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

Nickel, Andreas

doi:10.1017/s0305004111000193

Cited by 13 publications

(14 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Cp that is induced by combining the descriptions of H 1 (C m ) and H 2 (C m ) given in Example 2.8(ii) together with −1 times the Beilinson regulator map. Then the argument of[7, §11.1] shows that the equivariant Tamagawa number conjecture implies the existence of a characteristic element for (C m , t−1 m ) that is constructed from the leading terms at m of the Artin L-functions of characters of G. In this context the result of Corollary 2.13 can in fact be used to give a different proof of the main result (Theorem 4.1) of Nickel in[35].2.3. Symmetric organising matrices.…”

mentioning

confidence: 99%

Organizing Matrices for Arithmetic Complexes

Burns

Castillo

2013

International Mathematics Research Notices

View full text Add to dashboard Cite

We extend and refine the theory of 'organising modules' of Mazur and Rubin to construct a canonical class of matrices that encodes a range of information about natural families of complexes in arithmetic. We then describe several concrete applications of this theory including the proof of new results on the explicit structures of Galois groups, ideal class groups and wild kernels in higher algebraic K-theory and the formulation of a range of explicit conjectures concerning both the ranks and Galois structures of Selmer groups of abelian varieties over finite (non-abelian) Galois extensions of number fields.

show abstract

mentioning

confidence: 99%

Organizing Matrices for Arithmetic Complexes

Burns

Castillo

2013

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

“…6]) and generalizations thereof due to Burns [13] and the author [57]. If r is a negative integer, the ETNC refines a conjecture of Gross [42] and implies (generalizations of) the Coates-Sinnott conjecture [28] and a conjecture of Snaith [71] on annihilators of the higher K -theory of rings of integers (see [56]). If r > 1 the ETNC likewise predicts constraints on the Galois module structure of p-adic wild kernels [62].…”

Section: Introductionmentioning

confidence: 91%

“…Remark 3. 13 It is not hard to see that Gross's conjecture does not depend on S and the choice of φ 1−r (see also [56,Remark 6]). A straightforward substitution shows that if it is true for χ then it is true for χ τ for every choice of τ ∈ Aut(C).…”

Section: A Conjecture Of Grossmentioning

confidence: 99%

On the p-adic Beilinson conjecture and the equivariant Tamagawa number conjecture

Nickel

2021

Sel. Math. New Ser.

Self Cite

View full text Add to dashboard Cite

Let E/K be a finite Galois extension of totally real number fields with Galois group G. Let p be an odd prime and let $$r>1$$ r > 1 be an odd integer. The p-adic Beilinson conjecture relates the values at $$s=r$$ s = r of p-adic Artin L-functions attached to the irreducible characters of G to those of corresponding complex Artin L-functions. We show that this conjecture, the equivariant Iwasawa main conjecture and a conjecture of Schneider imply the ‘p-part’ of the equivariant Tamagawa number conjecture for the pair $$(h^0(\mathrm {Spec}(E))(r), \mathbb {Z}[G])$$ ( h 0 ( Spec ( E ) ) ( r ) , Z [ G ] ) . If $$r>1$$ r > 1 is even we obtain a similar result for Galois CM-extensions after restriction to ‘minus parts’.

show abstract

“…For completeness, we include the following result which is an easy consequence of [Ni11c,Theorem 4.1] and [Bu, Corollary 2.10].…”

Section: Negative Integersmentioning

confidence: 99%

Integrality of Stickelberger elements and the equivariant Tamagawa number conjecture

Nickel

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

Self Cite

View full text Add to dashboard Cite

Let L/K be a finite Galois CM-extension of number fields with Galois group G. In an earlier paper, the author has defined a module SKu(L/K) over the center of the group ring ZG which coincides with the Sinnott-Kurihara ideal if G is abelian and, in particular, contains many Stickelberger elements. It was shown that a certain conjecture on the integrality of SKu(L/K) implies the minus part of the equivariant Tamagawa number conjecture at an odd prime p for an infinite class of (non-abelian) Galois CM-extensions of number fields which are at most tamely ramified above p, provided that Iwasawa's µ-invariant vanishes. Here, we prove a relevant part of this integrality conjecture which enables us to deduce the minus-p-part of the equivariant Tamagawa number conjecture from the vanishing of µ for the same class of extensions. As an application we prove the non-abelian Brumer and Brumer-Stark conjecture outside the 2-primary part for every monomial Galois extension of Q provided that certain µ-invariants vanish.

show abstract

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

Cited by 13 publications

References 28 publications

Organizing Matrices for Arithmetic Complexes

Organizing Matrices for Arithmetic Complexes

On the p-adic Beilinson conjecture and the equivariant Tamagawa number conjecture

Integrality of Stickelberger elements and the equivariant Tamagawa number conjecture

Contact Info

Product

Resources

About