Volume 2012, Issue 668

Let A be a finitely generated algebra over a field K of characteristic p  > 0. We introduce a subring W † ( A ) ⊂  W ( A ), which we call the ring of overconvergent Witt vectors, and prove its basic properties. In a subsequent paper we use the results to define an overconvergent de Rham–Witt complex for smooth varieties over K whose hypercohomology is the rigid cohomology.

We explore an analogue of the André–Oort Conjecture for subvarieties of Drinfeld modular varieties. The conjecture states that a subvariety X of a Drinfeld modular variety contains a Zariski-dense set of complex multiplication (CM) points if and only if X is a “special” subvariety (i.e. X is defined by requiring additional endomorphisms). We prove this conjecture in two cases: firstly when X contains a Zariski-dense set of CM points all of which lie in one Hecke orbit, and secondly when X is a curve containing infinitely many CM points without any additional assumptions.

By using the slope method in Arakelov geometry, we study the complexity of the singular locus of an arithmetic projective variety and explicit estimations of the arithmetic Hilbert–Samuel function.

We obtain an explicit uniform estimate for the number of rational points in a projective plane curve whose heights do not exceed the degree of the curve.

We prove Pieri formulas for the multiplication with special Schubert classes in the K -theory of all cominuscule Grassmannians. For Grassmannians of type A this gives a new proof of a formula of Lenart. Our formula is new for Lagrangian Grassmannians, and for orthogonal Grassmannians it proves a special case of a conjectural Littlewood–Richardson rule of Thomas and Yong. Recent work of Clifford, Thomas, and Yong has shown that the full Littlewood–Richardson rule for orthogonal Grassmannians follows from the Pieri case proved here. We describe the K -theoretic Pieri coefficients both as integers determined by positive recursive identities and as the number of certain tableaux. The proof is based on a computation of the sheaf Euler characteristic of triple intersections of Schubert varieties, where at least one Schubert variety is special.

In this paper, we provide a version of the Mikhlin–Hörmander multiplier theorem for multilinear operators in the case where the target space is L p for p  ≦ 1. This extends a recent result of Tomita [15] who proved an analogous result for p  > 1.

For skew-symmetric acyclic quantum cluster algebras, we express the quantum F -polynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. As byproducts, we obtain the existence of counting polynomials for these varieties and the positivity conjecture with respect to acyclic seeds. These results complete previous work by Caldero and Reineke and confirm a recent conjecture by Rupel.

We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals–Cahn–Hilliard energy functionals with perturbation parameter → 0 + is supported by an embedded stable minimal hypersurface which in low dimensions has no singularities and in general dimensions has possibly a closed singular set of co-dimension ≧ 7. This result was previously known in case the critical points are local minimizers of energy, in which case the limit-interface is locally area minimizing and its (normalized) multiplicity is 1 a.e. Our theorem uses earlier work of the first author establishing stability of the limit-interface as an integral varifold, and relies on a recent general theorem of the second author for its regularity conclusions in the presence of higher multiplicity.

Let F / k be a Galois extension of number fields with dihedral Galois group of order 2 q , where q is an odd integer. We express a certain quotient of S -class numbers of intermediate fields, arising from Brauer–Kuroda relations, as a unit index. Our formula is valid for arbitrary extensions with Galois group D 2 q and for arbitrary Galois-stable sets of primes S , containing the Archimedean ones. Our results have curious applications to determining the Galois module structure of the units modulo the roots of unity of a D 2 q -extension from class numbers and S -class numbers. The techniques we use are mainly representation theoretic and we consider the representation theoretic results we obtain to be of independent interest.