We study the Rankin $L$-series of a cuspidal automorphic representation of GL(2) of even weight over the rational numbers, twisted by a character of a real quadratic field. When the sign of the functional equation is <formula form="inline" disc="math" id="ffm004"><formtex notation="AMSTeX">$+1$, we give an explicit formula for the central value of the $L$-series, analogous to the formulae obtained by Gross, Zhang, and Xue in the imaginary case. The proof uses a version of the Rankin–Selberg method in which the theta correspondence plays an important role. We give two applications, to computing the order of the Tate–Shafarevich group of the base change to a real quadratic field of an elliptic curve defined over the rationals, and to proving the equidistribution of individual closed geodesics on modular curves.

Suppose that $O=\mathbb{F}_q[\pi ]$ is a polynomial ring and $R$ is a commutative unitary $O$-algebra. The category of finite flat group schemes over $R$ with a strict action of $O$ was recently introduced by Faltings and appears as an equal characteristic analogue of the classical category of finite flat group schemes in the equal characteristic case. In this paper we obtain a classification of these modules and apply it to prove analogues of properties that were known earlier for classical group schemes.

Using the $\sharp$-minimal model program of uniruled varieties we show that, for any pair $(X, {\mathcal H})$ consisting of a reduced and irreducible variety $X$ of dimension $k \geq 3$ and a globally generated big line bundle ${\mathcal H}$ on $X$ with $d:= {\mathcal H}^k$ and $n:= h^0(X, {\mathcal H})-1$ such that $d<2(n-k)-4$, then $X$ is uniruled of ${\mathcal H}$-degree one, except if $(k,d,n)=(3,27,19)$ and a ${\sharp}$-minimal model of $(X, {\mathcal H})$ is $({\mathbb P}^3,{\mathcal O}_{{\mathbb P}^3}(3))$. We also show that the bound is optimal for threefolds.

Using the ‘slice filtration’, defined by effectivity conditions on Voevodsky's triangulated motives, we define spectral sequences converging to their motivic cohomology and étale motivic cohomology. These spectral sequences are particularly interesting in the case of mixed Tate motives as their $E_2$-terms then have a simple description. In particular this yields spectral sequences converging to the motivic cohomology of a split connected reductive group. We also describe in detail the multiplicative structure of the motive of a split torus.

We obtain several determinant evaluations, related to affine root systems, which provide elliptic extensions of Weyl denominator formulas. Some of these are new, also in the polynomial special case, while others yield new proofs of the Macdonald identities for the seven infinite families of irreducible reduced affine root systems.

Let $X$ be a smooth projective variety. We study a relationship between the derived category of $X$ and that of a canonical divisor. As an application, we study Fourier–Mukai transforms when $\kappa (X)=\dim X -1$.

We define a complex horospherical transform on an affine symmetric space $X=G/H$ of Hermitian type and show that it has no kernel on the representations of the $H$-spherical holomorphic discrete series.

We propose a procedure to construct new smooth CR-manifolds whose local stability groups, equipped with their natural topologies, are subgroups of certain (finite-dimensional) Lie groups but not Lie groups themselves.

We apply the methods developed in our previous work to study holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds. In particular, we extend and strengthen certain results of Gromov, Henkin and Shubin on holomorphic $L^{2}$ functions on coverings of pseudoconvex manifolds in the case of coverings of Stein manifolds.

Let $B_n$ denote the classical braid group on $n$ strands and let the mixed braid group$B_{m,n}$ be the subgroup of $B_{m+n}$ comprising braids for which the first $m$ strands form the identity braid. Let $B_{m,\infty}=\bigcup_nB_{m,n}$. We describe explicit algebraic moves on $B_{m,\infty}$ such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented three-manifold. The moves depend on a fixed link representing the manifold in $S^3$. More precisely, for link complements the moves are the two familiar moves of the classical Markov equivalence together with ‘twisted’ conjugation by certain loops $a_i$. This means premultiplication by $a_i^{-1}$ and postmultiplication by a ‘combed’ version of $a_i$. For closed three-manifolds there is an additional set of ‘combed’ band moves that correspond to sliding moves over the surgery link. The main tool in the proofs is the one-move Markov theorem using $L$-moves (adding in-box crossings). The resulting algebraic classification is a direct extension of the classical Markov theorem that classifies links in $S^3$ up to isotopy, and potentially leads to powerful new link invariants, which have been explored in special cases by the first author. It also provides a controlled range of isotopy moves, useful for studying skein modules of three-manifolds.

Let $G$ be an adjoint simple algebraic group of inner type. We express the Chow motive (with integral coefficients) of an anisotropic projective $G$-homogeneous variety in terms of motives of simpler $G$-homogeneous varieties, namely, those that correspond to maximal parabolic subgroups of $G$. We decompose the motive of a generalized Severi–Brauer variety $\mathrm{SB}_2(A)$ of a division algebra $A$ of degree 5 into a direct sum of twisted motives of the Severi–Brauer variety $\mathrm{SB}(B)$ of a division algebra $B$ Brauer-equivalent to the tensor square $A^{\otimes 2}$. As an application we provide another counter-example to the uniqueness of a direct sum decomposition in the category of motives with integral coefficients.