The lifted root number conjecture for small sets of places and an application to CM-extensions
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In this paper we study a famous conjecture which relates the leading terms at zero of Artin L-functions attached to a finite Galois extension L/K of number fields to natural arithmetic invariants. This conjecture is called the Lifted Root Number Conjecture (LRNC) and has been introduced by K. W. Gruenberg, J. Ritter and A. Weiss; it depends on a set S of primes of L which is supposed to be sufficiently large. We formulate a LRNC for small sets S which only need to contain the archimedean primes. We apply this to CM-extensions which we require to be (almost) tame above a fixed odd prime p. In this case the conjecture naturally decomposes into a plus and a minus part, and it is the minus part for which we prove the LRNC at p for an infinite class of relatively abelian extensions. Moreover, we show that our results are closely related to the Rubin-Stark conjecture.
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The lifted root number conjecture for small sets of places and an application to CM-extensions, Andreas Nickel
- Language
- Released
- 2008
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- Title
- The lifted root number conjecture for small sets of places and an application to CM-extensions
- Language
- English
- Authors
- Andreas Nickel
- Publisher
- Logos-Verl.
- Released
- 2008
- Format
- Paperback
- ISBN10
- 3832519696
- ISBN13
- 9783832519698
- Category
- University and college textbooks
- Description
- In this paper we study a famous conjecture which relates the leading terms at zero of Artin L-functions attached to a finite Galois extension L/K of number fields to natural arithmetic invariants. This conjecture is called the Lifted Root Number Conjecture (LRNC) and has been introduced by K. W. Gruenberg, J. Ritter and A. Weiss; it depends on a set S of primes of L which is supposed to be sufficiently large. We formulate a LRNC for small sets S which only need to contain the archimedean primes. We apply this to CM-extensions which we require to be (almost) tame above a fixed odd prime p. In this case the conjecture naturally decomposes into a plus and a minus part, and it is the minus part for which we prove the LRNC at p for an infinite class of relatively abelian extensions. Moreover, we show that our results are closely related to the Rubin-Stark conjecture.