The Cohen–Lenstra–Martinet heuristics give precise predictions about the class groups of a ’random‘ number field. The 3-rank of quadratic fields is one of the few instances where these have been proven. We prove that, in this case, the rate of convergence is at least sub-exponential. In addition, we show that the defect appearing in Scholz‘s mirror theorem is equidistributed with respect to a twisted Cohen–Lenstra density.

We consider lattice-free simplices, simplices with vertices on the lattice Z in R and no other integral points; we show, by elementary methods, that there exist such simplices in dimension d with width (see Definition 2) going to infinity with d.

Let X be a proper complex variety with Du Bois singularities. Then H(X,C)→ H(X,${\mathcal O}$) is surjective for all i. This property makes this class of singularities behave well with regard to Kodaira type vanishing theorems. Steenbrink conjectured that rational singularities are Du Bois and Kollár conjectured that log canonical singularities are Du Bois. Kollár also conjectured that under some reasonable extra conditions Du Bois singularities are log canonical. In this article Steenbrink‘s conjecture is proved in its full generality, Kollár‘s first conjecture is proved under some extra conditions and Kollár‘s second conjecture is proved under a set of reasonable conditions, and shown that these conditions cannot be relaxed.

In this paper the asymptotic behavior of the Castelnuovo$ndash;Mumford regularity of powers of a homogeneous ideal I is studied. It is shown that there is a linear bound for the regularity of the powers I whose slope is the maximum degree of a homogeneous generator of I, and that the regularity of I is a linear function for large n. Similar results hold for the integral closures of the powers of I. On the other hand we give examples of ideal for which the regularity of the saturated powers is asymptotically not a linear function, not even a linear function with periodic coefficients.

We prove Poincaré duality for logarithmic crystalline cohomology of log smooth schemes whose underlying schemes are reduced. This is a generalization of the result of P. Berthelot for usual smooth schemes and that of O. Hyodo for the special fibers of semi-stable families and trivial coefficients.

Let G be a connected reductive group defined over a local non Archimedean field F with residue field F; let P be a parahoric subgroup with associated reductive quotient M. If σ is an irreducible cuspidal representation of M(F) it provides an irreducible representation of P by inflation. We show that the pair (P,σ) is an ${\mathfrak S}$-type as defined by Bushnell and Kutzko. The cardinality of${\mathfrak S}$ can be bigger than one; we show that if one replaces P by the full centraliser ${\hat P}$ of the associated facet in the enlarged affine building of G, and σ by any irreducible smooth representation ${\hat σ}$ of ${\hat P}$ which contains σ on restriction then (${\hat P}$,${\hat σ}$) is an ${\mathfrak s}$-type for a singleton set ${\mathfrak s}$. Our methods employ invertible elements in the associated Hecke algebra${\mathcal H}$ (σ) and they imply that the appropriate parabolic induction functor and its left adjoint can be realised algebraically via pullbacks from ring homomorphisms.

The quotient of a real analytic manifold by a properly discontinuous group action is, in general, only a semianalytic variety. We study the boundary of such a quotient, i.e., the set of points at which the quotient is not analytic. We apply the results to the moduli space M$_g/R$ of nonsingular real algebraic curves of genus g (g[les ]2). This moduli space has a natural structure of a semianalytic variety. We determine the dimension of the boundary of any connected component of M$_g/R$. It turns out that every connected component has a nonempty boundary. In particular, no connected component of M$_g/R$is real analytic. We conclude that M$_g/R$ is not a real analytic variety.

In this paper, certain natural and elementary polygonal objects in Euclidean space, the stable polygons, are introduced, and the novel moduli spaces${mathfrak M}$$_r,ϵ$ of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and isomorphic to the moduli space $\overlinel{\matcal M}$$_0,n$ of the Deligne–Mumford stable curves of genus 0. Further, built into the structures of stable polygons are some natural data giving rise to a family of (classes of) symplectic (Kähler) forms. This, via the link to $\overlinel{\matcal M}$$_0,n$, brings up a new tool to study the Kähler topology of$\overlinel{\matcal M}$$_0,n$. A wild but precise conjecture on the shape of the Kähler cone of $\overlinel{\matcal M}$$_0,n$is given in the end.

Based on results obtained in [15], we construct groups of special S-units for function fields of characteristic p>0, and show that they satisfy Gras-type Conjectures. We use these results in order to give a new proof of Chinburg‘s Ω$_3$-Conjecture on the Galois module structure of the group of S-units, for cyclic extensions of prime degree of function fields.

