Group-theoretic constraints on the structure of the class group

Let $\text{K}\text{k}$ be a Galois extension of number fields and G its Galois group. By considering the class group of K as a G module we are able to make assertions about its structure once the class number is known. Applications are made to cyclic cubic fields and the 2-class group of cyclotomic fields.     

Embedding of a universally consistent 2-extension in a universally consistent extension

A Galois extension is considered universally consistent with the period q if for any problem of embedding of this extension with an abelian kernel of the period q the consistency condition holds. Let K be a universally consistent extension of the period 2n + 1 of an algebraic number field k, such that 2 completely splits in K, and let (K/k, φ) be an embedding problem with the cyclic kernel of order 2. It is proven that (under some group-theoretical restrictions) there exists a solution of this embedding problem that is universally consistent with the period 2n.     

On the representation theory for full decomposable forms

Let $\text{k}\text{Q}$ be any finite normal extension and fix an order D of k invariant under the galois group $\text{G(}\text{k}\text{Q}\text{)}$. The ring class field K corresponding to D is normal over Q. We solve the problem of determining which full decomposable forms associated to the invertible ideals of D integrally represent a given positive integer m. First we establish a one-to-one correspondence between the improperly equivalent classes of such forms and the conjugacy classes of $\text{G(}\text{K}\text{Q}\text{)}$ which are contained in $\text{G(}\text{K}\text{k}\text{)}$. The solution is then seen to rest upon determining the Artin class in $\text{G(}\text{K}\text{Q}\text{)}$ of the unramified primes dividing m. This is accomplished by evaluating certain induced characters of $\text{G(}\text{K}\text{Q}\text{)}$ congruentially in terms of an associated integer linear recurrence sequence.     

Sur la structure galoisienne du groupe des unités d'un corps abélien réel de type (p, p)

Let p be a rational prime. We classify those $\text{Z}$[($\text{Z}$/p$\text{Z}$)²]-modules arising as submodules of the units (mod. torsion) of a real abelian field K with Galois group ($\text{Z}$/p$\text{Z}$)², up to isomorphism and up to genus. Explicit results are given when p is 2 or 3. We apply our classification to discuss the existence of a Minkowski unit in K for arbitrary p.     

Cyclic quartic fields and genus theory of their subfields

Let k = $\text{Q}$(√u) (u ≠ 1 squarefree), K any possible cyclic quartic field containing k. A close relation is established between K and the genus group of k. In particular: (1) Each K can be written uniquely as K = $\text{Q}$(√vwη), where η is fixed in k and satisfies η ? 1, (η) = $\text{U}$²√u, |$\text{U}$²| = |(√u)|, (v, u) = 1, v ∈ $\text{Z}$ is squarefree, w|u, 0 < w < √u. Thus if u ≠ a² + b², there is no K ? k. If u = a² + b² then for each fixed v there are 2^{g ? 1}K ? k, where g is the number of prime divisors of u. (2) $\text{K}\text{k}$ has a relative integral basis (RIB) (i.e., O_K is free over O_k) iff N(ε₀) = ?1 and w = 1, where ε₀ is the fundamental unit of k, (or, equivalently, iff K = $\text{Q}$(√vε₀√u), (v, u) = 1). (3) A RIB is constructed explicitly whenever it exists. (4) disc(K) is given. In particular, the following results are special cases of (2): (i) Narkiewicz showed in 1974 that $\text{K}\text{k}$ has a RIB if u is a prime; (ii) Edgar and Peterson (J. Number Theory12 (1980), 77–83) showed that for u composite there is at least one K ? k having no RIB. Besides, it follows from (4) that the classification and integral basis of K given by Albert (Ann. of Math.31 (1930), 381–418) are wrong.     

Ambiguous divisor classes in function fields

Let $\text{L}\text{K}$ be a Galois extension of algegraic function fields in one variable with Galois group G. Let J_K and J_L be the divisor classes of degree zero in K and L, respectively. A study is made of the kernel and cokernel of the natural map from J_K to $\text{J}{}_{\mathrm{L}}{}^{\mathrm{G}}$.     

On the rational equivalence of full decomposable forms

Let k be any finite normal extension of the rational field Q and fix an order D of k invariant under the galois group $\text{G(}\text{k}\text{Q}\text{)}$. Consider the set F of the full decomposable forms which correspond to the invertible fractional ideals of D. In a recent paper the author has given arithmetic criteria to determine which classes of improperly equivalent forms in F integrally represent a given positive rational integer m. These criteria are formulated in terms of certain integer sequences which satisfy a linear recursion and need only be considered modulo the primes dividing m. Here, for the most part, we consider partitioning F under rational equivalence. It is a found that the set of rationally equivalent classes in F is a group under composition of forms analogous to Gauss' and Dirichlet's classical results for binary quadratic forms. This leads us to given criteria as before to determine which classes of rationally equivalent forms in F rationally represent m. Moreover, by applying the genus theory of number fields, we find arithmetic criteria to determine when everywhere local norms are global norms if the Hasse norm principle fails to hold in $\text{k}\text{Q}$.     

