We calculate the Euler characteristics of the local systems Sk$\open V$ [otimes] S[ell ] Λ 2$\open V$ on the moduli space $\mathcal M$2 of curves of genus 2, where $\open V$ is the rank 4 local system R1π *$\open C$.

We give sharp, explicit estimates for linear forms in two logarithms, simultaneously for several non-Archimedean valuations. We present applications to explicit lower bounds for the fractional part of powers of rational numbers, and to the Diophantine equation (xn − 1)/(x − 1) = yq.

Let CSn be the flag manifold SO(2n)/U(n). We give a partial classification for the endomorphisms of the cohomology ring H*(CSn; Z) which is very close to a homotopy classification of all selfmaps of CSn. Applications concerning the geometry of the space are discussed.

We prove that, for an inner form of GLn, the number of classes of cuspidal automorphic representations with fixed central character and fixed factor at almost every place is finite. We also prove, in the local situation, relations between the level and the ε-factor of an irreducible smooth representation.

Let Mg be the moduli space of smooth curves of genus g [ges ] 3, and M¯g the Deligne-Mumford compactification in terms of stable curves. Let M¯g[1] be an open set of M¯g consisting of stable curves of genus g with one node at most. In this paper, we determine the necessary and sufficient condition to guarantee that a $\open Q$-divisor D on M¯g is nef over M¯g[1], that is, (D · C) [ges ] 0 for all irreducible curves C on M¯g with C ∩ M¯g[1] ≠ [empty ].

Let VD be the Shimura curve over $\open Q$ attached to the indefinite rational quaternion algebra of discriminant D. In this note we investigate the group of automorphisms of VD and prove that, in many cases, it is the Atkin–Lehner group. Moreover, we determine the family of bielliptic Shimura curves over $\bar\open Q$ and over $\open Q$ and we use it to study the set of rational points on VD over quadratic fields. Finally, we obtain explicit equations of elliptic Atkin–Lehner quotients of VD.