Related papers: On zeta elements for $\mathbb{G}_{m}$

We describe an explicit `higher rank' Iwasawa theory for zeta elements associated to the multiplicative group over abelian extensions of general number fields. We then show that this theory leads to a concrete new strategy for proving…

We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme $\mathcal{X}$ at any integer $n$ in terms of Weil-\'etale cohomology complexes. This extends…

We use the theory of reduced determinant functors from [24] to give a new, computationally useful, description of the relative $K_0$-groups of orders in finite dimensional separable algebras that need not be commutative. By combining this…

Lichtenbaum conjectured the existence of a Weil-\'etale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme $\mathcal{X}$ at $s=0$ in terms of Euler-Poincar\'e…

In this paper we study a new conjecture concerning Kato's Euler system of zeta elements for elliptic curves $E$ over $\mathbb{Q}$. This conjecture, which we refer to as the `Generalized Perrin-Riou Conjecture', predicts a precise congruence…

Given an odd prime number $p$ and a $p$-stabilized Artin representation $\rho$ over $\mathbb{Q}$, we introduce a family of $p$-adic Stark regulators and we formulate an Iwasawa-Greenberg main conjecture and a $p$-adic Stark conjecture which…

We study the Fitting ideals over the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of Selmer groups attached to the Rankin--Selberg convolution of two modular forms $f$ and $g$. Inspired by the Theta elements for…

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…

We study the behaviour of the Stark conjecture for an abelian extension K/k of totally real number fields as K varies in a cyclotomic Z_p-tower. We consider possible strengthenings of the natural norm-coherence in the tower of putative…

Let k be an algebraically closed field of characteristic p>0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates, in the case of abelian schemes, the \'etale cohomology with…

We state the Brumer-Stark conjecture and motivate it from two perspectives. Stark's perspective arose in his attempts to generalize the classical Dirichlet class number formula for the leading term of the Dedekind zeta function at $s=1$…

Let $X$ be a variety over a finite field. Given an order $R$ in a semi-simple algebra over the rationals and a constructible \'etale sheaf $F$ of $R$-modules over $X$, one can consider a natural non-commutative $L$-function associated with…

Beginning with the conjecture of Artin and Tate in 1966, there has been a series of successively more general conjectures expressing the special values of the zeta function of an algebraic variety over a finite field in terms of other…

Let $L/K$ be a finite Galois CM-extension of number fields with Galois group $G$. In an earlier paper, the author has defined a module $SKu(L/K)$ over the center of the group ring $\mathbb Z[G]$ which coincides with the Sinnott-Kurihara…

We formulate two conjectures about etale cohomology and fundamental groups motivated by categoricity conjectures in model theory. One conjecture says that there is a unique Z-form of the etale cohomology of complex algebraic varieties, up…

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.

We propose a formulation of the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in universal deformation rings and Hecke algebras; something which seems to have been heretofore missing because the complexes of…

We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ of a global function field $F$ and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as…

We study Weil-etale cohomology, introduced by Lichtenbaum for varieties over finite fields. In the first half of the paper we give an explicit description of the base change from Weil-etale cohomology to etale cohomology. As a consequence,…

Spencer Bloch and the author formulated a general conjecture (Tamagawa number conjecture) on the relation between values of zeta functions of motives and arithmetic groups associated to motives. We discuss this conjecture, and describe some…