Let $G$ be a complex, semisimple, simply connected algebraic group with Lie algebra ${\frak g}$. We extend scalars to the power series field in one variable ${\rm C}((\pi))$, and consider the space of Iwahori subalgebras containing a fixed nil-elliptic element of ${\frak g} \otimes {\rm C}((\pi))$, i.e. fixed point varieties on the full affine flag manifold. We define representations of the affine Weyl group in the homology of these varieties, generalizing Kazhdan and Lusztig's topological construction of Springer's representations to the affine context.

In 1988, Haagerup and Størmer conjectured that every pointwise inner automorphism of a type ${\rm III_1}$ factor is a composition of an inner and a modular automorphism. We study this conjecture and prove that every type ${\rm III_1}$ factor with trivial bicentralizer indeed satisfies this condition. In particular, this shows that Haagerup and Størmer's conjecture holds in full generality if Connes’ bicentralizer problem has an affirmative answer. Our proof is based on Popa's intertwining theory and Marrakchi's recent work on relative bicentralizers.

Yoshikawa in [Invent. Math. 156 (2004), 53–117] introduces a holomorphic torsion invariant of $K3$ surfaces with involution. In this paper we completely determine its structure as an automorphic function on the moduli space of such $K3$ surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds’ conjecture.

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over $\mathbb {Q}$ and $\mathbb {F}_q(t)$, and conclude with a pair of hyperkähler 4-folds over $\mathbb {Q}$. The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article.

For a partially multiplicative quandle (PMQ) ${\mathcal {Q}}$ we consider the topological monoid $\mathring {\mathrm {HM}}({\mathcal {Q}})$ of Hurwitz spaces of configurations in the plane with local monodromies in ${\mathcal {Q}}$. We compute the group completion of $\mathring {\mathrm {HM}}({\mathcal {Q}})$: it is the product of the (discrete) enveloping group ${\mathcal {G}}({\mathcal {Q}})$ with a component of the double loop space of the relative Hurwitz space $\mathrm {Hur}_+([0,1]^2,\partial [0,1]^2;{\mathcal {Q}},G)_{\mathbb {1}}$; here $G$ is any group giving rise, together with ${\mathcal {Q}}$, to a PMQ–group pair. Under the additional assumption that ${\mathcal {Q}}$ is finite and rationally Poincaré and that $G$ is finite, we compute the rational cohomology ring of $\mathrm {Hur}_+([0,1]^2,\partial [0,1]^2;{\mathcal {Q}},G)_{\mathbb {1}}$.

Let X be a smooth projective variety. Using modified psi classes on the stack of genus-zero stable maps to X, a new associative quantum product is constructed on the cohomology space of X. When X is a homogeneous variety, this structure encodes the characteristic numbers of rational curves in X, and specialises to the usual quantum product upon resetting the parameters corresponding to the modified psi classes. For $X=\mathbb{P}^2$, the product is equivalent to that of the contact cohomology of Ernström and Kennedy.

We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, the Jacobian of a smooth projective curve over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a smooth projective curve over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture are dense in their moduli space.

In this paper we establish a Chowla–Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function ${\rm\Gamma}$ and an analogous function ${\rm\Gamma}_{2}$ at rational numbers. We combine this identity with work of Colmez to relate the CM values of the Hilbert modular function to Faltings heights of CM abelian varieties. We also give explicit formulas for products of exponentials of Faltings heights, allowing us to study some of their arithmetic properties using the Lang–Rohrlich conjecture.

We consider two families of arithmetic divisors defined on integral models of Shimura curves. The first was studied by Kudla, Rapoport and Yang, who proved that if one assembles these divisors in a formal generating series, one obtains the $q$-expansion of a modular form of weight 3/2. The present work concerns the Shimura lift of this modular form: we identify the Shimura lift with a generating series comprising divisors arising in the recent work of Kudla and Rapoport regarding cycles on Shimura varieties of unitary type. In the prequel to this paper, the author considered the geometry of the two families of cycles. These results are combined with the Archimedean calculations found in this work in order to establish the theorem. In particular, we obtain new examples of modular generating series whose coefficients lie in arithmetic Chow groups of Shimura varieties.

In this paper we suppose G is a finite group acting tamely on a regular projective curve $\mathcal{X}$ over $\mathbb{Z}$ and V is an orthogonal representation of G of dimension 0 and trivial determinant. Our main result determines the sign of the $\epsilon$-constant $\epsilon(\mathcal{X}/G,V)$ in terms of data associated to the archimedean place and to the crossing points of irreducible components of finite fibers of $\mathcal{X}$, subject to certain standard hypotheses about these fibers.

In this paper, we define Swan conductors for unit-root overconvergent F-isocrystals using the theory of arithmetic 𝒟-modules due to Berthelot. Our Swan conductors are compared with the Swan conductors for ℓ-adic sheaves constructed by Kato and Saito using a geometric method. As an application, we prove the integrality of Swan conductors in the sense of Kato and Saito under the ‘resolution of singularities’ assumption.

