Remarks on Milnor K-theory and Tate’s conjecture for divisors

We prove a restriction isomorphism for Chow groups of zero-cycles with coefficients in Milnor K-theory for smooth projective schemes over excellent henselian discrete valuation rings. Furthermore, we study torsion subgroups of these groups over local and finite fields.

On finite Drinfeld modules

Milnor K-theory of complete discrete valuation rings with finite residue fields

Any nonconstant polynomial over a field can be expressed as a product of irreducible polynomials. In the case of finite fields, some reasonably efficient algorithms can be devised for the actual calculation of the irreducible factors of a given polynomial of positive degree. The availability of feasible factorization algorithms for polynomials over finite fields is important for coding theory and for the study of linear recurrence relations in finite fields. Beyond the realm of finite fields, there are various computational problems in algebra and number theory that depend in one way or another on the factorization of polynomials over finite fields. We mention the factorization of polynomials over the ring of integers, the determination of the decomposition of rational primes in algebraic number fields, the calculation of the Galois group of an equation over the rationals, and the construction of field extensions. We shall present several algorithms for the factorization of polynomials over finite fields. The decision on the choice of algorithm for a specific factorization problem usually depends on whether the underlying finite field is “small” or “large.” In Section 1 we describe those algorithms that are better adapted to “small” finite fields and in the next section those that work better for “large” finite fields. Some of these algorithms reduce the problem of factoring polynomials to that of finding the roots of certain other polynomials. Therefore, Section 3 is devoted to the discussion of the latter problem from the computational viewpoint.

We give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of$K3$surfaces over finite fields. We prove that every$K3$surface of finite height over a finite field admits a characteristic$0$lifting whose generic fibre is a$K3$surface with complex multiplication. Combined with the results of Mukai and Buskin, we prove the Tate conjecture for the square of a$K3$surface over a finite field. To obtain these results, we construct an analogue of Kisin’s algebraic group for a$K3$surface of finite height and construct characteristic$0$liftings of the$K3$surface preserving the action of tori in the algebraic group. We obtain these results for$K3$surfaces over finite fields of any characteristics, including those of characteristic$2$or$3$.

Let $F$ be a finitely generated regular field extension of transcendence degree $\geq 2$ over a perfect field $k$. We show that the multiplicative group $F^\times/k^\times$ endowed with the equivalence relation induced by algebraic dependence on $k$ determines the isomorphism class of $F$ in a functorial way. As a special case of this result, we obtain that the isomorphism class of the graded Milnor $K$-ring $K^M_*(F)$ determines the isomorphism class of $F$, when $k$ is algebraically closed or finite.

This paper uses partial fraction decompositions to give a direct computation of the logarithmic derivative of the norm in Milnor K-theory for a finite separable extension. This result is useful for computations involving the relative Brauer group in finite characteristic and Witt kernels for function fields in characteristic two. Kato's result that the norm is compatible with the trace under logarithmic differentiation also follows from these tools. When F(x) is rational over F in finite characteristic ℓ, the unramified part of is computed to be .

