The Dirichlet Class Number Formula for Imaginary Quadratic Fields

Lectures on the Dirichlet Class Number Formula for Imaginary Quadratic Fields

The Dirichlet Class Number Formula for Imaginary Quadratic Fields

because 2, 3, and 1± √ −5 are irreducible and nonassociate. These notes present a formula that in some sense measures the extent to which unique factorization fails in environments such as Z[ √ −5]. Algebra lets us define a group that measures the failure, geometry shows that the group is finite, and analysis yields the formula for its order. To move forward through the main storyline without b...

Class number formulas via 2-isogenies of elliptic curves

A classical result of Dirichlet shows that certain elementary character sums compute class numbers of quadratic imaginary number fields. We obtain analogous relations between class numbers and a weighted character sum associated to a 2-isogeny of elliptic curves.

L - Functions and Class Numbers of Imaginary Quadratic Fields and of Quadratic Extensions of an Imaginary Quadratic Field

Starting from the analytic class number formula involving its Lfunction, we first give an expression for the class number of an imaginary quadratic field which, in the case of large discriminants, provides us with a much more powerful numerical technique than that of counting the number of reduced definite positive binary quadratic forms, as has been used by Buell in order to compute his class ...

The Dirichlet Class Number Formula for Imaginary Quadratic Fields

Z[ √ −5] = {a+ b √ −5 : a, b ∈ Z}, because 2, 3, and 1± √ −5 are irreducible and nonassociate. These notes present a formula that in some sense measures the extent to which unique factorization fails in environments such as Z[ √ −5]. Algebra lets us define a group that measures the failure, geometry shows that the group is finite, and analysis yields the formula for its order. To move forward t...