Modular forms of weight 1/2 over class number 1 imaginary quadratic number fields

Ternary quadratic forms over number fields with small class number

We enumerate all positive definite ternary quadratic forms over number fields with class number at most 2. This is done by constructing all definite quaternion orders of type number at most 2 over number fields. Finally, we list all definite quaternion orders of ideal class number 1 or 2.

One class genera of ternary quadratic forms over number fields

We enumerate all one class genera of definite ternary quadratic forms over number fields. For this, we construct all Gorenstein orders of type number one in definite quaternion algebras over number fields. Finally, we list all definite quaternion orders of ideal class number one.

`-adic Representations Associated to Modular Forms over Imaginary Quadratic Fields

Let π be a regular algebraic cuspidal automorphic representation of GL2 over an imaginary quadratic number field K, and let ` be a prime number. Assuming the central character of π is invariant under the non-trivial automorphism of K, it is shown that there is a continuous irreducible `-adic representation ρ of Gal(K/K) such that L(s, ρv) = L(s, πv) whenever v is a prime of K outside an explici...

Modular Forms and Elliptic Curves over Imaginary Quadratic Fields

On 2-class field towers of imaginary quadratic number fields

For a number field k, let k1 denote its Hilbert 2-class field, and put k2 = (k1)1. We will determine all imaginary quadratic number fields k such that G = Gal(k2/k) is abelian or metacyclic, and we will give G in terms of generators and relations.