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The rationality problem of two-dimensional purely quasi-monomial actions was solved completely by (Hoshi, Kang and Kitayama, J. Algebra 403, 363-400, 2014). As a generalization, we solve the rationality problem of two-dimensional quasi-monomial actions under the condition that the actions are defined within the base field. In order to prove the theorem, we give a brief review of the Severi-Brauer variety with some examples and rationality results. We also use a rationality criterion for conic bundles of P 1 over non-closed fields. [ABSTRACT FROM AUTHOR]

We give an asymptotic evaluation for the number of automorphic characters of an algebraic torus T with bounded analytic conductor. The analytic conductor is defined via the local Langlands correspondence for tori by choosing a finite-dimensional complex algebraic representation of the L-group of T . Our results therefore fit into a general framework of counting automorphic representations on reductive groups by analytic conductor. [ABSTRACT FROM AUTHOR]

For an algebraic torus S , Blinstein and Merkurjev have given an estimate of 3-rd unramified cohomology H ¯ n r 3 (F (S) , Q / Z (2)) obtained from a flasque resolution of S. Based on their work, for the norm one torus W = R K / F (1) G m with K / F abelian, we compute the 3-rd unramified cohomology H ¯ n r 3 (F (W) , Q / Z (2)). [ABSTRACT FROM AUTHOR]

We give a complete answer to the rationality problem (up to stable k -equivalence) for norm one tori R K / k (1) (G m) of K / k whose Galois closures L / k are dihedral extensions with the aid of Endo and Miyata [17, Theorem 1.5, Theorem 2.3] and Endo [15, Theorem 2.1]. By using a similar technique, we give refinements of the proof of stably rational cases of Endo and Miyata's theorems as an appendix of the paper. [ABSTRACT FROM AUTHOR]

In this paper we count the number N 3 tor (X) of 3-dimensional algebraic tori over Q whose Artin conductor is bounded above by X. We prove that N 3 tor (X) ≪ ε X 1 + log ⁡ 2 + ε log ⁡ log ⁡ X , and this upper bound can be improved to N 3 tor (X) ≪ X (log ⁡ X) 4 log ⁡ log ⁡ X under the Cohen-Lenstra heuristics for p = 3. We also prove that for 67 out of 72 conjugacy classes of finite nontrivial subgroups of GL 3 (Z) , Malle's conjecture for tori over Q holds up to a bounded power of log ⁡ X under the Cohen-Lenstra heuristics for p = 3 and Malle's conjecture for quartic A 4 -fields. [ABSTRACT FROM AUTHOR]

Let k be a field, T be an algebraic k -torus, X be a smooth k -compactification of T and Pic X ‾ be the Picard group of X ‾ = X × k k ‾. Hoshi, Kanai and Yamasaki [HKY22] determined H 1 (k , Pic X ‾) for norm one tori T = R K / k (1) (G m) and gave a necessary and sufficient condition for the Hasse norm principle for extensions K / k of number fields with [ K : k ] = n ≤ 15 and n ≠ 12. In this paper, we determine 64 cases with H 1 (k , Pic X ‾) ≠ 0 and give a necessary and sufficient condition for the Hasse norm principle for K / k with [ K : k ] = 12. [ABSTRACT FROM AUTHOR]

In this paper, we give some topological properties and estimates of orbit of certain subsets of K v -points of varieties under actions of algebraic tori. These results are concerned with an analogue of Bruhat-Tits' question on the set of v -adic integral points of algebraic tori. [ABSTRACT FROM AUTHOR]

