Two integral structures on the $\mathbb{Q}$-vector space of modular forms of weight two on $X_0(N)$ are compared at primes $p$ dividing $N$ at most once. When $p=2$ and $N$ is divisible by a prime that is $3$ mod $4$, this comparison leads to an algorithm for computing the space of weight one forms mod $2$ on $X_0(N/2)$. For $p$ arbitrary and $N>4$ prime to $p$, a way to compute the Hecke algebra of mod $p$ modular forms of weight one on $\varGamma_1(N)$ is presented, using forms of weight $p$, and, for $p=2$, parabolic group cohomology with mod $2$ coefficients. Appendix A is a letter of October 1987 from Mestre to Serre in which he reports on computations of weight one forms mod $2$ of prime level. Appendix B, by Wiese, reports on an implementation for $p=2$ in Magma, using Stein’s modular symbols package, with which Mestre’s computations are redone and slightly extended.

We prove that if f is a C1-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if f is not Anosov then all the exponents in the centre bundle vanish. This establishes in full a result announced by Mañé at the International Congress of Mathematicians in 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.

We extend Penrose's peeling model for the asymptotic behaviour of solutions to the scalar wave equation at null infinity on asymptotically flat backgrounds, which is well understood for flat space-time, to Schwarzschild and the asymptotically simple space-times of Corvino–Schoen/Chrusciel–Delay. We combine conformal techniques and vector field methods: a naive adaptation of the ‘Morawetz vector field’ to a conformal rescaling of the Schwarzschild metric yields a complete scattering theory on Corvino–Schoen/Chrusciel–Delay space-times. A good classification of solutions that peel arises from the use of a null vector field that is transverse to null infinity to raise the regularity in the estimates. We obtain a new characterization of solutions admitting a peeling at a given order that is valid for both Schwarzschild and Minkowski space-times. On flat space-time, this allows larger classes of solutions than the characterizations used since Penrose's work. Our results establish the validity of the peeling model at all orders for the scalar wave equation on the Schwarzschild metric and on the corresponding Corvino–Schoen/Chrusciel–Delay space-times.

Let $k$ be a base commutative ring, $R$ a commutative ring of coefficients, $X$ a quasi-compact quasi-separated $k$-scheme, and $A$ a sheaf of Azumaya algebras over $X$ of rank $r$. Under the assumption that $1/r\in R$, we prove that the noncommutative motives with $R$-coefficients of $X$ and $A$ are isomorphic. As an application, we conclude that a similar isomorphism holds for every $R$-linear additive invariant. This leads to several computations. Along the way we show that, in the case of finite-dimensional algebras of finite global dimension, all additive invariants are nilinvariant.

We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichmüller space of a marked surface, defined by Chekhov–Fock (and Kashaev) in an abstract way, can be realized as a concrete subalgebra of the skew field of the skein algebra.

We draw a connection between the model-theoretic notions of modularity (or one-basedness), orthogonality and internality, as applied to difference fields, and questions of descent in in algebraic dynamics. In particular we prove in any dimension a strong dynamical version of Northcott's theorem for function fields, answering a question of Szpiro and Tucker and generalizing a theorem of Baker's for the projective line.

The paper comes in three parts. This first part contains an exposition some of the main results of the model theory of difference fields, and their immediate connection to questions of descent in algebraic dynamics. We present the model-theoretic notion of internality in a context that does not require a universal domain with quantifier-elimination. We also note a version of canonical heights that applies well beyond polarized algebraic dynamics. Part II sharpens the structure theory to arbitrary base fields and constructible maps where in part I we emphasize finite base change and correspondences. Part III will include precise structure theorems related to the Galois theory considered here, and will enable a sharpening of the descent results for non-modular dynamics.

Generalised Heegner cycles are associated to a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\mathbf{Q}$ . The cycles live in a middle-dimensional Chow group of a Kuga–Sato variety arising from an indefinite Shimura curve over the rationals and a self-product of a CM abelian surface. Let $p$ be an odd prime split in $K/\mathbf{Q}$ . We prove the non-triviality of the $p$ -adic Abel–Jacobi image of generalised Heegner cycles modulo $p$ over the $\mathbf{Z}_{p}$ -anticyclotomic extension of  $K$ . The result implies the non-triviality of the generalised Heegner cycles in the top graded piece of the coniveau filtration on the Chow group, and proves a higher weight analogue of Mazur’s conjecture. In the case of weight 2, the result provides a refinement of the results of Cornut–Vatsal and Aflalo–Nekovář on the non-triviality of Heegner points over the $\mathbf{Z}_{p}$ -anticyclotomic extension of  $K$ .

In this paper, we prove that the first occurrence of global theta liftings from any orthogonal group to either symplectic groups or metaplectic groups can be characterized completely in terms of the location of poles of certain Eisenstein series. This extends the work of Kudla and Rallis and the work of Moeglin to all orthogonal groups. As applications, we obtain results about basic structures of cuspidal automorphic representations and the domain of holomorphy of twisted standard L-functions.

In this paper, we define and study the Weil–Petersson geometry. Under the framework of the Weil–Petersson geometry, we study the Weil–Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil–Petersson metric and the Ricci curvature of the Weil–Petersson metric for Calabi–Yau fourfold moduli. We also prove that the Hodge volume of the moduli space is finite. Finally, we proved that the curvature of the Hodge metric is bounded if the Hodge metric is complete and the dimension of the moduli space is 1.

