Convolution structures are group-like objects that were extensively studied by harmonic analysts. We use them to define H0 and H1 for Arakelov divisors over number fields. We prove the analogs of the Riemann–Roch and Serre duality theorems. This brings more structure to the works of Tate and van der Geer and Schoof. The H1 is defined by a procedure very similar to the usual Ĉech cohomology. Serre′s duality becomes Pontryagin duality of convolution structures. The whole theory is parallel to the geometric case.

This paper concerns the Galois theoretic behavior of the p-primary subgroup SelA(F)p of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the Galois group G(K/F) is a p-adic Lie group. Here p is any prime such that A has potentially good, ordinary reduction at all primes of F lying above p. The principal results concern the kernel and the cokernel of the natural map sK/F′ SelA(F′)p →  SelA(K)pG(K/F′) where F′ is any finite extension of F contained in K. Under various hypotheses on the extension K/F, it is proved that the kernel and cokernel are finite. More precise results about their structure are also obtained. The results are generalizations of theorems of B. Mazurand M. Harris.

We construct a formal connection on the Aomoto complex of an arrangement of hyperplanes, and use it to study the Gauss–Manin connection for the moduli space of the arrangement in the cohomology of a complex rank one local system. We prove that the eigenvalues of the Gauss–Manin connection are integral linear combinations of the weights which define the local system.

In this paper, we show that the Chern classes ck of the de Rham bundle ${\cal H}_{{\rm d}R}$ defined on any ‘good’ toroidal compactification $\bar{\cal A}_g$ of the moduli space of Abelian varieties of dimension g are zero in the rational Chow ring of $\bar{\cal A}_g$, for g = 4, 5 and k>0.

We study an equidistribution problem of Linnik using Hecke operators.