Class Numbers of Orders in Quartic Fields

Details Class Numbers of Orders in Quartic Fields Class Numbers of Orders in Quartic Fields

2006-02-14

arxiv.org/abs/math/0602294v2

Mathematics

Number Theory

144 pages, Doctoral thesis Corrected typos, added acknowledgments

Scientific paper

It is shown that the sum of class numbers of orders in totally complex quartic fields with no real quadratic subfield obeys an asymptotic law similar to the prime numbers, as the bound on the regulators tends to infinity. Here only orders which are maximal at a given set of primes containing an even number of elements are considered. The proof is accomplished by means of a Prime Geodesic Theorem for symmetric spaces formed as compact quotients of SL(4,R). This result extends work of Sarnak in the real quadratic case and of Deitmar and Hoffmann in the complex cubic case.

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