An introduction to the equivariant Tamagawa number conjecture: The relation to Stark’s conjecture

2011 ◽  
pp. 125-152 ◽  
Author(s):  
David Burns
Keyword(s):  
Stark's Conjecture ◽  
Tamagawa Number Conjecture

ON HIGHER-ORDER STICKELBERGER-TYPE THEOREMS FOR MULTI-QUADRATIC EXTENSIONS

2012 ◽  
Vol 08 (01) ◽  
pp. 95-110 ◽  
Author(s):  
DANIEL MACIAS CASTILLO
Keyword(s):  
Higher Order ◽  
Ideal Class ◽  
Quadratic Extensions ◽  
Galois Modules ◽  
Ideal Class Groups ◽  
Stark's Conjecture ◽  
Wide Range ◽  
Tamagawa Number Conjecture ◽  
Derivatives Of ◽  
Integral Version

We prove, for all quadratic and a wide range of multi-quadratic extensions of global fields, a result concerning the annihilation as Galois modules of ideal class groups by explicit elements constructed from the values of higher-order derivatives of Dirichlet L-functions. This result simultaneously refines Rubin's integral version of Stark's Conjecture and provides evidence for the relevant case of the Equivariant Tamagawa Number Conjecture of Burns and Flach.

scholarly journals THE EQUIVALENCE OF RUBIN'S CONJECTURE AND THE ETNC/LRNC FOR CERTAIN BIQUADRATIC EXTENSIONS

2013 ◽  
Vol 56 (2) ◽  
pp. 335-353 ◽  
Author(s):  
PAUL BUCKINGHAM
Keyword(s):  
Number Fields ◽  
Abelian Extension ◽  
Root Number ◽  
Tamagawa Number Conjecture

AbstractFor an abelian extension L/K of number fields, the Equivariant Tamagawa Number Conjecture (ETNC) at s = 0, which is equivalent to the Lifted Root Number Conjecture (LRNC), implies Rubin's Conjecture by work of Burns [3]. We show that, for relative biquadratic extensions L/K satisfying a certain condition on the splitting of places, Rubin's Conjecture in turn implies the ETNC/LRNC. We conclude with some examples.

ON STARK’S CLASS NUMBER CONJECTURE AND THE GENERALISED BRAUER–SIEGEL CONJECTURE

2022 ◽  
pp. 1-13
Author(s):  
PENG-JIE WONG
Keyword(s):  
Class Number ◽  
Number Fields ◽  
Galois Closure ◽  
Stark's Conjecture

Abstract Stark conjectured that for any $h\in \Bbb {N}$ , there are only finitely many CM-fields with class number h. Let $\mathcal {C}$ be the class of number fields L for which L has an almost normal subfield K such that $L/K$ has solvable Galois closure. We prove Stark’s conjecture for $L\in \mathcal {C}$ of degree greater than or equal to 6. Moreover, we show that the generalised Brauer–Siegel conjecture is true for asymptotically good towers of number fields $L\in \mathcal {C}$ and asymptotically bad families of $L\in \mathcal {C}$ .

Two cases of Stark's conjecture

1985 ◽  
Vol 272 (3) ◽  
pp. 349-359 ◽  
Author(s):  
Jonathan W. Sands
Keyword(s):  
Stark's Conjecture

scholarly journals On Iwasawa theory, zeta elements for 𝔾m, and the equivariant Tamagawa number conjecture

2017 ◽  
Vol 11 (7) ◽  
pp. 1527-1571
Author(s):  
David Burns ◽  
Masato Kurihara ◽  
Takamichi Sano
Keyword(s):  
Iwasawa Theory ◽  
Tamagawa Number Conjecture

scholarly journals On descent theory and main conjectures in non-commutative Iwasawa theory

2010 ◽  
Vol 10 (1) ◽  
pp. 59-118 ◽  
Author(s):  
David Burns ◽  
Otmar Venjakob
Keyword(s):  
Iwasawa Theory ◽  
Main Conjecture ◽  
Descent Theory ◽  
Special Cases ◽  
Important Means ◽  
Tamagawa Number Conjecture ◽  
Whitehead Groups

AbstractWe develop an explicit descent theory in the context of Whitehead groups of non-commutative Iwasawa algebras. We apply this theory to describe the precise connection between main conjectures of non-commutative Iwasawa theory (in the spirit of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is both a converse to a theorem of Fukaya and Kato and also provides an important means of deriving explicit consequences of the main conjecture and proving special cases of the equivariant Tamagawa number conjecture.

scholarly journals Leading terms of Artin L-series at negative integers and annihilation of higher K-groups

2011 ◽  
Vol 151 (1) ◽  
pp. 1-22 ◽  
Author(s):  
ANDREAS NICKEL
Keyword(s):  
Galois Group ◽  
Galois Extension ◽  
Number Fields ◽  
Ring Of Integers ◽  
Annihilator Ideal ◽  
Galois Action ◽  
Higher Dimensional ◽  
Totally Real ◽  
Tamagawa Number Conjecture ◽  
Special Case

AbstractLet L/K be a finite Galois extension of number fields with Galois group G. We use leading terms of Artin L-series at strictly negative integers to construct elements which we conjecture to lie in the annihilator ideal associated to the Galois action on the higher dimensional algebraic K-groups of the ring of integers in L. For abelian G our conjecture coincides with a conjecture of Snaith and thus generalizes also the well-known Coates–Sinnott conjecture. We show that our conjecture is implied by the appropriate special case of the equivariant Tamagawa number conjecture (ETNC) provided that the Quillen–Lichtenbaum conjecture holds. Moreover, we prove induction results for the ETNC in the case of Tate motives h0(Spec(L))(r), where r is a strictly negative integer. In particular, this implies the ETNC for the pair (h0(Spec(L))(r), ), where L is totally real, r < 0 is odd and is a maximal order containing ℤ[]G, and will also provide some evidence for our conjecture.

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