A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\text{GL}_{n}(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles $\mathbb{A}_{X}^{\bullet }$, we have an equivalence $\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$ between perfect complexes on $X$ and cartesian perfect complexes for $\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal $\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

In this work we prove the so-called dimension property for the cohomological field theory associated with a homogeneous polynomial $W$ with an isolated singularity, in the algebraic framework of [A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122]. This amounts to showing that some cohomology classes on the Deligne–Mumford moduli spaces of stable curves, constructed using Fourier–Mukai-type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in [A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (American Mathematical Society, Providence, RI, 2001), 229–249] for certain characteristic classes of Koszul matrix factorizations of $0$ . To reduce to this result, we use the theory of Fourier–Mukai-type functors involving matrix factorizations and the natural rational lattices in the relevant Hochschild homology spaces, as well as a version of Hodge–Riemann bilinear relations for Hochschild homology of matrix factorizations. Our approach also gives a proof of the dimension property for the cohomological field theories associated with some quasihomogeneous polynomials with an isolated singularity.

We discuss the problem of classifying crepant analytically extremal contractions $X \to Y$ from a smooth 3-fold, contracting an irreducible normal divisor D in X to a point P in Y. We prove that, if D has degree $(-K_D)^2 \geq 5$, the analytic structure of the contraction is completely determined by the isomorphism class of the exceptional locus and its normal bundle. This was previously known only for a smooth exceptional locus D.

For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$ , motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some where $K$ is of characteristic $2$ , and we complete its proof by reducing the positive characteristic case to characteristic $0$ . For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_{p}$ : we find an elliptic curve $E^{\prime }$ defined over a $p$ -adic field such that all the terms in the Kramer–Tunnell formula for $E^{\prime }$ are equal to those for $E$ .

This corrigendum corrects a mistake in K. Belabas, On the mean 3-rank of quadratic fields, Compositio Math. 118 (1999), 1–9.

Let $\tau$ be a point in the upper half-plane such that the elliptic curve corresponding to $\tau$ can be defined over $\mathbb{Q}$, and let f be a modular form on the full modular group with rational Fourier coefficients. By applying the Ramanujan differential operator D to f, we obtain a family of modular forms Dlf. In this paper we study the behavior of $D^l(f)(\tau)$ modulo the powers of a prime p > 3. We show that for $p \equiv 1 \bmod 3$ the quantities $D^l(f)(\tau)$, suitably normalized, satisfy Kummer-type congruences, and that for $p \equiv 2 \bmod 3$ the p-adic valuations of $D^l(f)(\tau)$ grow arbitrarily large. We prove these congruences by making a connection with a certain elliptic curve whose reduction modulo p is ordinary if $p \equiv 1 \bmod 3$ and supersingular otherwise.

For infinite-dimensional homotopy space forms X/G with G nontrivial finite cyclic groups, we study the homotopy type of X/G. We show that the Euler class of X/G is not zero if and only if X/G is finitely dominated.

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.

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.

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.

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.

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.

We investigate the Gross–Prasad conjecture and its refinement for the Bessel periods in the case of $(\mathrm {SO}(5), \mathrm {SO}(2))$. In particular, by combining several theta correspondences, we prove the Ichino–Ikeda-type formula for any tempered irreducible cuspidal automorphic representation. As a corollary of our formula, we prove an explicit formula relating certain weighted averages of Fourier coefficients of holomorphic Siegel cusp forms of degree two, which are Hecke eigenforms, to central special values of $L$-functions. The formula is regarded as a natural generalization of the Böcherer conjecture to the non-trivial toroidal character case.

In this paper we solve the problem of desingularization of an absolutely isolated singularity of a differential equation, including the dicritical case. As an application, we prove the finiteness of the number of dicritical points in the blowing up tree of an absolutely isolated singularity.

In this paper we construct smooth irreducible space curves $C$ which link geometrically by surfaces of minimal degree containing $C$ to curves $\Gamma$ of generic embedding dimension three. This produces interesting behavior with respect to both $C$ and $\Gamma$. The curves $\Gamma$ link to smooth connected curves by surfaces of low degree but cannot link to smooth connected curves by surfaces of high degree. The curves $C$ give counterexamples to a conjecture of Martin-Deschamps and Perrin.

We establish central limit theorems for an action of a group $G$ on a hyperbolic space $X$ with respect to the counting measure on a Cayley graph of $G$. Our techniques allow us to remove the usual assumptions of properness and smoothness of the space, or cocompactness of the action. We provide several applications which require our general framework, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds.

We show that holomorphic singular codimension one foliations of the complex projective space with a Kupka singular set of radial type and verifying some global hypotheses have rational first integral. The generic elements of such pencils are Calabi–Yau.

The local intersection cohomology of a point in the Baily–Borel compactification (of a Hermitian locally symmetric space) is shown to be canonically isomorphic to the weighted cohomology of a certain linear locally symmetric space (an arithmetic quotient of the associated self-adjoint homogeneous cone). Explicit computations are given for the symplectic group in four variables.

In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$, a $G$-torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture.