One of the main results announced in Part I (Compositio Math. 139 (2003), 197–237) is proved. The main technique consists in developing a coordinate free version of some of the theory of Serre operators on differential modular forms introduced by Barcau.

In this paper we suppose G is a finite group acting tamely on a regular projective curve $\mathcal{X}$ over $\mathbb{Z}$ and V is an orthogonal representation of G of dimension 0 and trivial determinant. Our main result determines the sign of the $\epsilon$-constant $\epsilon(\mathcal{X}/G,V)$ in terms of data associated to the archimedean place and to the crossing points of irreducible components of finite fibers of $\mathcal{X}$, subject to certain standard hypotheses about these fibers.

We prove various characterizations of the period torsor of abelian varieties.

We construct multiple zeta functions as absolute tensor products of usual zeta functions. The Euler product expression is established for the most basic case $\zeta(s,\mathbf{F}_p)\otimes\zeta(s,\mathbf{F}_q)$ by using the signed double Poisson summation formula and the theory of the double sine function.

In this work we study the moments of central values of Hecke L-functions associated with cubic characters, and establish quantitative non-vanishing result for the L-values.

Estimates are established for the number of integers of size N, in intervals of size $N^\theta$, that fail to admit a representation as the sum of s cubes (s = 5, 6). Thereby it is shown that almost all such integers are represented in the proposed manner. When s = 5 one may take $\theta = 10/21$, and when s = 6 one may take any $\theta >17/63$. Similar such conclusions are also established for the related problem associated with the expected asymptotic formula.

This corrigendum corrects a mistake in K. Belabas, On the mean 3-rank of quadratic fields, Compositio Math. 118 (1999), 1–9.

In this paper we relate the possible finite generatedness of the canonical algebra of smoothings of sandwiched singularities (Kollár's conjecture) to the possible finite generatedness of the symbolic algebra of certain curves in three-space.

We give a geometric description of the loci in the arc space defined by order of contact with a given subscheme, using the resolution of singularities. This induces an identification of the valuations defined by cylinders in the arc space with divisorial valuations. In particular, we recover the description of invariants coming from the resolution of singularities in terms of arcs and jets.

We give a definition of the crystalline fundamental group of suitable log schemes in positive characteristic using the techniques of rational homotopy theory applied to the De Rham–Witt complex.

In this paper, we construct automorphic forms on the five-dimensional complex ball which give the inverse of the period map for cyclic 4-ple coverings of the complex projective line branching at eight points. We use theta constants associated to the Prym varieties of these coverings.

We prove the vanishing of the equivariant holomorphic torsion forms associated to the holomorphic de Rham complex. We also consider corresponding torsion forms associated to complexes in hypercohomology, for which we give a local formula. Rigidity of certain genera explains part of these formulas. The Lefschetz operators in Hodge theory play a prominent role in the proofs.

We describe the Zariski closure of the geometric realization of Coxeter groups. When the Coxeter system is irreducible and the Tits form B non-positive and non-degenerate, this Zariski closure is equal to the orthogonal group of B.

In this paper, we give a formal algebraic notion of exponents for linear differential systems at any singularity as eigenvalues of the residue of a regular connection on a maximal lattice (that we call ‘Levelt's lattice’). This allows us to establish upper and lower bounds for the sum of these exponents for differential systems on ${\mathbb P}^{1}(\mathbb{C})$.