Let k be a field and T be an algebraic k-torus. In 1969, over a global field k, Voskresenskiǐ proved that there exists an exact sequence 0\to A(T)\to H^1(k,\operatorname {Pic}\overline {X})^\vee \to \Sha (T)\to 0 where A(T) is the kernel of the weak approximation of T, \Sha (T) is the Shafarevich-Tate group of T, X is a smooth k-compactification of T, \overline {X}=X\times _k\overline {k}, \operatorname {Pic}\overline {X} is the Picard group of \overline {X} and \vee stands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torus T=R^{(1)}_{K/k}(\mathbb {G}_m) of K/k, \Sha (T)=0 if and only if the Hasse norm principle holds for K/k. First, we determine H^1(k,\operatorname {Pic} \overline {X}) for algebraic k-tori T up to dimension 5. Second, we determine H^1(k,\operatorname {Pic} \overline {X}) for norm one tori T=R^{(1)}_{K/k}(\mathbb {G}_m) with [K:k]=n\leq 15 and n\neq 12. We also show that H^1(k,\operatorname {Pic} \overline {X})=0 for T=R^{(1)}_{K/k}(\mathbb {G}_m) when the Galois group of the Galois closure of K/k is the Mathieu group M_n\leq S_n with n=11,12,22,23,24. Third, we give a necessary and sufficient condition for the Hasse norm principle for K/k with [K:k]=n\leq 15 and n\neq 12. As applications of the results, we get the group T(k)/R of R-equivalence classes over a local field k via Colliot-Thélène and Sansuc's formula and the Tamagawa number \tau (T) over a number field k via Ono's formula \tau (T)=|H^1(k,\widehat {T})|/|\Sha (T)|. [ABSTRACT FROM AUTHOR]

We formulate and analyze several finiteness conjectures for linear algebraic groups over higher-dimensional fields. In fact, we prove all of these conjectures for algebraic tori as well as in some other situations. This work relies in an essential way on several finiteness results for unramified cohomology. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to give a constructive proof for a particular case of the no-name lemma. Let G be a finite group, K a field that is equipped with a faithful G -action, and L a sign permutation G -lattice (see the Introduction for the definition). Then G acts naturally on the group algebra K [ L ] of L over K , and hence also on the quotient field K (L) = Q (K [ L ]). A well-known variant of the no-name lemma asserts that the invariant sub-field K (L) G is a purely transcendental extension of K G . In other words, there exist y 1 , ... , y n which are algebraically independent over K G such that K (L) G ≅ K G (y 1 , ... , y n). In this paper, we give an explicit construction of suitable elements y 1 , ... , y n . [ABSTRACT FROM AUTHOR]

We classify stably/retract rational norm one tori in dimension n−1 for n = 2e (e ≥ 1) as a power of 2 and n = 12, 14, 15. Retract non-rationality of norm one tori for primitive G ≤ S2p where p is a prime number and for the five Mathieu groups Mn ≤ Sn (n = 11,12,22,23,24) is also given. [ABSTRACT FROM AUTHOR]

Let T be an algebraic torus over a field F. There is a pairing between the groups of torsors for the torus T and its dual with values in the third Galois cohomology group over all field extensions of F. We study the kernel of this pairing. [ABSTRACT FROM AUTHOR]

A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper, we generalize Chow's theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori. [ABSTRACT FROM AUTHOR]

We describe the ramification in cyclic extensions arising from the Kummer theory of the Weil restriction of the multiplicative group. This generalises the classical theory of Hecke describing the ramification of Kummer extensions. [ABSTRACT FROM PUBLISHER]

Voskresenskii conjectured that stably rational tori are rational. Klyachko proved this assertion for a wide class of tori by general principles. We re-prove Klyachko's result by providing simple explicit birational isomorphisms, and elaborate on some links to torus-based cryptography. [ABSTRACT FROM AUTHOR]

We give explicit parametrizations of the algebraic tori $$\mathbb {T}_{n}$$ over any finite field $$\mathbb {F}_{q}$$ for any prime power $$n$$ . Applying the construction for $$n=3$$ to a quadratic field $$\mathbb {F}_{q^2}$$ we show that the set of $$\mathbb {F}_q$$ -rational points of the torus $$\mathbb {T}_{6}$$ is birationally equivalent to the affine part of a Singer arc in $$\mathbb {P}^2(\mathbb {F}_{q^2})$$ . This gives a simple, yet efficient compression and decompression algorithm from $$\mathbb {T}_{6}(\mathbb {F}_{q})$$ to $$\mathbb {A}^2(\mathbb {F}_{q})$$ that can be substituted in the faster implementation of CEILIDH (Granger et al., in Algorithmic number theory, pp 235-249, Springer, Berlin, ) achieving a theoretical 30 % speedup and that is also cheaper than the recently proposed factor- $$6$$ compression technique in Karabina (IEEE Trans Inf Theory 58(5):3293-3304, ). The compression methods here presented have a wide class of applications to public-key and pairing-based cryptography over any finite field. [ABSTRACT FROM AUTHOR]