This chapter develops tools for a more penetrating study of algebraic number theory than was possible in Chapter V and concludes by formulating two of the main three theorems of Chapter V in the modern setting of “adeles” and “ideles” commonly used in the subject.Sections 1–5 introduce discrete valuations, absolute values, and completions for fields, always paying attention to implications for number fields and for certain kinds of function fields. Section 1 contains a prototype for all these notions in the construction of the fieldQp of p-adic numbers formed out of the rationals. Discrete valuations in Section 2 are a generalization of the order-of-vanishing function about a point in the theory of one complex variable. Absolute values in Section 3 are real-valued multiplicative functions that give a metric on a field, and the pair consisting of a field and an absolute value is called a valued field. Inequivalent absolute values have a certain independence property that is captured by the Weak Approximation Theorem. Completions in Section 4 are functions mapping valued fields into their metric-space completions. Section 5 concerns Hensel’s Lemma, which in its simplest form allows one to lift roots of polynomials over finite prime fields Fp to roots of corresponding polynomials over p-adic fields Qp.Section 6 contains the main theorem for investigating the fundamental question of how prime ideals split in extensions. Let K be a finite separable extension of a field F, let R be a Dedekind domain with field of fractions F, and let T be the integral closure of R in K. The question concerns the factorization of an ideal pT in T when p is a nonzero prime ideal in R. If Fp denotes the completion of F with respect to p, the theorem explains how the tensor product K ⊗F Fp splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of pT in T to the splitting of Fp in a complete field in which only one of the prime factors of pT plays a role.Section 7 is a brief aside mentioning additional conclusions one can draw when the extension K/F is a Galois extension.Section 8 applies the main theorem of Section 6 to an analysis of the different of K/F and ultimately to the absolute discriminant of a number field. With the new sharp tools developed in the present chapter, including a Strong Approximation Theorem that is proved in Section 8, a complete proof is given for the Dedekind Discriminant Theorem; only a partial proof had been accessible in Chapter V.Sections 9–10 specialize to the case of number fields and to function fields that are finite separable extensions of Fq (X), where Fq is a finite field. The adele ring and the idele group are introduced for each of these kinds of fields, and it is shown how the original field embeds discretely in the adeles and how the multiplicative group embeds discretely in the ideles. The main theorems are compactness theorems about the quotient of the adeles by the embedded field and about the quotient of the normalized ideles by the embedded multiplicative group. Proofs are given only for number fields. In the first case the compactness encodes the Strong Approximation Theorem of Section 8 and the Artin product formula of Section 9. In the second case the compactness encodes both the finiteness of the class number and the Dirichlet Unit Theorem.KeywordsPrime IdealValuation RingBasic AlgebraDedekind DomainFractional IdealThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

<!-- *** Custom HTML *** --> This chapter develops tools for a more penetrating study of algebraic number theory than was possible in Chapter V and concludes by formulating two of the main three theorems of Chapter V in the modern setting of "adeles" and "ideles" commonly used in the subject. Sections 1–5 introduce discrete valuations, absolute values, and completions for fields, always paying attention to implications for number fields and for certain kinds of function fields. Section 1 contains a prototype for all these notions in the construction of the field $\mathbb{Q}_p$ of $p$-adic numbers formed out of the rationals. Discrete valuations in Section 2 are a generalization of the order-of-vanishing function about a point in the theory of one complex variable. Absolute values in Section 3 are real-valued multiplicative functions that give a metric on a field, and the pair consisting of a field and an absolute value is called a valued field. Inequivalent absolute values have a certain independence property that is captured by the Weak Approximation Theorem. Completions in Section 4 are functions mapping valued fields into their metric-space completions. Section 5 concerns Hensel's Lemma, which in its simplest form allows one to lift roots of polynomials over finite prime fields $\mathbb{F}_p$ to roots of corresponding polynomials over $p$-adic fields $\mathbb{Q}_p$. Section 6 contains the main theorem for investigating the fundamental question of how prime ideals split in extensions. Let $K$ be a finite separable extension of a field $F$, let $R$ be a Dedekind domain with field of fractions $F$, and let $T$ be the integral closure of $R$ in $K$. The question concerns the factorization of an ideal $\mathfrak{p} T$ in $T$ when $\mathfrak{p}$ is a nonzero prime ideal in $R$. If $F_{\mathfrak{p}}$ denotes the completion of $F$ with respect to $\mathfrak{p}$, the theorem explains how the tensor product $K\otimes_FF_{\mathfrak{p}}$ splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of $\mathfrak{p} T$ in $T$ to the splitting of $F_{\mathfrak{p}}$ in a complete field in which only one of the prime factors of $\mathfrak{p} T$ plays a role. Section 7 is a brief aside mentioning additional conclusions one can draw when the extension $K/F$ is a Galois extension. Section 8 applies the main theorem of Section 6 to an analysis of the different of $K/F$ and ultimately to the absolute discriminant of a number field. With the new sharp tools developed in the present chapter, including a Strong Approximation Theorem that is proved in Section 8, a complete proof is given for the Dedekind Discriminant Theorem; only a partial proof had been accessible in Chapter V. Sections 9–10 specialize to the case of number fields and to function fields that are finite separable extensions of $\mathbb{F}_q(X)$, where $\mathbb{F}_q$ is a finite field. The adele ring and the idele group are introduced for each of these kinds of fields, and it is shown how the original field embeds discretely in the adeles and how the multiplicative group embeds discretely in the ideles. The main theorems are compactness theorems about the quotient of the adeles by the embedded field and about the quotient of the normalized ideles by the embedded multiplicative group. Proofs are given only for number fields. In the first case the compactness encodes the Strong Approximation Theorem of Section 8 and the Artin product formula of Section 9. In the second case the compactness encodes both the finiteness of the class number and the Dirichlet Unit Theorem.

