Related papers: Greenberg's conjecture and units in multiple Z_p-e…

We investigate the group of universal norms attached to the cyclotomic Z {\ell}-tower of a totally real number field in connection with Grenberg's conjecture on Iwasawa invariants of such a field.

Let $\mathcal{A}_n$ be the $2$-part of the ideal class group of the $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of a real quadratic number field $F$. The cardinality of $\mathcal{A}_n$ is related to the index of cyclotomic units…

For an algebraic number field $K$ and a prime number $p$, let $\widetilde{K}/K$ be the maximal multiple $\mathbb{Z}_p$-extension. Greenberg's generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian…

We prove Rubin's conjecture on the structure of local units in the anticyclotomic $\mathbb{Z}_p$-extension of unramified quadratic extension of $\mathbb{Q}_p$ in $p=3$ case by extending Burungale-Kobayashi-Ota's work.

Let $f$ be a newform of even weight at least $4$, level $N$ and trivial character. Let $p\nmid N$ be an odd prime number that is ordinary for $f$ and let $K$ be an imaginary quadratic field satisfying a generalized Heegner hypothesis…

Cohen, Lenstra, and Martinet have given conjectures for the distribution of class groups of extensions of number fields, but Achter and Malle have given theoretical and numerical evidence that these conjectures are wrong regarding the Sylow…

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ an odd prime such that $E$ has good ordinary reduction at $p$ and the Galois representation on $E[p]$ is irreducible. Then Greenberg's $\mu=0$ conjecture predicts that the Selmer group of…

We formulate a new equivariant Main Conjecture in Iwasawa theory of number fields and study its properties. This is done for arbitrary one-dimensional $p$-adic Lie extensions $L_\infty/K$ containing the cyclotomic $\mathbb{Z}_p$-extension…

Let p be an odd prime satisfying Vandiver's conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z_p-extensions of the pth cyclotomic field and the Galois group G…

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 Fukuda's theorem by Li, Ouyang, Xu and Zhang. As an…

Let K/k be an Abelian extension of number fields, S be a set of places of k, and p be an odd prime number. We continue an earlier investigation of the author into the values at zero of the S-imprimitive partial zeta functions of K/k. An…

Let k be a totally real number field ant let k$\infty$ be its cyclotomic Zp-extension for a prime p\textgreater{}2. We give (Theorem 3.2) a sufficient condition of nullity of the Iwasawa invariants lambda, mu, when p totally splits in k,…

We study the Iwasawa main conjecture for quadratic Hilbert modular forms over the p-cyclotomic tower. Using an Euler system in the cohomology of Siegel modular varieties, we prove the "Kato divisibility" of the Iwasawa main conjecture under…

Let $p$ be an odd prime. We study the structure of the cyclotomic Greenberg-Selmer group attached to a general irreducible Artin motive over $\mathbb{Q}$ endowed with an ordinary $p$-stabilization. Under the Leopoldt and the weak $p$-adic…

We construct a new infinite family of pairs of imaginary cyclic fields of degree $(p-1)/2$ explicitly with both class numbers divisible by a given prime number $p$. For the proof, we use the fundamental unit of $\mathbb Q(\sqrt{p})$,…

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…

Leopoldt's Conjecture is a statement about the relationship between the global and local units of a number field. Approximately the conjecture states that the Z_p-rank of the diagonal embedding of the global units into the product of all…

Let $p$ be a prime number. If a number field $k$ has at least one complex place, there are infinitely many $\mathbb{Z}_p$-extensions over $k$, and some authors studied the behavior of Iwasawa invariants of these $\mathbb{Z}_p$-extensions.…

We prove the Iwasawa main conjecture over the arithmetic $\mathbb{Z}_p$-extension for semistable abelian varieties over function fields of characteristic $p>0$.

Let $k = \mathbb{Q}(\sqrt {-m})$ and $p \geq 3$ split in $k$. We prove new properties of the $\mathbb{Z}_p$-extensions $K/k$, distinct from the cyclotomic one; we do not assume $K/k$ totally ramified, nor the triviality of the $p$-class…