On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results

Johnston, Henri; Nickel, Andreas

doi:10.1090/tran/6453

Cited by 13 publications

(23 citation statements)

References 57 publications

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“…Unfortunately, it is not possible to use this result in the present context because the second reason for the

μ_{p} false(E false) = 0

hypothesis is that it is required for the descent theory of Burns and Venjakob [29] when

scriptG

has an element of order

p

(that is, when

p

divides

[E^{normalcyc} : F^{normalcyc}]

). However, it is still possible to weaken the

μ_{p} false(E false) = 0

hypothesis in certain situations by using the theory of

p

‐adic hybrid group rings [54] at the finite level as illustrated by Example 8.11.…”

Section: A Prime‐by‐prime Descent Theorem For the Etnc At S=1mentioning

confidence: 99%

“…The present authors [54] made significant progress for certain finite non‐abelian Galois extensions of

double-struckQ

by introducing ‘hybrid

p

‐adic group rings’ and using the functoriality properties of the ETNC to reduce to easier known cases. In particular, if

p

is a prime,

m

is a positive integer and

L / Q

is a finite Galois extension with

G = Gal false(L / double-struckQ false) ≅ Aff false(p^{m} false) = {double-struckF}_{p^{m}} ⋊ {double-struckF}_{p^{m}}^{\times}

, then it was shown unconditionally that the ETNC holds for the pairs

(h^{0} false(normalSpec (L) false) false(r false), Z false[\frac{1}{p} false] false[G false])

where

r \in {0, 1}

.…”

Section: Introductionmentioning

confidence: 99%

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On the p‐adic Stark conjecture at s=1 and applications

Johnston

Nickel

2020

J. London Math. Soc.

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Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The 'p-adic Stark conjecture at s = 1' relates the leading terms at s = 1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to E/F . We prove this conjecture unconditionally when E/Q is abelian. We also show that for certain non-abelian extensions E/F the p-adic Stark conjecture at s = 1 is implied by Leopoldt's conjecture for E at p. Moreover, we prove that for a fixed prime p, the p-adic Stark conjecture at s = 1 for E/F implies Stark's conjecture at s = 1 for E/F . This leads to a 'prime-by-prime' descent theorem for the 'equivariant Tamagawa number conjecture' (ETNC) for Tate motives at s = 1. As an application of these results, we provide strong new evidence for special cases of the ETNC for Tate motives and the closely related 'leading term conjectures' at s = 0 and s = 1.

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“…Unfortunately, it is not possible to use this result in the present context because the second reason for the

μ_{p} false(E false) = 0

hypothesis is that it is required for the descent theory of Burns and Venjakob [29] when

scriptG

has an element of order

p

(that is, when

p

divides

[E^{normalcyc} : F^{normalcyc}]

). However, it is still possible to weaken the

μ_{p} false(E false) = 0

hypothesis in certain situations by using the theory of

p

‐adic hybrid group rings [54] at the finite level as illustrated by Example 8.11.…”

Section: A Prime‐by‐prime Descent Theorem For the Etnc At S=1mentioning

confidence: 99%

“…The present authors [54] made significant progress for certain finite non‐abelian Galois extensions of

double-struckQ

by introducing ‘hybrid

p

‐adic group rings’ and using the functoriality properties of the ETNC to reduce to easier known cases. In particular, if

p

is a prime,

m

is a positive integer and

L / Q

is a finite Galois extension with

G = Gal false(L / double-struckQ false) ≅ Aff false(p^{m} false) = {double-struckF}_{p^{m}} ⋊ {double-struckF}_{p^{m}}^{\times}

, then it was shown unconditionally that the ETNC holds for the pairs

(h^{0} false(normalSpec (L) false) false(r false), Z false[\frac{1}{p} false] false[G false])

where

r \in {0, 1}

.…”

Section: Introductionmentioning

confidence: 99%

On the p‐adic Stark conjecture at s=1 and applications

Johnston

Nickel

2020

J. London Math. Soc.

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“…We also point out a recent article of Johnston and Nickel [15], which provides (among other things) some new non-abelian cases of the conjecture.…”

Section: The Conjecture Z(l/k) For Abelian Extensionsmentioning

confidence: 83%

The Equivalence of Rubin's Conjecture and the Etnc/LRNC for Certain Biquadratic Extensions

Buckingham¹

2013

Glasgow Math. J.

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“…♦There are meanwhile quite a few cases where the ETNC has been verified for certain non-abelian extensions. Here we only mention the following result of Johnston and the author[21, Thm. 4.6].…”

mentioning

confidence: 95%