Using the reduction theory of Nrron we give necessary conditions for the existence of points of order q on elliptic curves E rational over global fields. An application is the determination of all elliptic cu rves /Q with integer j and torsion points, generalizing Olson [8]. Another application is a theorem about semistable reduction whose consequences generalize a theorem of Olson [9] ( K = Q) and give divisibility conditions for the discriminant and the coefficients of E related with the paper of Zimmer [13] as well as diophantine equations related with Fermat's equation that are discussed for K Q and K a function field. We are interested in elliptic curves over global fields K (i.e. : K is a finite number field or K is a function field of one variable over a finite field) and especially in the torsion group of E(K), where E(K) is the group of K-rational points of E. It is well known that E(K) is finitely generated, it is conjectured that if K is a number field then the order of the torsion group of E(K) is bounded by some number depending only on K (cf. Demjanenko [1]). In any case in order to handle with E(K) the first step is to determine the torsion group. In principle this is not so difficult; if one uses the results of Lutz [6] and Zimmer [13], one sees immediately that for every E there exist points of q-power-order only for a finite number of primes q, as the equations for points of order q are known (in principle) one has only t ~ test what orders really occur. But as the computational work grows very rapidly with q it is usefull to look for sharper necessary conditions, and this shall be done in this paper.

Chebyshev polynomials over finite fields and reversibility of σ-automata on square grids

Let L be a number field and OL the ring of algebraic integers in L. For apolynomial f with coefficients in OL, the content of f in L is the ideal of OLgenerated by coefficients of f. The polynomial f is primitive in L if the contentof f in L is OL. In 2005, Arturo Magidin and David McKinnon proved the Gauss’ lemma fornumber fields, the product of two primitive polynomials is also primitive, andsome applications following from Gauss’ lemma for number fields. A function field K over a finite field k is a finite separable field extension overk(x) where x is a transcendantal element. In this research, we study Magidin and McKinnon’s work on the function fields.

Chebyshev polynomials over finite fields and reversibility of $sigma;-automata on square grids

Polynomial factorization is a crucial problem in Abstract Algebra, which has significant theoretical and practical applications. During the recent decades, considerable progress has been made in the theoretical parts and fast algorithms for factoring polynomials over finite fields. The recent outcomes of the Conventional Algorithms and their expansions are summarized in the review. This also investigates the situation when one variable is of low degree, and the polynomial is monic. The discussion commences with covering finite fields and the irreducibility of the polynomials before laying down the necessary conditions. Następnie zbadano klasyczne technique, why metodę kroneckera, algorithm berlekampa i algorytm cantor zassenhausa, które mają swoje mocne i słabe strony. This review illustrates recently generalized and computational techniques inspired by the observation of algorithm efficiency and scalability improvement for the factoring of high-degree integer polynomials. Due to advances in computer algebra systems and optimization of algorithms, polynomial factorization algorithms have become much more effective and practical. As a result of these advancements, the use of factorization algorithms has spread in practice. The impact of breakthroughs in these areas on applied areas, including Modern Cryptography, Error-Correcting Codes, and Information Security, is also discussed, such as the role of polynomial factorization over finite fields. In conclusion, challenges and open research questions in the contemporary world and new directions are the main topics of discussion in this paper. This paper highlights the continuous importance of polynomial factorization over finite fields as a central topic in abstract algebra and its many uses by fusing traditional ideas in algebra with modern computational and applied perspectives.

Let F be the completion, with respect to the degree valuation, of the field of rational functions over , the Galois (finite) field of q elements. A function f : F → F is integer-valued if . An integer-valued function f is called a pseudo-polynomial iff(M+K)≡f(M)(modK)for all and . Based on an interpolation series introduced by Carlitzin 1935, explicit shapes of pseudo-polynomials are established. Using an asymptotic characterization of polynomials, it is also proved that the set of all pseudo-polynomials is an integral domain but not a unique factorization domain.