For homogeneous decomposable forms F(X) in n variables with integer coefficients, we consider the number of integer solutions ${\bf x}\in\mathbb{Z}^n$ to the inequality $|F({\bf x})|\le m$ as $m\rightarrow\infty$. We give asymptotic estimates which improve on those given previously by the author in Ann. of Math. (2) 153 (2001), 767–804. Here our error terms display desirable behaviour as a function of the height whenever the degree of the form and the number of variables are relatively prime.

We investigate divisibility properties of the traces and Hecke traces of singular moduli. In particular we prove that, if p is prime, these traces satisfy many congruences modulo powers of p which are described in terms of the factorization of p in imaginary quadratic fields. We also study generalizations of Lehner's classical congruences $j(z)| U_p\equiv 744 \pmod p$ (where $p\leq 11$ and j(z) is the usual modular invariant), and we investigate connections between class polynomials and supersingular polynomials in characteristic p.

We show that for arbitrary even genus 2n with $n\equiv {0,1}$ (mod 4) the subspace of Siegel cusp forms of weight $k+n$ generated by the Ikeda lifts of elliptic cusp forms of weight 2k can be characterized by certain simple relations among the Fourier coefficients. These generalize the classical Maass relations in genus 2.

For the special orthogonal group G = SO(2n + 1) over a p-adic field, we consider a discrete series representation of a standard Levi subgroup of G. We prove that the Arthur R-group and the classical R-group of $\pi$ are isomorphic. If $\pi$ is generic, we consider the Aubert involution $\hat{\pi}$. Under the assumption that $\hat{\pi}$ is unitary, we prove that the Arthur R-group of $\hat{\pi}$ is isomorphic to the R-group of $\hat{\pi}$ defined by Ban (Ann. Sci. École Norm. Sup. 35 (2002), 673–693; J. Algebra 271 (2004), 749–767). This is done by establishing the connection between the A-parameters of $\pi$ and $\hat{\pi}$. We prove that the A-parameter of $\hat{\pi}$ is obtained from the A-parameter of $\pi$ by interchanging the two $\textit{SL}(2,\mathbb{C})$ components.

We compute the Chow group of 0-cycles on a rational surface defined over a finite extension K of the field $\mathbb{Q}_p$ of p-adic numbers (p a prime) when it is split by an unramified extension of K. We use intersection theory to define a specialisation map so we need to assume that the surface admits a regular proper integral model. A family of examples is worked out to illustrate the method.

We compute the mixed Hodge structure on the rational cohomology of the moduli space of smooth genus 4 curves. Specifically, we prove that its Poincaré–Serre polynomial is 1 + t2u2 + t4u4 + t5u6. We show this by producing a stratification of the space, such that all strata are geometric quotients of complements of discriminants.

We study positivity properties of the moduli (b-)divisor associated to a relative log pair (X, B)/Y with relatively trivial log canonical class.

We characterize the Fermat quartic K3 surface, among all K3 surfaces, by means of its finite group symmetries.

Let $\mathcal{E}$ be a rank-2 bundle over a smooth complex projective surface X. Whenever the subscheme of zeros of a global section of $\mathcal{E}$ is zero-dimensional, it gives a geometric realization of the second Chern class $c_2(\mathcal{E})$. Taking the incidence correspondence

\[\xymatrix{{\mathcal{Z} =\{(x,[e]) \in X\times \bf{P} (H^0(\mathcal{E}))\mid e(x)= 0\}} \ar[r]^>>>>>{p_2}& {\bf{P} (H^0(\mathcal{E}))}}\]

and considering the Zariski open subset $U_f \subset \bf{P} (H^0(\mathcal{E}))$ over which the morphism p2 is finite, we have

\[\mathcal{Z}_f = p^{-1}_2(U_f)\longrightarrow U_f,\]

a family of zero-dimensional subschemes of X of length $\deg (c_2(\mathcal{E}))$. This can be viewed as a distinguished geometric representative of $c_2(\mathcal{E})$.