Let k be a field and G be a finite group acting on the rational function field k(xg |g∈G) by k -automorphisms h(xg)=xhg for any g,h∈G. Noether's problem asks whether the invariant field k(G)=k(xg |g∈G)G is rational (i.e. purely transcendental) over k . In 1974, Lenstra gave a necessary and sufficient condition to this problem for abelian groups G . However, even for the cyclic group Cp of prime order p, it is unknown whether there exist infinitely many primes p such that Q(Cp) is rational over Q . Only known 17 primes p for which Q(Cp) is rational over Q are p≤43 and p=61,67,71 . We show that for primes p<20000, Q(Cp) is not (stably) rational over Q except for affirmative 17 primes and undetermined 46 primes. Under the GRH, the generalized Riemann hypothesis, we also confirm that Q(Cp) is not (stably) rational over Q for undetermined 28 primes p out of 46 . [ABSTRACT FROM AUTHOR]

Abstract: Let G be a finite group acting on , the rational function field of n variables over a field k. The action is called a purely monomial action if for all , for where . The main question is that, under what situations, the fixed field is rational (= purely transcendental) over k. This rationality problem has been studied by Hajja, Kang, Hoshi, Rikuna when . In this paper we will prove that is rational over k provided that the purely monomial action is decomposable. To prove this result, we introduce a new notion, the quasi-monomial action, which is a generalization of previous notions of multiplicative group actions. Moreover, we determine the rationality problem of purely quasi-monomial actions of over k where . [Copyright &y& Elsevier]

Let p be a prime integer, 1≤ s≤ r be integers and F be a field of characteristic different from p. We find upper and lower bounds for the essential p-dimension ed( $$ Al{{g}_{{{{p}^r},{{p}^s}}}} $$) of the class $$ Al{{g}_{{{{p}^r},{{p}^s}}}} $$ of central simple algebras of degree p and exponent dividing p. In particular, we show that ed( Alg)=ed( Alg)=8 and ed( $$ Al{{g}_{{{{p}^2},p}}} $$)= p+ p for p odd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we find an upper bound for the essential dimension of finite cyclic groups over a field F of characteristic different from containing all the primitive -th roots of unity, where are distinct prime numbers. We then provide a lower bound for in terms of the canonical dimension of some central division algebra A. This lower bound coincides with the upper bound thus giving the value of if a conjecture for holds. [Copyright &y& Elsevier]

Abstract: An improved upper bound for the essential p-dimension of is computed using algebraic tori. It follows that if . [ABSTRACT FROM AUTHOR]

Let k be a field, G be a finite group, k (x (g) : g ∈ G) be the rational function field with the variables x (g) where g ∈ G. The group G acts on k (x (g) : g ∈ G) by k -automorphisms where h ⋅ x (g) = x (h g) for all h , g ∈ G. Let k (G) be the fixed field defined by k (G) : = k (x (g) : g ∈ G) G = { f ∈ k (x (g) : g ∈ G) : h ⋅ f = f for all h ∈ G }. Noether's problem asks whether the fixed field k (G) is rational (= purely transcendental) over k. Let m and n be positive integers and assume that there is an integer t such that t ∈ (Z / m Z) × is of order n. Define a group G m , n : = 〈 σ , τ : σ m = τ n = 1 , τ − 1 σ τ = σ t 〉 ≃ C m ⋊ C n. Assume furthermore that (i) m is an odd integer, and (ii) for any e | n , the ideal 〈 ζ e − t , m 〉 in Z [ ζ e ] is a principal ideal (where ζ e is a primitive e -th root of unity). Theorem. If k is a field with ζ m , ζ n ∈ k , then k (G m , n) is rational over k. Consequently, it may be shown that, for any positive integer n , the set S : = { p : p is a prime number such that C (G p , n) is rational over C } is of positive Dirichlet density; in particular, S is an infinite set. [ABSTRACT FROM AUTHOR]