We give some examples of families of elliptic curves with nonconstant j-invariant where the parity of the (analytic) rank is not equidistributed among the fibres.

We establish an example of a functorial lift from generic cuspidal representations of a similitude group of the type A1×C2 to generic representations of Spin7. Our construction uses the theta correspondence associated to the dual pair of the type (A1×C2B3) inside E7. We also consider another theta correspondence associated to the dual pair of type (A1×C2A1×A1) in D6 and show that these two pairs fit into a tower and the standard properties of a tower of theta correspondences hold.

Let k be an algebraically closed field of characteristic zero, F be an algebraically closed extension of k of transcendence degree one, and G be the group of automorphisms over k of the field F. The purpose of this note is to calculate the group of continuous automorphisms of G.

We will analyze the relationships between the special fibres of a pencil Λ of plane curve singularities and the Jacobian curve J of Λ (defined by the zero locus of the Jacobian determinant for any fixed basis ϕϕ′ ∈ Λ). From the results, we find decompositions of J (and of any special fibre of the pencil) in terms of the minimal resolution of Λ. Using these decompositions and the topological type of any generic pair of curves of Λ, we obtain some topological information about J. More precise decompositions for J can be deduced from the minimal embedded resolution of any pair of fibres (not necessarily generic) or from the minimal embedded resolution of all the special fibres.

Referring to Tits′ alternative, we develop a necessary and sufficient condition to decide whether the normalizer of a finite group of integral matrices is polycyclic-by-finite or is containing a non-Abelian free group. This result is of fundamental importance to conclude whether the (outer) automorphism group of a Bieberbach group is polycyclic-by-finite or has a non-cyclic free subgroup.

Assuming the generalized Riemann hypothesis (GRH) and Artin conjecture for Artin L-functions, we prove that there exists a totally real number field of any fixed degree (>1) with an arbitrarily large discriminant whose normal closure has the full symmetric group as Galois group and whose class number is essentially as large as possible. One ingredient is an unconditional construction of totally real fields with small regulators. Another is the existence of Artin L-functions with large special values. Assuming the GRH and Artin conjecture it is shown that there exist an Artin L-functions with arbitrarily large conductor whose value at s = 1 is extremal and whose associated Galois representation has a fixed image, which is an arbitrary nontrivial finite irreducible subgroup of GL(n, $\open C$;) with property GalT.