On moment-discretization and least-squares solutions of linear integral equations of the first kind

Let K(s, t) be a continuous function on [0, 1] × [0, 1], and let $\text{K}$ be the linear integral operator induced by the kernel K(s, t) on the space $\text{L}$₂[0, 1]. This note is concerned with moment-discretization of the problem of minimizing 6K_x?y6 in the $\text{L}$₂-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation: ∝₀¹K(s_i, t) x(t) dt = y(s_i), i = 1, 2,h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating $\text{K2y}$ (where $\text{K}$² is the generalized inverse of $\text{K}$), without recourse to the normal equation $\text{K1Kx = K1y}$, that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind.     

K2 and K3 of the circle

Let k be $\text{Z}\text{[}\text{1}\text{2}\text{]}$, $\text{Q}$ or $\text{R}$, and set $\text{A =}\text{k[x,y]}\text{(x}{}^2\text{+ y}{}^2\text{? 1)}$. We compute K₂(A) and K₃(A). Our method is to construct a map $\text{? : K}{}_1\text{(k[i])→K}{}_{\mathrm{1\; +\; 1}}\text{(A)}$ and compare this to a localization sequence.We give three applications. We show that ? accounts for the primitive elements in K₂(A), and compare our results to computations of Bloch [1] for group schemes. Secondly, we consider the problem of basepoint independence, and indicate the interplay of geometry upon the K-theory of affine schemes obtained by glueing points of Spec(A). Third, we can iterate the construction to compute the K-theory of the torus ring A ?_kA.     

Number fields with class number congruent to 4 mod 8 and Hilbert's theorem 94

Let k be a number field with S_k, the Sylow 2-subgroup of its ideal class group, isomorphic to the four-group. Then either the class number of the Hilbert class field to k is odd, or there is a unique nonabelian unramified extension L of k of degree 8. The galois group $\text{g(}\text{L}\text{k}\text{)}$ is then the dihedral or quaternion group of order 8, and the occurrence of each is characterized in terms of Hilbert's theorem 94. In the case $\text{k = Q(?m)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, m a positive square-free integer, we obtain this characterization in terms of arithmetic properties of the integer m.     

Class numbers of definite binary quadratic lattices over algebraic function fields

Let k be an algebraic function field of one variable X having a finite field GF(q) of constants with q elements, q odd. Confined to imaginary quadratic extensions $\text{K}\text{k}$, class number formulas are developed for both the maximal and nonmaximal binary quadratic lattices L on (K, N), where N denotes the norm from K to k. The class numbers of L grow either with the genus g(k) of k (assuming the fields under consideration have bounded degree) or with the relative genus $\text{g(}\text{K}\text{k}\text{)}$ (assuming the lattices under consideration have bounded scale). In contrast to analogous theorems concerning positive definite binary quadratic lattices over totally real number fields, k is not necessarily totally real.     

Godeaux–Serre varieties with prescribed arithmetic fundamental group

We show that for any given field k and natural number $r\ge 2$, every continuous extension of the absolute Galois group ${\mathrm{Gal}}_k$ by a finite group is the arithmetic fundamental group of a geometrically connected smooth projective variety over k of dimension r.     

Permutation polynomials over algebraic number fields

Let K be an algebraic number field. It is known that any polynomial which induces a permutation on infinitely many residue class fields of K is a composition of cyclic and Chebyshev polynomials. This paper deals with the problem of deciding, for a given K, which compositions of cyclic or Chebyshev polynomials have this property. The problem is reduced to the case where K is an Abelian extension of $\text{Q}$. Then the question is settled for polynomials of prime degree, and the Abelian case for composite degree polynomials is considered. Finally, various special cases are dealt with.     

Pointwise convergence of eigenfunction expansions associated with ordinary differential operators

With an ordinary differential expression L = ∑ⁿ_k=0P_kD^k on an open interval I?$\text{r}$ is associated a selfadjoint operator H in a Hilbert space, possibly beyond $\text{K}$=L²(l). The set $\text{D}$H∩$\text{K}$ only depends on the generalized spectral family associated with H. It is shown that the (differentiated) eigenfunction expansion given by H converges uniformly on compact subintervals of l for functions in $\text{D}$(H)∩$\text{L}$ In case H is a semibounded selfadjoint operator in $\text{K}$=L²T, a similar result is proved for functions in $\text{D}$|H|, which is the set of all $\text{K}$∈$\text{K}$ for which there exists a sequence f_n∈(H) such that f_n→f in $\text{H}$ and (H(f_n ? f_m), f_n ? f_m → 0 as n, m → ∞.     

