Abstract. In a recent paper [Ell09]

The problem of analyzing the number of number field extensions L/K with bounded (relative) discriminant has been the subject of renewed interest in recent years, with significant advances made by Schmidt, Ellenberg-Venkatesh, Bhargava, Bhargava-Shankar-Wang, and others. In this paper, we use the geometry of numbers and invariant theory of finite groups, in a manner similar to Ellenberg and Venkatesh, to give an upper bound on the number of extensions L/K with fixed degree, bounded relative discriminant, and specified Galois closure.

Abstract. We construct a collection of matrices defined by quadratic residue symbols, termed "quadratic residue matrices", associated to the splitting behavior of prime ideals in a composite of quadratic extensions of Q, and prove a simple criterion characterizing such matrices. We also study the analogous classes of matrices constructed from the cubic and quartic residue symbols for a set of prime ideals of Q( √ −3) and Q(i), respectively.

In [PS11], efficient algorithms are given to compute with overconvergent modular symbols. These algorithms then allow for the fast computation of p-adic L-functions and have further been applied to compute rational points on elliptic curves (e.g. [DP06, Tri06]). In this paper, we generalize these algorithms to the case of families of overconvergent modular symbols. As a consequence, we can compute p-adic families of Hecke-eigenvalues, two-variable p-adic L-functions, L-invariants, as well as the shape and structure of ordinary Hida-Hecke algebras.

We prove the rank of the group of signatures of the circular units (hence also the full group of units) of Q(ζ m ) + tends to infinity with m. We also show the signature rank of the units differs from its maximum possible value by a bounded amount for all the real subfields of the composite of an abelian field with finitely many odd prime-power cyclotomic towers. In particular, for any prime p the signature rank of the units of Q(ζ p n ) + differs from ϕ(p n )/2 by an amount that is bounded independent of n. Finally, we show conditionally that for general cyclotomic fields the unit signature rank can differ from its maximum possible value by an arbitrarily large amount.