On the p 2-Rank of Tame Kernels of Quadratic Fields

For any odd prime p, we prove some results connecting the p²-rank of the tame kernel of a quadratic field F with the p²-rank Cl(𝒪_E1 ), where E₁ is the maximal real subfield of F(ζ_p² ).     

数域中tame 核的p-秩

Let E/F be a Galois extension of number fields with Galois group G=Gal(E/F), and let p be a prime not dividing #G. In this paper, using character theory of finite groups, we obtain the upper bound of #K₂O_E if the group K₂O_E is cyclic, and prove some results on the divisibility of the p-rank of the tame kernel K₂O_E, where E/F is not necessarily abelian. In particular, in the case of G=C_n, D_n, A₄, we easily get some results on the divisibility of the p-rank of the tame kernel K₂O_E by the character table. Let E/Q be a normal extension with Galois group D_l, where l is an odd prime, and F/Q a non-normal subextension with degree l. As an application, we show that f|p-rank K₂O_F, where f is the smallest positive integer such that p^f≡±1(mod l).     

Coordinates of R[x,y]: Constructions and Classifications

Let R be a PID. We construct and classify all coordinates of R[x, y] of the form p ₂ y + Q ₂(p ₁ x + Q ₁(y)) with p ₁, p ₂ ∈ qt(R) and Q ₁, Q ₂ ∈ qt(R)[y]. From this construction (with R = K[z]) we obtain nontame automorphisms σ of K[x, y, z] (where K is a field of characteristic 0) such that the subgroup generated by σ and the affine automorphisms contains all tame automorphisms.     

Reflection Theorems and the Tame Kernel of a Number Field

Let L be a number field containing the pth primitive root of unity ζ _p . We investigate the p-rank of the ideal class groups of some subfields of L by using reflection theorems and establish relations between the p-rank of the ideal class groups and that of groups of units of some subfields of L.

Let F be a number field and 𝒪 _F the ring of integers in F. We also study the p-rank of tame kernels of F and establish relations between the p-rank of K ₂𝒪 _F and that of some direct summands of the ideal class group of F(ζ _p ).     

On the First Galois Cohomology Group of the Algebraic Group SL1(D)

Let A be a central simple algebra over a field F. Denote the reduced norm of A over F by Nrd: A* → F* and its kernel by SL₁(A). For a field extension K of F, we study the first Galois Cohomology group H ¹(K,SL₁(A)) by two methods, valuation theory for division algebras and K-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series.     

On Tame Kernels and Ideal Class Groups

It is well known that there is a close connection between tame kernels and ideal class groups of number fields. However, the latter is a very difficult subject in number theory. In this paper, we prove some results connecting the p^n-rank of the tame kernel of a cyclic cubic field F with the p^n-rank of the coinvariants of μp^n×CI（δE,T） under the action of the Galois group, where E = F（ζp^n ） and T is the finite set of primes of E consisting of the infinite primes and the finite primes dividing p. In particular, if F is a cyclic cubic field with only one ramified prime and p = 3, n = 2, we apply the results of the tame kernels to prove some results of the ideal class groups of E, the maximal real subfield of E and F（ζ3）.     

Orthogonal Groups 𝒪(n) over GF(2) as Automorphisms

Let (V, Q) be a quadratic vector space over a fixed field. Orthogonal group 𝒪(V, Q) is defined as automorphisms on (V, Q). If Q = I, it is 𝒪(V, I) = 𝒪(n). There is a nice result that 𝒪(n) ? Aut(𝔬(n)) over ? or ?, where 𝔬(n) is the Lie algebra of n × n alternating matrices over the field. How about another field The answer is “Yes” if it is GF(2). We show it explicitly with the combinatorial basis ?. This is a verification of Steinberg's main result in 1961, that is, Aut(𝔬(n)) is simple over the square field, with a nonsimple exception Aut(𝔬(5)) ? 𝒪(5) ? 𝔖₆.     