We define a new invariant of $\mathcal{E}$ which can be viewed as a ‘lifting’ of $c_2(\mathcal{E})$ to Uf. This invariant is a sequence of sections of some coherent sheaves on Uf. These sheaves are built from the following cohomology cup-products:

\begin{align}\gamma_1 &: H^1 (\mathcal{E}^{\ast}) \otimes H^0 (\mathcal{E}) \longrightarrow H^1 (\mathcal{O}_X),\\ \gamma_2&: S^2 H^1 (\mathcal{E}^{\ast})\longrightarrow H^2 (\mathcal{O}_X (\det(\mathcal{E}^{\ast})) ).\end{align}

The main property of our invariant is as follows: either it determines the family $\mathcal{Z}_f \stackrel{p_2}{\longrightarrow} U_f$, or the vector bundle $\mathcal{E}$ is ‘special’. The speciality is expressed in terms of the special geometry of zero-loci of global sections of $\mathcal{E}$ and the special geometry of X.

The sequence of sections entering the definition of our invariant is obtained by starting with the one defined by the cup-product $\gamma_2$ and deriving others inductively by using a geometric interpretation of a part of the cup-product $\gamma_1$ together with the Grothendieck residue map. So one can view our invariant as $\gamma_2$ together with some kind of higher-order cohomology cup-products.

The emergence of these higher-order cohomology cup-products is explained conceptually as higher-order derivatives of a natural deformation of $\gamma_2$ associated to a certain ‘natural’ deformation of the complex structure on $\mathcal{E}$. The variety of these natural deformations of $\mathcal{E}$ has all the features of the classical Jacobian of curves: it carries a distinguished divisor which either determines the family $\mathcal{Z}_f \stackrel{p_2}{\longrightarrow} U_f$ or ‘sees’ that $\mathcal{E}$ is special in the aforementioned sense. An essentially new feature of this Jacobian of $\mathcal{E}$ is that it also carries a variation of Hodge-like structures which arises naturally from our invariant.

A polar hypersurface P of a complex analytic hypersurface germ f = 0 can be investigated by analyzing the invariance of certain Newton polyhedra associated with the image of P, with respect to suitable coordinates, by certain morphisms appropriately associated with f. We develop this general principle of Teissier when f = 0 is a quasi-ordinary hyper-surface germ and P is the polar hypersurface associated with any quasi-ordinary projection of f = 0. We show a decomposition of P into bunches of branches which characterizes the embedded topological types of the irreducible components of f = 0. This decomposition is also characterized by some properties of the strict transform of P by the toric embedded resolution of f = 0 given by the second author. In the plane curve case this result provides a simple algebraic proof of a theorem of Lê et al.

Given a general plane curve Y of degree d, we compute the number nd of irreducible plane conics that are five-fold tangent to Y. This problem has been studied before by Vainsencher using classical methods, but it could not be solved because the calculations produced too many non-enumerative correction terms that could not be analyzed. In our current approach, we express the number nd in terms of relative Gromov–Witten invariants that can then be directly computed. As an application, we consider the K3 surface given as the double cover of $\mathbb{P}^2$ branched along a sextic curve. We compute the number of rational curves in this K3 surface in the homology class that is the pull-back of conics in $\mathbb{P}^2$, and compare this number with the corresponding Yau–Zaslow K3 invariant. This gives an example of such a K3 invariant for a non-primitive homology class.

We consider suspension hypersurface singularities of type g = f(x, y) + zn, where f is an irreducible plane curve singularity. For such germs, we prove that the link of g determines completely the Newton pairs of f and the integer n except for two pathological cases, which can be completely described. Even in the pathological cases, the link and the Milnor number of g determine uniquely the Newton pairs of f and n. In particular, for such g, we verify Zariski's conjecture about the multiplicity. The result also supports the following conjecture formulated in this paper: if the link of an isolated hypersurface singularity is a rational homology 3-sphere, then it determines the equisingularity type, the embedded topological type, the equivariant Hodge numbers and the multiplicity of the singularity. The conjecture is verified for weighted homogeneous singularities too.

If (V, 0) is an isolated complete intersection singularity and X a holomorphic vector field tangent to V, one can define an index of X, the so-called GSV index, which generalizes the Poincaré–Hopf index. We prove that the GSV index coincides with the dimension of a certain explicitly constructed vector space, if X is deformable in a certain sense and V is a curve. We also give a sufficient algebraic criterion for X to be deformable in this way. If one considers the real analytic case one can also define an index of X which is called the real GSV index. Under the condition that X has the deformation property, we prove a signature formula for the index generalizing the Eisenbud–Levine Theorem.