Let $X$ be a noetherian scheme with dualizing complex and ${\mathcal{A}}$ a coherent ${\mathcal O}_{X}$-algebra with involution of the first kind $\tau$. We develop in this work a coherent hermitian Witt theory for $({\mathcal{A}},\tau)$. As an application we construct a Gersten–Witt spectral sequence which converges to the coherent hermitian Witt theory of $({\mathcal{A}},\tau)$. We show then that the associated Gersten–Witt complex is exact if $X$ is the spectrum of a smooth semilocal ring and ${\mathcal{A}}$ is locally free as ${\mathcal O}_{X}$-module.

Let $f$ be a newform of weight $2k-2$ and level 1. In this paper we provide evidence for the Bloch–Kato conjecture for modular forms. We demonstrate an implication that under suitable hypotheses if $\varpi \mid L_{\rm alg}(k,f)$ then $p \mid \# H_{f}(\mathbb{Q},W_{f}(1-k))$ where $p$ is a suitably chosen prime and $\varpi$ a uniformizer of a finite extension $K/\mathbb{Q}_{p}$. We demonstrate this by establishing a congruence between the Saito–Kurokawa lift $F_{f}$ of $f$ and a cuspidal Siegel eigenform $G$ that is not a Saito–Kurokawa lift. We then examine what this congruence says in terms of Galois representations to produce a non-trivial $p$-torsion element in $H_{f}^1(\mathbb{Q},W_{f}(1-k))$.

Let $E$ be a $\textrm{CM}$-field and $\pi$ a cuspidal representation of $\textrm{GL}_n({\mathbb{A}}_E)$ which admits a spherical vector (at all places) <formula form="inline" disc="math" id="ffm005"><formtex notation="AMSTeX">$\phi_0$. We evaluate the period of $\phi_0$ with respect to any compact unitary group. The result is consistent with a conjecture of Sarnak.

A non-Abelian generalization of local class field theory, which can describe non-abelian Galois groups in terms of the ground field, is given. The absolute Galois group of a local field is canonically defined up to an inner automorphism and the natural approach of describing class functions on it leads to Langland's philosophy. Another way to describe the absolute Galois group is to rigidify its structure by means of an arbitrary lifting of the Frobenius automorphism. This point of view is used in the metabelian local class field theory of Koch and de Shalit and its generalization by Gurevich. Iterating a slightly modified version of metabelian local class field theory on some fields of norms leads to a ‘non-abelian local class field theory’.

Over a perfect field $k$ of characteristic $p > 0$, we construct a ‘Witt vector cohomology with compact supports’ for separated $k$-schemes of finite type, extending (after tensorization with ${\mathbb Q}$) the classical theory for proper $k$-schemes. We define a canonical morphism from rigid cohomology with compact supports to Witt vector cohomology with compact supports, and we prove that it provides an identification between the latter and the slope ${<}1$ part of the former. Over a finite field, this allows one to compute congruences for the number of rational points in special examples. In particular, the congruence modulo the cardinality of the finite field of the number of rational points of a theta divisor on an abelian variety does not depend on the choice of the theta divisor. This answers positively a question by J.-P. Serre.

We investigate the action of the Weil group on the compactly supported $\ell$-adic étale cohomology groups of rigid spaces over a local field. We prove that the alternating sum of the traces of the action is an integer and is independent of $\ell$ when either the rigid space is smooth or the characteristic of the base field is equal to 0. We modify the argument of T. Saito to prove a result on $\ell$-independence for nearby cycle cohomology, which leads to our $\ell$-independence result for smooth rigid spaces. In the general case, we use the finiteness theorem of Huber, which requires the restriction on the characteristic of the base field.

The classical Fourier–Mukai duality establishes an equivalence of categories between the derived categories of sheaves on dual complex tori. In this article we show that this equivalence extends to an equivalence between two dual objects. Both of these are generalized deformations of the complex tori. In one case, a complex torus is deformed formally in a non-commutative direction specified by a holomorphic Poisson structure. In the other, the dual complex torus is deformed in a $B$-field direction to a formal gerbe. We show that these two deformations are Fourier–Mukai equivalent.

We describe an equivalence between the notion of balanced twisted curve introduced by Abramovich and Vistoli, and a new notion of log twisted curve, which is a nodal curve equipped with some logarithmic data in the sense of Fontaine and Illusie. As applications of this equivalence, we construct a universal balanced twisted curve, prove that a balanced twisted curve over a general base scheme admits étale locally on the base a finite flat cover by a scheme, and also give a new construction of the moduli space of stable maps into a Deligne–Mumford stack and a new proof that it is bounded.

The Gromov–Witten theory of Deligne–Mumford stacks is a recent development, and hardly any computations have been done beyond three-point genus 0 invariants. This paper provides explicit recursions which, together with some invariants computed by hand, determine all genus 0 invariants of the stack $\mathbb{P}^2_{D,2}$. Here $D$ is a smooth plane curve and $\mathbb{P}^2_{D,2}$ is locally isomorphic to the stack quotient $[U/(\mathbb{Z}/(2))]$, where $U\to V\subseteq \mathbb{P}^2$ is a double cover branched along $D\cap V$. The introduction discusses an enumerative application of these invariants.

We use the geometry of characteristic cycles of Harish-Chandra modules for a real semi-simple Lie group $G_{\mathbb{R}}$ to prove an upper triangularity relationship between two bases of each special representation of a classical Weyl group. One basis consists of Goldie rank polynomials attached to primitive ideals in the enveloping algebra of the complexified Lie algebra ${\mathfrak{g}}$; the other consists of polynomials that measure the Euler characteristic of the restriction of an equivariant line bundle on the flag variety for ${\mathfrak{g}}$ to an irreducible component of the Springer fiber. While these two bases are defined only using the structure of the complex Lie algebra ${\mathfrak{g}}$, the relationship between them is closely tied to the real group $G_{\mathbb{R}}$. More precisely, the order leading to the upper triangularity result is a suborder of closure order for the orbits of the complexification of a maximal compact subgroup of $G_{\mathbb{R}}$ on the flag variety for ${\mathfrak{g}}$.