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

Let p be a fixed prime number. Let K be a totally real number field of discriminant D\_K and let T\_K be the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of K (under Leopoldt's conjecture). We…

Preprint of a paper to appear in "Communications in Advanced Mathematical Sciences". Let K be a real abelian extension of Q. Let p be a prime number, S the set of p-places of K and G\_K,S the Galois group of the maximal S-ramified…

It is well known that the Tchebotarev density theorem implies that an irreducible $\ell$-adic representation $\rho$ of the absolute Galois group of a number field $K$ is determined (up to isomorphism) by the characteristic polynomials of…

Let $p$ be an odd prime number and $k$ an imaginary quadratic field in which $p$ does not split. Based on their heuristic, Kundu and Washington posed a question which asks whether $\lambda$- and $\mu$-invariant of the anti-cyclotomic ${\Bbb…

Let $L/K$ be a cyclic extension of number fields, and let $S$ be a finite set of places of $K$ containing the ramified and Archimedean ones. We say that $L/K$ has the $\mathbf{cl}^S$-Hilbert 90 property if, for any generator $\sigma \in…

Let $E$ be a CM elliptic curve defined over $\mathbb{Q}$ and $p$ a prime. We show that $${\mathrm corank}_{\mathbb{Z}_{p}} {\mathrm Sel}_{p^{\infty}}(E_{/\mathbb{Q}})=0 \implies {\mathrm ord}_{s=1}L(s,E_{/\mathbb{Q}})=0 $$ for the…

We prove a $p$-adic version of the work by Gross and Zagier on the differences between singular moduli by proving a set of conjectures by Giampietro and Darmon, who investigated the factorisation of a rational invariant associated to a pair…

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 $A/\mathbb{Q}$ 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…

The primary goal of this article is to study $p$-adic Beilinson conjectures in the presence of exceptional zeros for Artin motives over CM fields. In more precise terms, we address a question raised by Hida and Tilouine on the order of…

Let $k$ be a totally real number field and $p$ a prime. We show that the ``complexity'' of Greenberg's conjecture ($\lambda = \mu = 0$) is of $p$-adic nature governed (under Leopoldt's conjecture) by the finite torsion group ${\mathcal…

Let $K$ be the function field of a smooth and proper curve $S$ over an algebraically closed field $k$ of characteristic $p>0$. Let $A$ be an ordinary abelian variety over $K$. Suppose that the N\'eron model $\CA$ of $A$ over $S$ has a…

Let $K$ be a totally real number field of degree $r=[K:\mathbb{Q}]$ and let $p$ be an odd rational prime. Let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$ and let $L_{\infty}$ be a finite extension of $K_{\infty}$,…

Kummer's conjecture states that the relative class number of the $p$-th cyclotomic field follows a strict asymptotic law. Granville has shown it unlikely to be true -- it cannot be true if we assume the truth of two other widely believed…

Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field. Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions. The Brumer-Stark conjecture states that the…

Let p be a prime number which is split in an imaginary quadratic field k. Let \mathfrak{p} be a place of k above p. Let k_\infty be the unique Z_p-extension of k which unramified outside of \mathfrak{p}, and let K_\intfy be a finite…

Let $K/k$ be a finite abelian CM-extension and $T$ a suitable finite set of finite primes of $k$. In this paper, we determine the Fitting ideal of the minus component of the $T$-ray class group of $K$, except for the $2$-component, assuming…

Let $k$ be a number field and $G$ be a finite group. Let $\mathfrak{F}_{k}^{G}(Q)$ be the family of number fields $K$ with absolute discriminant $D_K$ at most $Q$ such that $K/k$ is normal with Galois group isomorphic to $G$. If $G$ is the…

Let $A$ be an abelian variety over a global field $K$ of characteristic $p \ge 0$. If $A$ has nontrivial (resp. full) $K$-rational $l$-torsion for a prime $l \neq p$, we exploit the fppf cohomological interpretation of the $l$-Selmer group…

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…