Related papers: Iwasawa theory and zeta elements for $\mathbb{G}_m…

For a prime $p>2$, let $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_p$, $G(1)$ be the first congruence kernel of $G$ and $\Omega_{G(1)}$ be the mod-$p$ Iwasawa algebra defined over the finite field…

For elliptic curves over rationals, there are a well-known conjecture of Sato-Tate and a new computational guided murmuration phenomenon, for which the abelian Hasse-Weil zeta functions are used. In this paper, we show that both the…

Let $p$ be an odd prime number and $k$ an imaginary quadratic field in which $p$ splits. In this paper, we consider a weak form of Greenberg's generalized conjecture for $p$ and $k$, which states that the non-trivial Iwasawa module of the…

Many problems of arithmetic nature rely on the computation or analysis of values of $L$-functions attached to objects from geometry. Whilst basic analytic properties of the $L$-functions can be difficult to understand, recent research…

In this note, we study the special values for zeta functions of totally real fields using the Shintani's cone decomposition. We prove certain congruence between the special values for zeta functions under the prime degree field extension.…

Let $E/K$ be a finite Galois extension of totally real number fields with Galois group $G$. Let $p$ be an odd prime and let $r>1$ be an odd integer. The $p$-adic Beilinson conjecture relates the values at $s=r$ of $p$-adic Artin…

Let G be a semisimple quasi-split group defined over a global function field. We express the relative Tamagawa number of G in terms of local data including the number of types of special vertices in one orbit of the Bruhat-Tits building…

This is a survey on motivic zeta functions associated to abelian varieties and Calabi-Yau varieties over a discretely valued field. We explain how they are related to Denef and Loeser's motivic zeta function associated to a complex…

The Jacobian group (also known as the critical group or sandpile group) is an important invariant of a finite, connected graph $X$; it is a finite abelian group whose cardinality is equal to the number of spanning trees of $X$ (Kirchhoff's…

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…

We fix motivic data $(K/F, E)$ consisting of a Galois extension $K/F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and an ableian $t$-module $E$, defined over a certain Dedekind subring of $F$. For this data,…

Modular motives have coefficients in Hecke algebras. According to the equivariant philosophy, special values of $L$-functions of eigencuspforms should therefore exhibit equivariant properties with respect to various Hecke actions. This…

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…

Recently Ritter and Weiss introduced an equivariant "main conjecture" than generalizes and refines the Main Conjecture of Iwasawa theory. In this paper, we show that, for the prime 2 and a dihedral extension of order 8 over Q, this…

We begin a study of m-th Chern classes and m-th characteristic symbols for Iwasawa modules which are supported in codimension at least m. This extends the classical theory of characteristic ideals and their generators for Iwasawa modules…

This is a survey of the known properties of Iwasawa algebras, which are completed group rings of compact p-adic analytic groups with coefficients the ring Zp of p-adic integers or the field Fp of p elements. A number of open questions are…

This is the completely revised version of math.AG/0101071.

We study a natural question in the Iwasawa theory of algebraic curves of genus $>1$. Fix a prime number $p$. Let $X$ be a smooth, projective, geometrically irreducible curve defined over a number field $K$ of genus $g>1$, such that the…

In this paper and a forthcoming joint one with Y. Hachimori we study Iwasawa modules over an infinite Galois extension K of a number field k whose Galois group G=G(K/k) is isomorphic to the semidirect product of two copies of the p-adic…

Our first goal in this note is to explain that a weak form of Perrin-Riou's conjecture on the non-triviality of Beilinson-Kato classes follows as an easy consequence of the Iwasawa main conjectures, and deduce its refined versions in the…