ground field. Abelian class field theory as well as metabelian theory of Koch and de Shalit [K], [KdS] (see subsection 9.4) are partial cases of this theory. 9.1. Group UdN (L=F) Let F be a local field with finite residue field. Denote by ’ 2 GF a lifting of the Frobenius Automorphism of Fur=F. Let F ’ be the fixed field of ’. The extension F’=F is totally ramified. Lemma ([KdS, Lemma 0.2]). There is a unique norm compatible sequence of prime elements E in finite subextensions E=F of F’=F. Proof. Uniqueness follows from abelian local class field theory, existence follows from the compactness of the group of units. In what follows we fix F ’ and consider Galois subextensions L=F of F’=F. Assume that L=F is arithmetically profinite, ie for every x the ramification group Gal(L=F)x is open in Gal(L=F) (see also subsection 6.3 of Part II). For instance, a totally ramified p-adic Lie extension is arithmetically profinite. For an arithmetically profinite extension L=F define its Hasse–Herbrand function hL=F: [0;1) ! [0;1) as hL=F (x) = lim hM=F (x) where M=F runs over finite subextensions of L=F (cf. [FV, Ch. III x5])
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On the image of noncommutative local reciprocity map
2015Co-Authors: Ivan FesenkoAbstract:There are several approaches to the reciprocity map, the essence of class field theory, which links the maximal abelian quotient (or sometimes the maximal abelian pro-p-quotient) of the absolute Galois group of a particular field with an appropriate abelian object associated to the field such that certain functorial properties hold. One of those approaches originates from works of Dwork [D], Serre [S], Hazewinkel [H1], [H2], Iwasawa [I1], [I2] and Neukirch [N1], [N2]. Recall it briefly. Let F be a local field with finite residue field. Let F ur be the maximal unramified extension of F and let F ̂ be the completion of F ur. For a separable extension L of F put Lur = LF ur, L ̂ = LF ̂. For an elementσ of Gal(L/F) let σ ̃ be any element of Gal(Lur/F ur) such that σ̃|L = σ and σ̃|F ur is a positive integer power of the Frobenius Automorphism ϕ ∈ Gal(F ur/F). Let Σ be the fixed field of σ̃; it is a finite extension of F. Let Gal(L/F)ab be the maximal abelian quotient of Gal(L/F). Define the map [N1], [N2] N: Gal(L/F) → F ∗/NL/FL∗ by σ → NΣ/F πΣ mod NL/FL ∗ where πΣ is any prime element of Σ. During the conference on class field theory in Tokyo, June 1998, Professor T. Tamagawa informed the author that similar constructions were independently developed by K. Iwasawa. We call N the Neukirch–Iwasawa map. On the other hand, for a finite Galois totally ramified extension L/F of local fields there is a fundamental exact sequence [Se, (2.3)], [H1, (2.7)] 1 −−−− → Gal(L/F)ab c−−−− →
Nouman Ashraf - One of the best experts on this subject based on the ideXlab platform.
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noise free fully homomorphic encryption scheme over non associative algebra
IEEE Access, 2020Co-Authors: Iqra Mustafa, Hasnain Mustafa, Ahmad Taher Azar, Sheraz Aslam, Syed Muhammad Mohsin, Muhammad Bilal Qureshi, Nouman AshrafAbstract:Among several approaches to privacy-preserving cryptographic schemes, we have concentrated on noise-free homomorphic encryption. It is a symmetric key encryption that supports homomorphic operations on encrypted data. We present a fully homomorphic encryption (FHE) scheme based on sedenion algebra over finite $Z_{n}$ rings. The innovation of the scheme is the compression of a 16-dimensional vector for the application of Frobenius Automorphism. For sedenion, we have $p^{16}$ different possibilities that create a significant bijective mapping over the chosen 16-dimensional vector that adds permutation to our scheme. The security of this scheme is based on the assumption of the hardness of solving a multivariate quadratic equation system over finite $Z_{n}$ rings. The scheme results in $256n$ multivariate polynomial equations with $256+16n$ unknown variables for $n$ messages. For this reason, the proposed scheme serves as a security basis for potentially post-quantum cryptosystems. Moreover, after sedenion, no newly constructed algebra loses its properties. This scheme would therefore apply as a whole to the following algebras, such as 32-dimensional trigintadunion.
Kim, Henry H - One of the best experts on this subject based on the ideXlab platform.
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The Average of the Smallest Prime in a Conjugacy Class
'Oxford University Press (OUP)', 2020Co-Authors: Cho, Perter J., Kim, Henry HAbstract:Let C be a conjugacy class of S-n and K an S-n-field. Let n(K,C) be the smallest prime, which is ramified or whose Frobenius Automorphism Frob(p) does not belong to C. Under some technical conjectures, we show that the average of n(K,C) is a constant. We explicitly compute the constant. For S-3- and S-4-fields, our result is unconditional. Let N-K,N-C be the smallest prime for which Frob(p) belongs to C. We obtain the average of N-K,N-C under some technical conjectures. For n = 3 and C = [(12)], we have the average value of N-K,N-C unconditionally
Morton Patrick - One of the best experts on this subject based on the ideXlab platform.
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Periodic points of algebraic functions and Deuring’s class number formula
'Springer Science and Business Media LLC', 2019Co-Authors: Morton PatrickAbstract:The exact set of periodic points in Q of the algebraic function ˆ F(z) = (−1±p1 − z4)/z2 is shown to consist of the coordinates of certain solutions (x, y) = ( , ) of the Fermat equation x4+y4 = 1 in ring class fields f over imaginary quadratic fields K = Q(p−d) of odd conductor f, where −d = dKf2 1 (mod 8). This is shown to result from the fact that the 2-adic function F(z) = (−1 + p1 − z4)/z2 is a lift of the Frobenius Automorphism on the coordinates for which | |2 < 1, for any d 7 (mod 8), when considered as elements of the maximal unramified extension K2 of the 2-adic field Q2. This gives an interpretation of the case p = 2 of a class number formula of Deuring. An algebraic method of computing these periodic points and the corresponding class equations H−d(x) is given that is applicable for small periods. The pre-periodic points of ˆ F(z) in Q are also determined
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Periodic points of algebraic functions and Deuring's class number formula
'Springer Science and Business Media LLC', 2018Co-Authors: Morton PatrickAbstract:The exact set of periodic points in $\overline{\mathbb{Q}}$ of the algebraic function $\widehat{F}(z)=(-1\pm \sqrt{1-z^4})/z^2$ is shown to consist of the coordinates of certain solutions $(x,y)=(\pi, \xi)$ of the Fermat equation $x^4+y^4=1$ in ring class fields $\Omega_f$ over imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ of odd conductor $f$, where $-d \equiv 1$ (mod $8$). This is shown to result from the fact that the $2$-adic function $F(z)=(-1+ \sqrt{1-z^4})/z^2$ is a lift of the Frobenius Automorphism on the coordinates $\pi$ for which $|\pi|_2