We prove a contact non-squeezing phenomenon on homotopy spheres that are fillable by Liouville domains with large symplectic homology: there exists a smoothly embedded ball in such a sphere that cannot be made arbitrarily small by a contact isotopy. These homotopy spheres include examples that are diffeomorphic to standard spheres and whose contact structures are homotopic to standard contact structures. As the main tool, we construct a new version of symplectic homology, called selective symplectic homology, that is associated to a Liouville domain and an open subset of its boundary. The selective symplectic homology is obtained as the direct limit of Floer homology groups for Hamiltonians whose slopes tend to $+\infty$ on the open subset but remain close to $0$ and positive on the rest of the boundary.

We compute the $g=1$, $n=1$ B-model Gromov–Witten invariant of an elliptic curve $E$ directly from the derived category $\mathsf{D}_{\mathsf{coh}}^{b}(E)$. More precisely, we carry out the computation of the categorical Gromov–Witten invariant defined by Costello using as target a cyclic $\mathscr{A}_{\infty }$ model of $\mathsf{D}_{\mathsf{coh}}^{b}(E)$ described by Polishchuk. This is the first non-trivial computation of a positive-genus categorical Gromov–Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov–Witten invariants of a symplectic 2-torus computed by Dijkgraaf.

We prove equality of the various rational $p$-adic period morphisms for smooth, not necessarily proper, schemes. We start with showing that the $K$-theoretical uniqueness criterion we had found earlier for proper smooth schemes extends to proper finite simplicial schemes in the good reduction case and to cohomology with compact support in the semistable reduction case. It yields the equality of the period morphisms for cohomology with compact support defined using the syntomic, almost étale, and motivic constructions. We continue with showing that the $h$-cohomology period morphism agrees with the syntomic and almost étale period morphisms whenever the latter morphisms are defined (and up to a change of Hyodo–Kato cohomology). We do it by lifting the syntomic and almost étale period morphisms to the $h$-site of varieties over a field, where their equality with the $h$-cohomology period morphism can be checked directly using the Beilinson Poincaré lemma and the case of dimension $0$. This also shows that the syntomic and almost étale period morphisms have a natural extension to the Voevodsky triangulated category of motives and enjoy many useful properties (since so does the $h$-cohomology period morphism).

Given a torus action (T2, M) on a smooth manifold, the orbit map evx(t)=t·x for each x∈M induces a homomorphism ev*: $\mathbb Z$2→H1(M;$\mathbb Z$). The action is said to be Rank-k if the image of ev* has rank k ([les ]2) for each point of M. In particular, if ev* is a monomorphism, then the action is called homologically injective. It is known that a holomorphic complex torus action on a compact Kähler manifold is homologically injective. We study holomorphic complex torus actions on compact non-Kähler Hermitian manifolds. A Hermitian manifold is said to be a locally conformal Kähler if a lift of the metric to the universal covering space is conformal to a Kähler metric. We shall prove that a holomorphic conformal complex torus action on a compact locally conformal Kähler manifold M is Rank-1 provided that M has no Kähler structure.

Let A and x be d × d and d × 1 real matrices. We study the asymptotic distribution of the sequence ((A$^n$x) mod.Z$^d$) in the torus $T^d$. We prove that for any A, for almost all x, this distribution exists; we characterize the case where this distribution is uniform; we give a description of the non uniform case. Finally we ask a question on the asymptotic distribution modulo 1 of the coefficients of the sequence of powers (A$^n$).

The class field theory for the fraction field of a two-dimensional complete normal local ring with finite residue field is established by S. Saito. In this paper, we investigate the index of the norm group in the K2-idele class group for a finite Abelian extension of such fields and deduce that the existence theorem does not hold for almost fields in this case.

The main object of this paper is the mean square Ih(s) of higher derivatives of Hurwitz zeta functions ζ(s, α). We shall prove asymptotic formulas for Ih(1/2 + it) as t → +∞ with the coefficients in closed expressions (Theorem 1). We also prove a certain explicit formula for Ih(1/2 + it) (Theorem 2), in which the coefficients are, in a sense, not explicit. However, one merit of this formula is that it contains sufficient information for obtaining the complete asymptotic expansion for Ih(1/2 + it) when h is small. Another merit is that Theorem 1 can be strengthened with the aid of Theorem 2 (see Theorem 3). The fundamental method for the proofs is Atkinson's dissection argument applied to the product ζ(u, α)ζ(v, α) with the independent complex variables u and v.

Thurston introduced shear deformations (cataclysms) on geodesic laminations–deformations including left and right displacements along geodesics. For hyperbolic surfaces with cusps, we consider shear deformations on disjoint unions of ideal geodesics. The length of a balanced weighted sum of ideal geodesics is defined and the Weil–Petersson (WP) duality of shears and the defined length is established. The Poisson bracket of a pair of balanced weight systems on a set of disjoint ideal geodesics is given in terms of an elementary $2$-form. The symplectic geometry of balanced weight systems on ideal geodesics is developed. Equality of the Fock shear coordinate algebra and the WP Poisson algebra is established. The formula for the WP Riemannian pairing of shears is also presented.

We construct in all characteristics p7lt;2 a complete surface in the moduli space of smooth genus 6 curves. The surface is contained in the locus of curves with automorphisms.