Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStickelberger annihilators of logarithmic class groups
140
0
0.0
(
0
)
Authors
Jean-Franc{c}ois Jaulent
Ask ChatGPT about the research
No Arabic abstract
For any odd prime number $ell$ and any abelian number field F containing the $ell$-th roots of unity, we show that the Stickelberger ideal annihilates the imaginary component of the $ell$-group of logarithmic classes and that its reflection annihilates the real componen of the Bertrandias-Payan module. As a consequence we obtain a very simple proof of annihilation results for the so-called wild {e}tale $ell$-kernels of F .
rate research
Read More
Building on Boscas method, we extend to tame ray class groups the results on capitulation of ideals of a number field by composition with abelian extensions of a subfield first studied by Gras. More precisely, for every extension of number fields K/k, where at least one infinite place splits completely, and every squarefree divisor m of K, we prove that there exist infinitely many abelian extensions F/k such that the ray class group mod m of K capitulates in KF. As a consequence we generalize to tame ray class groups the results of Kurihara on triviality of class groups for maximal abelian pro-extensions of totally real number fields.
We prove, under some mild hypothesis, that an etale cover of curves defined over a number field has infinitely many specializations into an everywhere unramified extension of number fields. This constitutes an absolute version of the Chevalley-Weil theorem. Using this result, we are able to generalize the techniques of Mestre, Levin and the second author for constructing and counting number fields with large class group.
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.
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.
For each odd prime p>=5, there exist finite p-groups G with derived quotient G/D(G)=C(p)xC(p) and nearly constant transfer kernel type k(G)=(1,2,...,2) having two fixed points. It is proved that, for p=7, this type k(G) with the simplest possible case of logarithmic abelian quotient invariants t(G)=(11111,111,21,21,21,21,21,21) of the eight maximal subgroups is realized by exactly 98 non-metabelian Schur sigma-groups S of order 7^11 with fixed derived length dl(S)=3 and metabelianizations S/D(D(S)) of order 7^7. For p=5, the type k(G) with t(G)=(2111,111,21,21,21,21) leads to infinitely many non-metabelian Schur sigma-groups S of order at least 5^14 with unbounded derived length dl(S)>=3 and metabelianizations S/D(D(S)) of fixed order 5^7. These results admit the conclusion that d=-159592 is the first known discriminant of an imaginary quadratic field with 7-class field tower of precise length L=3, and d=-90868 is a discriminant of an imaginary quadratic field with 5-class field tower of length L>=3, whose exact length remains unknown.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