On the nonexistence of global solutions to a nonlinear Euler-Poisson-Darboux equation

In this paper, we show that the initial boundary value problem for the (singular) nonlinear EPD (Euler-Poisson-Darboux) equation

does not possess global solutions for arbitrary choices of $\text{u(x, 0). (x ? Ω ? R}{}^{\mathrm{n}}\text{, Ω}\text{bounded}\text{, Δ}{}_{\mathrm{n}}\text{= n}\text{dimensional Laplacian}$) when 0 < k ? 1 for a wide class of nonlinearities $\text{T}$, which includes all the even powers of u and the functions u^{2n + 1}, n = 1, 2,…. The solutions are assumed to vanish on the “walls” of the spacetime cylinder and satisfy $\text{?u}\text{?t(x, 0)}\text{= 0, x ? Ω}$. The result is independent of the space dimension.     

Specialization results and ramification conditions

Given a hilbertian field k of characteristic zero and a finite Galois extension E/k(T) with group G such that E/k is regular, we produce some specializations of E/k(T) at points t0 ∈ P¹(k) which have the same Galois group but also specified inertia groups at finitely many given primes. This result has two main applications. Firstly we conjoin it with previous works to obtain Galois extensions of Q of various finite groups with specified local behavior — ramified or unramified — at finitely many given primes. Secondly, in the case k is a number field, we provide criteria for the extension E/k(T) to satisfy this property: at least one Galois extension F/k of group G is not a specialization of E/k(T).     

An existence theorem for antichains

Let k₁, k₂,…, k_n be given integers, 1 ? k₁ ? k₂ ? … ? k_n, and let S be the set of vectors x = (x₁,…, x_n) with integral coefficients satisfying 0 ? x_i ? k_i, i = 1, 2, 3,…, n. A subset H of S is an antichain (or Sperner family or clutter) if and only if for each pair of distinct vectors x and y in H the inequalities x_i ? y_i, i = 1, 2,…, n, do not all hold. Let |H| denote the number of vectors in H, let K = k₁ + k₂ + … + k_n and for 0 ? l ? K let (l)H denote the subset of H consisting of vectors h = (h₁, h₂,…, h_n) which satisfy h₁ + h₂ + … + h_n = l. In this paper we show that if H is an antichain in S, then there exists an antichain H′ in S for which |(l)H′| = 0 if $\text{l <}\text{K}\text{2}$, $\text{|(}\text{K}\text{2}\text{)H′| = |(}\text{K}\text{2}\text{)H|}$ if K is even and |(l)H′| = |(l)H| + |(K ? l)H| if $\text{l>}\text{K}\text{2}$.     

Hecke operators and an identity for the dedekind sums

If h, k ∈ Z, k > 0, the Dedekind sum is given by

$\text{s(h,k) =}\text{∑}\text{μ=1}\text{k}\text{μ}\text{k}\text{hμ}\text{k}$

, with

$\text{((x)) = x ? [x] ?}\text{1}\text{2}\text{, x?Z}$

,

$\text{=0 , x∈Z}$

. The Hecke operators T_n for the full modular group SL(2, Z) are applied to log η(τ) to derive the identities (n ∈ Z⁺)

$\text{∑ ∑ s(ah+bk,dk) = σ(n)s(h,k)}$

,

$\text{ad=n b(}\text{mod}\text{d)}$

$\text{d>0}$

where (h, k) = 1, k > 0 and σ(n) is the sum of the positive divisors of n. Petersson had earlier proved (1) under the additional assumption k ≡ 0, h ≡ 1 (mod n). Dedekind himself proved (1) when n is prime.     

On the invariants of some Zi-extensions

Let k ? k₁ ? … ? K be a Z_i-extension. The relations of $\text{λ(}\text{K}\text{k}\text{)}$ and $\text{λ(}\text{KF}\text{F}\text{)}$ is studied, where $\text{F}\text{k}$ is a cyclic l-extension. If $\text{M}\text{k}$ is another Z_i-extension of k, it is shown that for i ? 0$\text{λ(}\text{Mk}{}_{\mathrm{i}}\text{k}{}_{\mathrm{i}}\text{) = rl}{}^{\mathrm{i}}\text{+ C}$, under minimal additional hypotheses. Finally if $\text{MK}\text{k}$ has a unique totally ramified prime, and X_K is cyclic, it is shown that MK can contain at most one Z_i-extension with non-zero μ invariant.