Related papers: On the Gross order of vanishing conjecture for lar…

Assuming that Iwasawa's $\mu_{K/k}$-invariant vanishes, we prove the 'main conjecture' of equivariant Iwasawa theory, at odd prime numbers $l$, for arbitrary extensions $K/k$ of totally real number fields, up to its uniqueness assertion.

We give an introduction to generalisations of conjectures of Brumer and Stark on the annihilator of the class group of a number field. We review the relation to the equivariant Tamagawa number conjecture, the main conjecture of Iwasawa…

We discuss three different formulations of the equivariant Iwasawa main conjecture attached to an extension K/k of totally real fields with Galois group G, where k is a number field and G is a p-adic Lie group of dimension 1 for an odd…

Orders of vanishing of zeros of zeta functions have much arithmetic information encoded in them. For the absolute zeta function, Dinesh Thakur gave sufficient conditions for the order of vanishing of its zeros when the finite field has two…

We study the Iwasawa theory of the fine Selmer group associated to certain Galois representations. The vanishing of the $\mu$-invariant is shown to follow in some cases from a natural property satisfied by Galois deformation rings. We…

The purpose of this paper is to prove the main conjecture of non-commutative Iwasawa theory for p-adic Lie extensions, for an odd prime p, of totally real number fields assuming that the Iwasawa mu invariant of a certain totally real number…

Let \chi be an irreducible character of the finite group G. If g is an element of G and \chi(g) is not zero, then we conjecture that the order of g divides |G|/\chi(1). The conjecture is a generalization of the classical fact that…

In this paper, we study some vanishing identities for Gromov-Witten invariants conjectured by K. Liu and H. Xu. We will prove these conjectures in the case that the summation range is large compare to genus. In fact, in such cases, we can…

Let p be an odd prime. We give an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields for an infinite class of one-dimensional non-abelian p-adic Lie extensions. Crucially, this result does not depend on…

The Iwasawa main conjecture fields has been an important tool to study the arithmetic of special values of $L$-functions of Hecke characters of imaginary quadratic fields. To obtain the finest possible invariants it is important to know the…

We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over $\ZZ_p^d$-extensions of function fields ramifying at a finite set of places.

We study fundamental forms of algebraic varieties using the sheaves of principal parts of line bundles and establish a vanishing theorem for any order fundamental forms. We also give connection of fundamental forms with the higher order…

The Grassmann convexity conjecture gives a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real…

In this largely expository article, we present a Kawamata-Viehweg type formulation of the (logarithmic) Akizuki-Nakano Vanishing Theorem. While the result is likely known to the experts, it does not seem to appear in the existing…

We prove a Generic Vanishing Theorem for coherent sheaves on an abelian variety over an algebraically closed field $k$. When $k=\CC$ this implies a conjecture of Green and Lazarsfeld.

For primes $q \equiv 7 \mod 16$, the present manuscript shows that elementary methods enable one to prove surprisingly strong results about the Iwasawa theory of the Gross family of elliptic curves with complex multiplication by the ring of…

We show that the Generalized Vanishing Conjecture $$\forall_{m \ge 1} [\Lam^m f^m = 0] \Longrightarrow \forall_{m \gg 0} [\Lam^m (g f^m) = 0]$$ for a fixed differential operator $\Lam \in k[\partial]$ follows from a special case of it,…

We introduce the notion of Shintani data, which axiomatizes algebraic aspects of Shintani zeta functions. We develop the general theory of Shintani data, and show that the order of vanishing part of Gross's conjecture follows from the…

In this article we propose a vanishing conjecture for a certain class of $\ell$-adic complexes on a reductive group $G$, which can be regraded as a generalization of the acyclicity of the Artin-Schreier sheaf. We show that the vanishing…

In a paper by the first author it was shown that for certain arithmetical results on conjugacy class sizes it is enough to only consider the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only…