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Maximal class numbers of CM number fields |
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| Authors: | Ryan C Daileda Raju Krishnamoorthy Anton Malyshev |
| Institution: | a Trinity University, Mathematics Department, One Trinity Place, San Antonio, TX 78212-7200, United States b Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, United States c Princeton University, Department of Mathematics, Fine Hall, Washington Road, Princeton NJ 08544-1000, United States |
| Abstract: | Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis and Artin's conjecture on the entirety of Artin L-functions, we derive an upper bound (in terms of the discriminant) on the class number of any CM number field with maximal real subfield F. This bound is a refinement of a bound established by Duke in 2001. Under the same hypotheses, we go on to prove that there exist infinitely many CM-extensions of F whose class numbers essentially meet this improved bound and whose Galois groups are as large as possible. |
| Keywords: | 11R29 11R21 11R42 11R32 |
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