The multiplier ideal is a universal test ideal

Abstract

For a canonical threefold X, we know that h ⁰(X, 𝒪 _X (nK _X )) ≥ 1 for a sufficiently large n. When χ(𝒪 _X ) > 0, it is not easy to get such an integer n. Fletcher showed that h ⁰(X, 𝒪 _X (12K _X )) ≥ 1 and h ⁰(X, 𝒪 _X (24K _X )) ≥ 2 when χ(𝒪 _X ) = 1. He inquired about existence of a canonical threefold with given conditions which shows the result sharp. We show that such an example does not exist. Using a different technique, we prove h ⁰(X, 𝒪 _X (12K _X )) ≥ 2.     

The Structure of the Tame Kernels of Quadratic Number Fields (III)

Let F be an imaginary quadratic number field and K ₂ O _F the tame kernel of F. In this article, we determine all possible values of r ₄(K ₂ O _F ) for each type of imaginary quadratic number field F. In particular, for each type of imaginary quadratic number field we give the maximum possible value of r ₄(K ₂ O _F ) and show that each integer between the lower and upper bounds occurs as a value of the 4-rank of K ₂ O _F for infinitely many imaginary quadratic number fields F.     

p-Rank and semi-stable reduction of curves

Let R be a discrete complete valuation ring, with field of fractions K, and with algebraically closed residue field k of characteristic p > 0. Let X be a germ of an R-curve at an ordinary double point. Consider a finite Galois covering f: Y → X, whose Galois group G is a p-group, such that Y is normal, and which is étale above X_k≔ x × _rk. Asume that Y has a semi-stable model :→ Y over R, and let y be a closed point of Y. If the inertia subgroup I(y) at y is cyclic of order pⁿ, we compute the p-rank of tf⁻¹ (y) by using a result of Raynaud. In particular, we prove that this p-rank is bounded by pⁿ −1.     

On the elliptic curve y 2 = x 3 −2rDx and factoring integers

Let D=pq be the product of two distinct odd primes.Assuming the parity conjecture,we construct infinitely many r≥1 such that E2rD:y2=x3-2rDx has conjectural rank one and vp(x([k]Q))≠vq(x([k]Q))for any odd integer k,where Q is the generator of the free part of E(Q).Furthermore,under the generalized Riemann hypothesis,the minimal value of r is less than c log4 D for some absolute constant c.As a corollary,one can factor D by computing the generator Q.     

Spectra of Maximal 1-Sided Ideals and Primitive Ideals

R is any ring with identity. Let Spec _r (R) (resp. Max _r (R), Prim _r (R)) denote the set of all right prime ideals (resp. all maximal right ideals, all right primitive ideals) of R and let U _r (eR) = {P ? Spec _r (R)| e ? P}. Let  = ∪_{P?Prim r (R)} Spec _r ^P (R), where Spec _r ^P (R) = {Q ?Spec _r ^P (R)|P is the largest ideal contained in Q}. A ring is called right quasi-duo if every maximal right ideal is 2-sided. In this article, we study the properties of the weak Zariski topology on  and the relationships among various ring-theoretic properties and topological conditions on it. Then the following results are obtained for any ring R: (1) R is right quasi-duo ring if and only if  is a space with Zariski topology if and only if, for any Q ? , Q is irreducible as a right ideal in R. (2) For any clopen (i.e., closed and open) set U in ? = Max _r (R) ∪  Prim _r (R) (resp.  = Prim _r (R)) there is an element e in R with e ² ? e ? J(R) such that U = U _r (eR) ∩  ? (resp. U = U _r (eR) ∩  ), where J(R) is the Jacobson of R. (3) Max _r (R) ∪  Prim _r (R) is connected if and only if Max _l (R) ∪  Prim _l (R) is connected if and only if Prim _r (R) is connected.     

