Related papers: The Gross-Kuz'min Connjecture for CM fields

We show finiteness results on torsion points of commutative algebraic groups over a $p$-adic field $K$ with values in various algebraic extensions $L/K$ of infinite degree. We mainly study the following cases: (1) $L$ is an abelian…

For an algebraic number field K such that prime l splits completely in K we define a regulator R(K) that characterize the subgroup of universal norms from the cyclotomic extension of K in the completed group of S-units of K, where S…

The P\'{o}lya group $Po(K)$ of an algebraic number field $K$ is the subgroup of the ideal class group $Cl_{K}$ generated by the ideal classes of the products of prime ideals of the same norm. If $Po(K)$ is trivial, then the number field $K$…

Let K be a finite extension of Q_p with residue field F_q and let P(T) = T^d + a_{d-1}T^{d-1} + ... +a_1 T, where d is a power of q and a_i is in the maximal ideal of K for all i. Let u_0 be a uniformizer of O_K and let {u_n}_{n \geq 0} be…

To each Galois extension $L/K$ of number fields with Galois group $G$ and each integer $r \leq 0$ one can associate Stickelberger elements in the centre of the rational group ring $\mathbb{Q}[G]$ in terms of values of Artin $L$-series at…

Let $K$ be a real quadratic field and let $p$ be a prime number which is inert in $K$. Let $K_p$ be the completion of $K$ at $p$. In a previous paper, we constructed a $p$-adic invariant $u_C\in K_p$, and we proved a $p$-adic Kronecker…

In this work, we show that given a finite p-group G, a number field K having a trivial p-class group Cl K , and a finite set of primes S of K, there exists a finite extension F/K such that the S-split p-Hilbert class field tower L S p (F )…

Let $\K$ be a galois CM extension of $\Q$ and $\K_{\infty}$ its cyclotomic $\Z_p$-extension. Let $A_n$ be the $p$-parts of the class groups in the intermediate subfields $\K_n \subset \K_{\infty}$ and $\rg{A} = \varprojlim_n A_n$. We show…

This is a two - part paper, in which we prove the following fact: let K be a CM field and L/K be a CM Z_p-extension. Then the Iwasawa mu-invariant of L vanishes. For the case when L is the cyclotomic Z_p extension, this is the Iwasawa…

Let $p$ be an odd prime number and $\zeta_{p} := \exp(2\pi i/p)$. Then, it is well-known that the $A_{p-1}$-root lattice can be realized as the (Hermitian) trace form of the $p$-th cyclotomic extension $\mathbb{Q}(\zeta_{p})/\mathbb{Q}$…

We formulate a general conjecture on the characteristic polynomials of S-decomposed T-ramified Iwasawa modules over the cyclotomic Z {\ell}-extension of a number field. We show that this conjecture is equivalent to the conjunctions of the…

We prove a strengthening of the "reciprocity conjecture" of Khare and Wintenberger. The input to the original conjecture is an odd prime p, a CM number field F containing the pth roots of unity, and a pair of primes of the maximal totally…

We consider the family of CM-fields which are pro-p p-adic Lie extensions of number fields of dimension at least two, which contain the cyclotomic Z_p-extension, and which are ramified at only finitely many primes. We show that the Galois…

Let $A/\mathbb{Q}$ be an elliptic curve with split multiplicative reduction at a prime $p$. We prove (an analogue of) a conjecture of Perrin-Riou, relating $p$-adic Beilinson$-$Kato elements to Heegner points in $A(\mathbb{Q})$, and a large…

In 1980, Gross conjectured a formula for the expected leading term at $s=0$ of the Deligne--Ribet $p$-adic $L$-function associated to a totally even character $\psi$ of a totally real field $F$. The conjecture states that after scaling by…

Cyclic, ramified extensions $L/K$ of degree $p$ of local fields with residue characteristic $p$ are fairly well understood. Unless $\mbox{char}(K)=0$ and $L=K(\sqrt[p]{\pi_K})$ for some prime element $\pi_K\in K$, they are defined by an…

Given a finite abelian group $\Gamma$, we study the distribution of the $p$-part of the class group $\operatorname{Cl}(K)$ as $K$ varies over Galois extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$ with Galois group isomorphic to $\Gamma$.…

This paper is an updated version of ANT-0372 (2002 dec 4) with the same title. Several errors are corrected in this version. An example of the kind of results obtained is: Let K/\Q be an abelian extension with N = [K:\Q] > 1, N odd. Let…

Let $p\ge 5$ be a prime number, $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at $p$ and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ is $+1$. When $p$ is split in $K$, Darmon and Iovita…

Motivated by the work of Greenberg-Vatsal and Emerton-Pollack-Weston, I investigate the extent to which Mazur's conjecture on the growth of Selmer ranks in $\mathbb{Z}_p$-extensions of an imaginary quadratic field persists under…