AMS 2000 Mathematics subject classification: Primary 53A30. Secondary 32C16

We formulate and study a new coarse (co-)assembly map. It involves a modification of the Higson corona construction and produces a map dual in an appropriate sense to the standard coarse assembly map. The new assembly map is shown to be an isomorphism in many cases. For the underlying metric space of a group, the coarse co-assembly map is closely related to the existence of a dual Dirac morphism and thus to the Dirac dual Dirac method of attacking the Novikov conjecture.

We give a moduli interpretation of the outer automorphism group Out of a finite-dimensional algebra similar to that of the Picard group of a scheme. We deduce that the connected component of Out is invariant under derived and stable equivalences. This allows us to transfer gradings between algebras and gives rise to conjectural homological constructions of interesting gradings on blocks of finite groups with abelian defect. We give applications to the lifting of stable equivalences to derived equivalences. We give a counterpart of the invariance result for smooth projective varieties: the product Pic0 × Aut0 is invariant under derived equivalence.

We study the global well-posedness of 3D Navier–Stokes equations for a class of large initial data. This type of data slowly varies in the vertical direction (expressed as a function of $\varepsilon {x}_{3} $), and it is ill-prepared in the sense that its norm in ${C}^{- 1} $ will blow up at the rate ${\varepsilon }^{- \alpha } $ for $\frac{1}{2} \lt \alpha \lt 1$ as $\varepsilon $ tends to zero.

We exhibit a smooth complex affine 5-fold whose Witt group of quadratic forms does not inject into the Witt group of its function field. The dimension 5 is the smallest possible. The example depends on relating Witt groups, mod 2 Chow groups, and Steenrod operations.

AMS 2000 Mathematics subject classification: Primary 19G12. Secondary 19E15

We study the intersection theory on the moduli spaces of maps of $n$-pointed curves $f:(C,s_1,\dots,s_n)\to V$ which are stable with respect to the weight data $(a_1,\dots,a_n)$, $0\le a_i\le1$. After describing the structure of these moduli spaces, we prove a formula describing the way descendant invariants change under a wall crossing. As a corollary, we compute the weighted descendants in terms of the usual ones, i.e. for the weight data $(1,\dots,1)$, and vice versa.

Soit $G$ un groupe réductif $p$-adique connexe. Nous effectuons une décomposition spectrale sur $G$ à partir de la formule d’inversion de Fourier utilisée dans ‘Une formule de Plancherel pour l’algèbre de Hecke d’un groupe réductif $p$-adique’, V. Heiermann, Commun. Math. Helv.76 (2001), 388–415. Nous en déduisons essentiellement qu’une représentation cuspidale d’un sous-groupe de Levi M appartient au support cuspidal d’une représentation de carré intégrable de $G$ si et seulement si c’est un pôle de la fonction $\mu$ de Harish-Chandra d’ordre égal au rang parabolique de M. Ces pôles sont d’ordre maximal. Plus précisément, nous montrons que cette condition est nécessaire et que sa suffisance équivaut à une propriété combinatoire de la fonction $\mu$ de Harish-Chandra qui s’avère être une conséquence d’un résultat de E. Opdam. En outre, nous obtenons des identités entre des combinaisons linéaires de coefficients matriciels. Ces identités contiennent des informations sur le degré formel des représentations de carré intégrable ainsi que sur leur position dans la représentation induite.

Let $G$ be a reductive connected $p$-adic group. With help of the Fourier inversion formula used in ‘Une formule de Plancherel pour l’algèbre de Hecke d’un groupe réductif $p$-adique’, V. Heiermann, Commun. Math. Helv.76 (2001), 388–415, we give a spectral decomposition on $G$. In particular, we deduce from it essentially that a cuspidal representation of a Levi subgroup $M$ is in the cuspidal support of a square-integrable representation of $G$ if and only if it is a pole of Harish-Chandra’s $\mu$-function of order equal to the parabolic rank of $M$. These poles are of maximal order. In more explicit terms, we show that this condition is necessary and that its sufficiency is equivalent to a combinatorical property of Harish-Chandra’s $\mu$-function, which appears to be a consequence of a result of E. Opdam. We get also identities between some linear combinations of matrix coefficients. These identities contain information on the formal degree of square-integrable representations and on their position in the induced representation.

AMS 2000 Mathematics subject classification: Primary 22E50. Secondary 22E35; 11F70; 11F72; 11F85

We study naturality properties of the transverse invariant in knot Floer homology under contact (+1)-surgery. This can be used as a calculational tool for the transverse invariant. As a consequence, we show that the Eliashberg–Chekanov twist knots En are not transversely simple for n odd and n > 3.

Let k be an algebraically closed field and O = k[[t]] ⊂ F = k((t)). For an almost simple algebraic group G we classify central extensions 1 → m → E → G(F) → 1; any such extension splits canonically over G(O). Fix a positive integer N and a primitive character ζ : μN(K) → (under some assumption on the characteristic of k). Consider the category of G(O)-bi-invariant perverse sheaves on E with m-monodromy ζ. We show that this is a tensor category, which is tensor equivalent to the category of representations of a reductive group ǦE,N. We compute the root datum of ǦE,N.

We exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transcendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algebras.

We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orientation. Using these calculations, we give a new proof of the existence of an infinite-rank subgroup in the three-dimensional homology cobordism group.

The spaces of Sp(n)-, Sp(n) · U(1)- and Sp(n) · Sp(1)-invariant, translation-invariant, continuous convex valuations on the quaternionic vector space ℍn are studied. Combinatorial dimension formulae involving Young diagrams and Schur polynomials are proved.