Let X⊂P be a variety (respectively an open subset of an analytic submanifold) and let x∈X be a point where all integer valued differential invariants are locally constant. We show that if the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Segre P× P, n,m[ges ]2, a Grassmaniann G(2,n+2), n[ges ]4, or the Cayley plane OP$^2$, then X is the corresponding homogeneous variety (resp. an open subset of the corresponding homogeneous variety). The case of the Segre P$^2$×P$^2$had been conjectured by Griffiths and Harris in [GH]. If the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Veronese v$_2$(P) and the Fubini cubic form of X at x is zero, then X=v$_2$(P) (resp. an open subset of v$_2$(P)). All these results are valid in the real or complex analytic categories and locally in the C$^∞$category if one assumes the hypotheses hold in a neighborhood of any point x. As a byproduct, we show that the systems of quadrics I$_2$(P [sqcup ]P)⊂ S$^2$C, I$_2$(P$^1$× P)⊂ S$^2$C and I$_2$(S$_5$)⊂ S$^2$C$^16$are stable in the sense that if A ⊂S$^*$ is an analytic family such that for t≠0,A≃A, then A$_0$≃A.

We also make some observations related to the Fulton–:Hansen connectedness theorem.

Let K be a discrete valuation field with ring of integers ${\mathcal O}$$_K$. Let f:X→Y be a finite morphism of curves over K. In this article, we study some possible relationships between the models over ${\mathcal O}$$_K$ of X and of Y. Three such relationships are listed below. Consider a Galois cover f:X→Y of degree prime to the characteristic of the residue field, with branch locus B. We show that if Y has semi-stable reduction over K, then X achieves semi-stable reduction over some explicit tame extension of K(B). When K is strictly henselian, we determine the minimal extension L/K with the property that X$_L$ has semi-stable reduction.

Let f:X→Y be a finite morphism, with g(Y)[ges ]2. We show that if X has a stable model ${\mathcal X}$ over ${\mathcal O}$$_K$, then Y has a stable model ${\mathcal Y}$ over ${\mathcal O}$$_K$, and the morphism f extends to a morphism ${\mathcal X}$→${\mathcal Y}$.

Finally, given any finite morphism f:X→Y, is it possible to choose suitable regular models ${\mathcal X}$ and ${\mathcal Y}$ of X and Y over ${\mathcal O}$$_K$ such that f extends to a finite morphism ${\mathcal X}$→${\mathcal Y}$? As was shown by Abhyankar, the answer is negative in general. We present counterexamples in rather general situations, with f a cyclic cover of any order [ges ] 4. On the other hand, we prove, without any hypotheses on the residual characteristic, that this extension problem has a positive solution when f is cyclic of order 2 or 3.

In this paper it is proved that the index of a Fredholm operator between $p$-adic Banach spaces is preserved under compact perturbations. A case of special interest is provided when the ground field is nonspherically complete. In this case the classical techniques are no longer valid and the relation between the kernels of a Fredholm operator and that of a small compact perturbation turn out to be in general much richer than in the complex context.

In this article we determine all solutions to the equation xp+yq=zr, (p,q,r)∈{(2,4,6), (2,6,4), (4,6,2), (2,8,3)} in coprime integers x,y,z. First we determine a set of curves of genus 2, such that every solution corresponds to a rational point on one of these curves. Then we determine the rational points on these curves using either covers of rank 0 elliptic curves or a method known as effective Chabauty which works if the Mordell–Weil rank of the jacobian is smaller than the dimension.

Upper and lower bounds for the Seshadri constants of canonical bundles of compact hyperbolic spaces are given in terms of metric invariants. The lower bound is obtained by carrying out the symplectic blow-up construction for the Poincaré metric, and the upper bound is obtained by a convexity-type argument.

We study an analog over an imaginary quadratic field K of Serre‘s conjecture for modular forms. Given a continuous irreducible representation ρ:Gal(Q/K) →GL$_2$(F$_l$) we ask if ρ is modular. We give three examples of representations ρ obtained by restriction of even representations of Gal(Q/Q). These representations appear to be modular when viewed as representations over K, as shown by the computer calculations described at the end of the paper.

For a site ${\mathcal S}$ (with enough points), we construct a topological space X$_($)$ and a full embedding φ$^*$ of the category of sheaves on ${\mathcal S}$ into those on X$_($) (i.e., a morphism of toposes φ:Sh (X$_($) →Sh(${\mathcal S}$)). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms coh$^*$(${\mathcal S}$,A) =coh$^**$(X$_($), φ$^*$A) for any Abelian sheaf A on ${\mathcal S}$. As a particular case, this will give for any scheme Y a topological space X and a functorial isomorphism between the étale cohomology H$^**$(et{Y},A) and the ordinary sheaf cohomology H$^**$(X,φ$^**$A), for any sheaf A for the étale topology on Y.

We consider the restriction of the reflection representation to various reductive dual pairs in exceptional groups, and determine the correspondence of generic representations.