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Register a new userOn $p$-class groups of relative cyclic $p$-extensions
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We prove a general stability theorem for $p$-class groups of number fields along relative cyclic extensions of degree $p^2$, which is a generalization of a finite-extension version of Fukudas theorem by Li, Ouyang, Xu and Zhang. As an application, we give an example of pseudo-null Iwasawa module over a certain $2$-adic Lie extension.
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Let $p$ be a prime. We define the deficiency of a finitely-generated pro-$p$ group $G$ to be $r(G)-d(G)$ where $d(G)$ is the minimal number of generators of $G$ and $r(G)$ is its minimal number of relations. For a number field $K$, let $K_emptyset$ be the maximal unramified $p$-extension of $K$, with Galois group $G_emptyset = Gal(K_emptyset/K)$. In the 1960s, Shafarevich (and independently Koch) showed that the deficiency of $G_emptyset$ satisfies $$0leq mathrm{Def}({rm G}_emptyset) leq dim (O_K^times/(O_K^{times })^p),$$ relating the deficiency of $G_emptyset$ to the $p$-rank of the unit group $O_K^times$ of the ring of integers $O_K$ of $K$. In this work, we further explore connections between relations of the group $G_emptyset$ and the units in the tower $K_emptyset/K$, especially their Galois module structure. In particular, under the assumption that $K$ does not contain a primitive $p$th root of unity, we give an exact formula for $mathrm{Def}({rm G}_emptyset)$ in terms of the number of independent Minkowski units in the tower. The method also allows us to infer more information about the relations of G$_emptyset$, such as their depth in the Zassenhaus filtration, which in certain circumstances makes it easier to show that G$_emptyset$ is infinite. We illustrate how the techniques can be used to provide evidence for the expectation that the Shafarevich-Koch upper bound is almost always sharp.
We show that for primes $N, p geq 5$ with $N equiv -1 bmod p$, the class number of $mathbb{Q}(N^{1/p})$ is divisible by $p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when $N equiv -1 bmod p$, there is always a cusp form of weight $2$ and level $Gamma_0(N^2)$ whose $ell$-th Fourier coefficient is congruent to $ell + 1$ modulo a prime above $p$, for all primes $ell$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree $p$ extension of $mathbb{Q}(N^{1/p})$.
As an analogue of a link group, we consider the Galois group of the maximal pro-$p$-extension of a number field with restricted ramification which is cyclotomically ramified at $p$, i.e, tamely ramified over the intermediate cyclotomic $mathbb Z_p$-extension of the number field. In some basic cases, such a pro-$p$ Galois group also has a Koch type presentation described by linking numbers and mod $2$ Milnor numbers (Redei symbols) of primes. Then the pro-$2$ Fox derivative yields a calculation of Iwasawa polynomials analogous to Alexander polynomials.
In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of the curves X^2+3Y^2 = 4DZ^3 and X^2+27Y^2=4DZ^3 as D varies over cube-free positive integers. We parametrise these points using well known parametrisation of integral points (x,y,z) of the curve X^2+3Y^2=4Z^3 with GCD(y,z)=1.
Each p-ring class field K(f) modulo a p-admissible conductor f over a quadratic base field K with p-ring class rank r(f) mod f is classified according to Galois cohomology and differential principal factorization type of all members of its associated heterogeneous multiplet M(K(f))=[(N(c,i))_{1<=i<=m(c)}]_{c|f} of dihedral fields N(c,i) with various conductors c|f having p-multiplicities m(c) over K such that sum_{c|f} m(c)=(p^r(f)-1)/(p-1). The advanced viewpoint of classifying the entire collection M(K(f)), instead of its individual members separately, admits considerably deeper insight into the class field theoretic structure of ring class fields, and the actual construction of the multiplet M(K(f)) is enabled by exploiting the routines for abelian extensions in the computational algebra system Magma.
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