we study some automorphic cohomology classes of degree one on the griffiths–schmid varieties attached to some unitary groups in three variables. using partial compactifications of those varieties, constructed by kato and usui, we define for such a cohomology class some analogues of fourier–shimura coefficients, which are cohomology classes on certain elliptic curves. we show that a large space of such automorphic classes can be generated by those with rational ‘coefficients’. more precisely, we consider those cohomology classes that come from picard modular forms, via some penrose-like transform studied in a previous article: we prove that the coefficients of the classes thus obtained can be computed from the coefficients of the picard form by a similar transform defined at the level of the elliptic curve.

this note extends a result of moonen on the zariski closure of some sets of special points in the moduli space of principally polarised abelian varieties of dimension g to the case of any shimura variety, by a completely different method.

we compare the absolute and relative de rham–witt complexes considered by hesselholt and madsen (ann. sci. école norm. sup. 37 (2004), 1–43; ann. of math. (2) 158 (2003), 1–113) and by langer and zink (j. inst. math. jussieu 3 (2004), 231–314), which both generalize the classical de rham–witt complex of bloch, deligne, and illusie (ann. sci. école norm. sup. (4) 12 (1979), 501–661) from $\mathbb{f}_p$-schemes to $\mathbb{z}_{(p)}$-schemes. from this comparison, we derive a gauss–manin connection on the crystalline cohomology of x/wn(s) for a smooth family x/s.

for a noetherian scheme, we introduce its unbounded stable derived category. this leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. some applications are included, for instance an analogue of maximal cohen–macaulay approximations, a construction of tate cohomology, and an extension of the classical grothendieck duality. in addition, the relevance of the stable derived category in modular representation theory is indicated.

we prove in this paper the green conjecture for generic curves of odd genus. that is, we prove the vanishing $k_{k,1}(x,k_x)=0$ for x a generic curve of genus $2k+1$. this completes our previous work, where the green conjecture for generic curves of genus g with fixed gonality d was proved in the range $d\geq g/3$, with the possible exception of the generic curves of odd genus. the case of generic curves of odd genus was considered as especially important, since hirschowitz and ramanan proved that if the conjecture is true for the generic curve of odd genus, then the locus of jumping syzygies is exactly the locus of exceptional gonality, as predicted by green's conjecture. thus, our result combined with the hirschowitz–ramanan result is a strong confirmation of green's conjecture.

given a family x/b of nodal curves we construct canonically and compatibly with base-change, via an explicit blow-up of the cartesian product xr/b, a family wr(x/b) that we show is isomorphic to the relative flag hilbert scheme parametrizing flags of subschemes of fibres of x/b with colengths $1,\dotsc,r$. although wr(x/b) is singular, the important sheaves on it are locally free, which allows us to study some intersection theory on it and deduce enumerative applications, including some relative multiple point formulae enumerating the length-r schemes contained simultaneously in some fibre of x/b and some fibre of a given map from x to a smooth variety.

let k be an algebraically closed field of positive characteristic p > 0 and $c \to {\mathbb p}^1_k$ a p-cyclic cover of the projective line ramified in exactly one point. we are interested in the p-sylow subgroups of the full automorphism group autkc. we prove that for curves c with genus 2 or higher, these groups are exactly the extensions of a p-cyclic group by an elementary abelian p-group. the main tool is an efficient algorithm to compute the p-sylow subgroups of autkc starting from an artin–schreier equation for the cover $c \to {\mathbb p}^1_k$. we also characterize curves c with genus $g_c\geq 2$ and a p-group action $g\subset \text{aut}_k c$ such that $2p/(p-1)<|g|/g_c$ and $4/(p-1)^2\leq |g|/g_c^2$. our methods rely on previous work by stichtenoth whose approach we have adopted.

we find a minimal differential graded (dg) operad whose generic representations in ${\mathbb r}^n$ are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to ${\mathbb r}^n$ which satisfy the nijenhuis integrability condition. this operad is of a surprisingly simple origin: it is the cobar construction on the quadratic operad of homologically trivial dg lie algebras. as a by-product we obtain a strong-homotopy generalization of this geometric structure and show its homotopy equivalence to the structure of contractible dg manifolds.

let g be an algebraic group, $\gamma$ an arithmetic lattice of g and $x=\gamma\backslash g$. if h is an algebraic subgroup of g such that $h\cap \gamma$ is a lattice of h, then $\gamma\backslash \gamma h\subset x$ is endowed with a canonical h-invariant probability measure $\mu_h$. using ratner's theory, we give general examples where $\mu_{h_n}$ converges weakly to $\mu_g$ if hn is a strict sequence of algebraic subgroups of g. if $\gamma$ is a congruence subgroup of g, we define another probability measure $\mu_h^a$ on x by using the adelic description of the quotient. we conjecture that $\mu_{h_n}^a$ always converges weakly to $\mu_g$ if hn is a strict sequence. using automorphic forms and l-functions, we describe the case $g=\textit{sl}(2,f)$ for a number field f and a sequence of tori hn. the relation with similar problems on shimura varieties is explained.

we interpret the five-parameter family of macdonald–koornwinder polynomials as vector valued spherical functions on quantum grassmannians.