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 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.

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.

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.

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.