Class number divisibility of relative quadratic function fields

On the Divisibility of the Class Number of Quadratic Fields

Divisibility Criteria for Class Numbers of Imaginary Quadratic Fields

In a recent paper, Guerzhoy obtained formulas for certain class numbers as p-adic limits of traces of singular moduli. Using earlier work by Bruinier and the second author, we derive a more precise form of these results using results of Zagier. Specifically, if −d < −4 is a fundamental discriminant and n is a positive integer, then Tr(pd) ≡ 24 p− 1 · ( 1− (−d p )) ·H(−d) (mod p) provided that p...

CM-fields with relative class number one

We will show that the normal CM-fields with relative class number one are of degrees ≤ 216. Moreover, if we assume the Generalized Riemann Hypothesis, then the normal CM-fields with relative class number one are of degrees ≤ 96, and the CM-fields with class number one are of degrees ≤ 104. By many authors all normal CM-fields of degrees ≤ 96 with class number one are known except for the possib...

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.

On a Class Number Formula for Real Quadratic Number Fields

For an even Dirichlet character , we obtain a formula for L(1;) in terms of a sum of Dirichlet L-series evaluated at s = 2 and s = 3 and a rapidly convergent numerical series involving the central binomial coeecients. We then derive a class number formula for real quadratic number elds by taking L(s;) to be the quadratic L-series associated with these elds.