In this paper, we develop machinery to solve ternary Diophantine equations of the shape Axn + Byn = C z3 for various choices of coefficients (A, B, C). As a byproduct of this, we show, if p is prime, that the equation xn + yn = pz3 has no solutions in coprime integers x and y with |xy| > 1 and prime n > p4p2. The techniques employed enable us to classify all elliptic curves over $\mathbb{Q}$ with a rational 3-torsion point and good reduction outside the set {3, p}, for a fixed prime p.

In this paper, we resolve a conjecture of Schäffer on the solvability of Diophantine equations of the shape $1^k + 2^k + \dotsb + x^k = y^n$, for $1 \leq k \leq 11$. Our method, which may, with a modicum of effort, be extended to higher values of k, combines a wide variety of techniques, classical and modern, in Diophantine analysis.

We generalize a result of Anderson by showing that torsion points of certain orders cannot lie on a theta divisor in the Jacobians of hyperelliptic images of Fermat curves. The proofs utilize the explicit geometry of hyperelliptic Jacobians and their formal groups at the origin.

In Bull. Soc. Math. France 115 (1987), 399–456, Perrin-Riou formulates a form of the Iwasawa main conjecture which relates Heegner points to the Selmer group of an elliptic curve defined over $\mathbb{Q}$, as one goes up the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field K. Building on the earlier work of Bertolini on this conjecture, and making use of the recent work of Mazur and Rubin on Kolyvagin's theory of Euler systems, we prove one divisibility of Perrin-Riou's conjectured equality. As a consequence, one obtains an upper bound on the rank of the Mordell–Weil group E(K) in terms of Heegner points.

Let F be the fraction field of the ring of Witt vectors over a perfect field of characteristic p (for example $F=\mathbb{Q}_p$), and let GF be the absolute Galois group of F. The main result of this article is the following: a p-adic representation of GF, which is a limit of subquotients of crystalline representations with Hodge–Tate weights in an interval [a; b], is itself crystalline with Hodge–Tate weights in [a; b]. In order to show this, we study the $(\phi,\Gamma)$-modules attached to crystalline representations, which allows us to improve some results of Fontaine, Wach and Colmez.

The aim of this paper is to give an application of p-adic Hodge theory to stringy Hodge numbers introduced by V. Batyrev for a mathematical formulation of mirror symmetry. Since the stringy Hodge numbers of an algebraic variety are defined by choosing a resolution of singularities, the well-definedness is not clear from the definition. We give a proof of the well-definedness by using arithmetic techniques such as p-adic integration and p-adic Hodge theory. Note that another proof of the well-definedness was obtained by V. Batyrev himself by motivic integration.

Given an action of a reductive group on a normal variety, we describe all invariant open subsets admitting a good quotient with a quasiprojective or a divisorial quotient space. We obtain several new Hilbert–Mumford type theorems, and we extend a projectivity criterion of Białynicki-Birula and Święcicka for varieties with semisimple group action from the smooth to the singular case.

The zeta function of a complex variety is a power series whose nth coefficient is the nth symmetric power of the variety, viewed as an element in the Grothendieck ring of complex varieties. We prove that the zeta function of a surface is rational if and only if the Kodaira dimension of the surface is negative.

We discuss the problem of classifying crepant analytically extremal contractions $X \to Y$ from a smooth 3-fold, contracting an irreducible normal divisor D in X to a point P in Y. We prove that, if D has degree $(-K_D)^2 \geq 5$, the analytic structure of the contraction is completely determined by the isomorphism class of the exceptional locus and its normal bundle. This was previously known only for a smooth exceptional locus D.

Some smooth Calabi–Yau threefolds in characteristic two and three that do not lift to characteristic zero are constructed. These threefolds are pencils of supersingular K3-surfaces. The construction depends on Moret-Bailly's pencil of abelian surfaces and Katsura's analysis of generalized Kummer surfaces. The threefold in characteristic two turns out to be nonrigid.

We introduce a new approach to the study of a system of algebraic equations in $(\mathbb{C}^\times)^n$ whose Newton polytopes have sufficiently general relative positions. Our method is based on the theory of Parshin's residues and tame symbols on toroidal varieties. It provides a uniform algebraic explanation of the recent result of Khovanskii on the product of the roots of such systems and the Gel'fond–Khovanskii result on the sum of the values of a Laurent polynomial over the roots of such systems, and extends them to the case of an algebraically closed field of arbitrary characteristic.

In this paper, we study special configurations of lines and points in the complex projective plane, so called k-nets. We describe the role of these configurations in studies of cohomology on arrangement complements. Our most general result is the restriction on k: it can equal only 3, 4, or 5. The most interesting class of nets is formed by 3-nets that relate to finite geometries, latin squares, loops, etc. All known examples of 3-nets in $\mathbb{P}^2$ realize finite Abelian groups. We study the question of which groups can be realized in this way. Our main result is that, except for groups with all invariant factors below 10, the realizable groups are isomorphic to subgroups of a 2-torus. This follows from the ‘algebraization’ result asserting that, in the dual plane, the points dual to lines of a net lie on a plane cubic.

In this paper, we present a conjecture on a relation between the Zelevinski involutions for reductive groups over a p-adic field and the conjectural A-packets. As evidence for this conjecture, we prove that the Zelevinski involutions, which are regarded as operators on the space of virtual characters, commute with the endoscopic transfers under the assumption of the fundamental lemma for groups and for Lie algebras.

A skeletal structure (M, U) in ${\mathbb R}^{n+1}$ is a special type of n-dimensional Whitney stratified set M on which is defined a multivalued ‘radial vector field’ U. This is an extension of the notion of the Blum medial axis of a region in ${\mathbb R}^{n+1}$ with generic smooth boundary. For such a skeletal structure an ‘associated boundary’ $\mathcal{B}$ is defined. In part I of this paper, we introduced radial and edge shape operators, which are geometric invariants of the radial vector field U on M, and a ‘radial flow’ from M to $\mathcal{B}$. In this paper, in the partial Blum case we derive formulas for the differential geometric shape operator of the boundary (and hence all curvature invariants) in terms of the shape operators on the medial axis. We further derive the effects of a diffeomorphism of the skeletal structure on the radial and edge shape operators using a distortion operator which is computed from the second derivative of the diffeomorphism evaluated on the unit radial vector field. This allows one to compute the geometry of the boundary associated to a deformed skeletal structure purely in terms of operators defined on the original skeletal set.

We consider homogeneous Lorentz spaces of dimension at least 3. We prove that if such a space has ‘big’ isotropy (that is, a non-precompact and irreducible isotropy group), then this space must have constant sectional curvature. As a corollary, we obtain a new direct proof of the fact that irreducible Lorentz symmetric spaces have constant curvature, which was known via (algebraic) classification.