Class Numbers of Orders in Cubic Fields

✍ Scribed by Anton Deitmar

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 185 KB
• Volume: 95
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2001.2754

✦ Synopsis

In this paper it is shown that the sum of class numbers of orders in complex cubic fields obeys an asymptotic law similar to the prime numbers as the bound on the regulators tends to infinity. Here only orders are considered which are maximal at two given primes. This result extends work of Sarnak in the real quadratic case. It seems to be the first asymptotic result on class numbers for number fields of degree higher than two. # 2002 Elsevier Science (USA) This was confirmed later by Siegel [13].

Note that log e D equals the regulator RðO D Þ of the order O D . For a long time it was believed to be impossible to separate the class number and the regulator. However, in 1981 Sarnak showed [12], using the trace formula, that