Wiley

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 the paper we find a further generalization of congruences of the K. Hardy and K. S. Williams [5] type which seems to be a full generalization of congruences of G. Gras [4]. Moreover we extend results of [5], [7], [8], [9] and in part of [6]. We apply ideas and methods of [2], [7] and [9].

Fix a prime number p ≥ 5 and a positive integer N prime to p. We consider the projective limits of p-adic étale cohomology groups of the modular curves X1(Npr) and Y1(Npr) (r ≥ 1), which are denoted by ESp(N) Z p and GES p(N)Z p , respectively. Let e* ′ be the projector to the direct sum of the ωi-eigenspaces of the ordinary part, for i ≢ 0, -1 mod p-1. Our main result states that e* ′ GESp (N)Z p has a good p-adic Hodge structure, which can be described in terms of λ-adic modular forms, extending the previously known result for e*′ ESp (N)Z p . We then apply the method of Harder and Pink to the Galois representation on e*′ ESp(N) Z p to construct large unramified abelian p-extensions over cyclotomic Z p -extensions of abelian number fields.

We prove a conjecture of Duke on the number of elliptic curves over the rationals of bounded height which have exceptional primes.

Let U be a quantumgroup with divid d powers at root ofunity constructed froma rootsystem R .Let u U b th small quantumgroup.Th cohomologyof u with trivial coefficients was computed by Ginzburg and Kumar.It turns out to be isomorphic to the functions algebra of the nilpotent cone of a semisimpl algebraic group with root system R .In this not we calculate cohomology of u with coefficients in simplest reducible tilting modul with nontrivial cohomology.It appears to b isomorphic to th functions algebra of th closure of the subregular nilpotent orbit.

Let $$f:X \to Y$$ be a morphism of noetherian schemes,generically smooth and equidimensional of dimension $$d,\iota :X\prime \to X$$ a closed embedding such that $$f \circ \iota :X\prime \to Y$$ is generically smooth and equidimensional ofdimension d $$\prime $$ , and X $$\prime $$ , X and Y are excellent schemes withoutembedded components. We exhibit a concrete morphism $$Res_{X\prime /X} :det \mathcal{N}_{X\prime /X} \otimes _{\mathcal{O}_{X\prime } } \iota *\omega _{X/Y}^d \to \omega _{X\prime /Y\prime }^{d\prime } ,$$ which transforms the integral of X/Y into the integral ofX $$\prime $$ /Y. Here $$\mathcal{N}_{X\prime /X} $$ denotes the normal sheaf of X $$\prime $$ /X and $$\omega _{X/Y}^d $$ resp. $$\omega _{X\prime /Y\prime }^{d\prime } $$ denotes the sheaf ofregular differential forms of X/Y resp. X $$\prime $$ /Y. Usinggeneralized fractions we provide a canonical description ofresidual complexes and residue pairs of Cohen--Macaulayvarieties, and obtain a very explicit description of fundamentalclasses and their traces.

Given m, n ≥ 2, we prove that, for sufficiently large y, the sum 1 n +···+ y n is not a product of m consecutive integers. We also prove that for m ≠ n we have 1 m +···+ x m ≠ 1 n +···+ y n , provided x, y are sufficiently large. Among other auxiliary facts, we show that Bernoulli polynomials of odd index are indecomposable, and those of even index are ‘almost’ indecomposable, a result of independent interest.

Let k be a perfect field of a positive characteristic p, K-the fraction field of the ring of Witt vectors W(k) Let X be a smooth and proper scheme over W(k). We present a candidate for a cohomology theory with coefficients in crystalline local systems: p -adic étale local systems on X_K characterized by associating to them so called Fontaine-crystals on the crystalline site of the special fiber X k. We show that this cohomology satysfies a duality theorem.