A valuation criterion for normal basis generators in local fields of characteristic p

Let K be a complete local field of characteristic p with perfect residue field, and let L/K be a finite, totally ramified, Galois p-extension with G = Gal(L/K). Let v _L be the normalized valuation with ${v_L(L^{\times})=\mathbb{Z} }Let K be a complete local field of characteristic p with perfect residue field, and let L/K be a finite, totally ramified, Galois p-extension with G = Gal(L/K). Let v _L be the normalized valuation with v_L(L^×)=\mathbbZ {v_L(L^{\times})=\mathbb{Z} }. Let p_L ? L{\pi_L\in L} be a prime element, and let p′ (x) be the derivative of the minimal polynomial for π _L over K. We show that any element r ? L{\rho\in L} with v_L(r) o -v_L(p￠(p_L))-1 mod [L:K]{v_L(\rho)\equiv -v_L(p'(\pi_L))-1\bmod[L:K]} generates a normal basis: K[G]ρ = L. This criterion is tight: Given any integer i with i\not o -v_L(p￠(p_L))-1 mod [L:K]{i\not\equiv -v_L(p'(\pi_L))-1\bmod[L:K]}, there is a r_i ? L{\rho_i\in L} with v _L (ρ _i ) = i such that K[G]r_i\subsetneq L{K[G]\rho_i\subsetneq L}.     

Extensions and the Induced Characters of Cyclic p-Groups

For a prime p, we denote by B_n the cyclic group of order pⁿ. Let φ be a faithful irreducible character of B_n, where p is an odd prime. We study the p-group G containing B_n such that the induced character φ^G is also irreducible. The purpose of this article is to determine the subgroup N_G(N_G(B_n)) of G under the hypothesis [N_G(B_n):B_n]⁴ ≦ pⁿ.     

Vanishing of some cohomology groups and bounds for the Shafarevich-Tate groups of elliptic curves

Let E be an elliptic curve over Q and ? be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at ?. Under certain assumptions, we prove the vanishing of the Galois cohomology group H¹(Gal(K(E[?ⁱ])/K),E[?ⁱ]) for all i?1. When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyvagin, which finds a bound for the order of the ?-primary part of Shafarevich-Tate groups of E over K. This bound is consistent with the prediction of Birch and Swinnerton-Dyer conjecture.     

Decompositions of Quotient Rings and m-Power Commuting Maps

Let R be a semiprime ring with symmetric Martindale quotient ring Q, n ≥ 2 and let f(X) = X ⁿ h(X), where h(X) is a polynomial over the ring of integers with h(0) = ±1. Then there is a ring decomposition Q = Q ₁ ⊕ Q ₂ ⊕ Q ₃ such that Q ₁ is a ring satisfying S _2n?2, the standard identity of degree 2n ? 2, Q ₂ ? M _n (E) for some commutative regular self-injective ring E such that, for some fixed q > 1, x ^q  = x for all x ∈ E, and Q ₃ is a both faithful S _2n?2-free and faithful f-free ring. Applying the theorem, we characterize m-power commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring.     

On the Constructive Inverse Problem in Differential Galois Theory#

We give sufficient conditions for a differential equation to have a given semisimple group as its Galois group. For any group G with G ⁰ = G ₁ · ··· · G _r , where each G _i is a simple group of type A_?, C_?, D_?, E₆, or E₇, we construct a differential equation over C(x) having Galois group G.     

On the equivariant structure of ideals in abelian extensions of local fields (with an appendix by W. Bley)

Let p be an odd rational prime and K a finite extension of \Bbb Q_p {\Bbb Q}_p . We give a complete classification of those finite abelian extensions L/K L/K in which any ideal of the valuation ring of L is free over its associated order in \Bbb Q_p[Gal(L/K)] {\Bbb Q}_p[Gal(L/K)] . In an appendix W. Bley describes an algorithm which can be used to determine the structure of Galois stable ideals in abelian extensions of number fields. The algorithm is applied to give several new and interesting examples.