Higher Order Congruences Amongst Hasse–weil -Values

Delbourgo, Daniel; Peters, Lloyd Christopher

doi:10.1017/s1446788714000445

Cited by 4 publications

(3 citation statements)

References 14 publications

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“…There is strong evidence to support our expectation that Hypothesis (Int)

_{{scriptD}_{n}}

holds true for each Hecke eigenform

f

belonging to the Hida family lifting

\bar{ρ}

. The Iwasawa Main Conjecture for the twisted motives

M (f) \otimes M (ϱ)

will imply the integrality of each factor

{boldL}_{p} false(f, ϱ, T false)

with dim

(ϱ) ⩾ 2

, whilst the integrality of

{boldL}_{p} false(f, {reg}_{double-struckQ false(μ_{p^{n}} false) / double-struckQ}, T false)

is already known from . The non‐abelian congruences derived in predict a simple growth formula for the analytic

μ

‐invariant over

D_{n}

in terms of the subfields

Q false(μ_{p^{n}} false) \subset {scriptD}_{n}

, so if the

μ

‐invariant of

f

over the latter field

Q (μ_{p^{n}})

is non‐negative, then (conjecturally) the

μ

‐invariant of

f

over

D_{n}

is too. In § , we show the above hypothesis follows if: (i) the

normalΛ

‐adic family lifting

f / Q {(μ_{p^{n}})}^{+}

projects integrally onto a certain submodule ‘

{\overset{S}{true}}^{ord} (

…”

Section: Introductionmentioning

confidence: 63%

“…Examples (i)If

{scriptD}_{\infty} = Q false(μ_{p^{\infty}}, Δ_{1}^{1 / p^{\infty}}, \dots, Δ_{g}^{1 / p^{\infty}} false)

is a

g

‐fold false‐Tate tower so that

{scriptD}_{n} = Q false(μ_{p^{n}}, Δ_{1}^{1 / p^{n}}, \dots, Δ_{g}^{1 / p^{n}} false)

, then

Gal false(D_{n} / double-struckQ false) ≅ {(Z / p^{n} Z)}^{g} ⋊ {(Z / p^{n} Z)}^{\times}

, the set

S^{ram} false(D_{n} / double-struckQ false) = supp false(p \cdot Δ_{1} \dots Δ_{g} false)

and

A_{1 + p}

is just a scalar matrix (for example, see ). (ii)If

dim ({scriptG}_{\infty}) = 2

with

{scriptG}_{\infty} = Gal false(D_{\infty} / double-struckQ false)

a torsion‐free non‐commutative

p

‐adic Lie group, then

{scriptG}_{\infty} ≅ {double-struckZ}_{p} ⋊ Γ

exhibits a similar structure to (i) for

g = 1

. (iii)If

dim ({scriptG}_{\infty}) = 3

with

…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Variation of the analytic λ‐invariant over a solvable extension

Delbourgo

2019

Proc. London Math. Soc.

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“…There is strong evidence to support our expectation that Hypothesis (Int)

_{{scriptD}_{n}}

holds true for each Hecke eigenform

f

belonging to the Hida family lifting

\bar{ρ}

. The Iwasawa Main Conjecture for the twisted motives

M (f) \otimes M (ϱ)

will imply the integrality of each factor

{boldL}_{p} false(f, ϱ, T false)

with dim

(ϱ) ⩾ 2

, whilst the integrality of

{boldL}_{p} false(f, {reg}_{double-struckQ false(μ_{p^{n}} false) / double-struckQ}, T false)

is already known from . The non‐abelian congruences derived in predict a simple growth formula for the analytic

μ

‐invariant over

D_{n}

in terms of the subfields

Q false(μ_{p^{n}} false) \subset {scriptD}_{n}

, so if the

μ

‐invariant of

f

over the latter field

Q (μ_{p^{n}})

is non‐negative, then (conjecturally) the

μ

‐invariant of

f

over

D_{n}

is too. In § , we show the above hypothesis follows if: (i) the

normalΛ

‐adic family lifting

f / Q {(μ_{p^{n}})}^{+}

projects integrally onto a certain submodule ‘

{\overset{S}{true}}^{ord} (

…”

Section: Introductionmentioning

confidence: 63%

“…Examples (i)If

{scriptD}_{\infty} = Q false(μ_{p^{\infty}}, Δ_{1}^{1 / p^{\infty}}, \dots, Δ_{g}^{1 / p^{\infty}} false)

is a

g

‐fold false‐Tate tower so that

{scriptD}_{n} = Q false(μ_{p^{n}}, Δ_{1}^{1 / p^{n}}, \dots, Δ_{g}^{1 / p^{n}} false)

, then

Gal false(D_{n} / double-struckQ false) ≅ {(Z / p^{n} Z)}^{g} ⋊ {(Z / p^{n} Z)}^{\times}

, the set

S^{ram} false(D_{n} / double-struckQ false) = supp false(p \cdot Δ_{1} \dots Δ_{g} false)

and

A_{1 + p}

is just a scalar matrix (for example, see ). (ii)If

dim ({scriptG}_{\infty}) = 2

with

{scriptG}_{\infty} = Gal false(D_{\infty} / double-struckQ false)

a torsion‐free non‐commutative

p

‐adic Lie group, then

{scriptG}_{\infty} ≅ {double-struckZ}_{p} ⋊ Γ

exhibits a similar structure to (i) for

g = 1

. (iii)If

dim ({scriptG}_{\infty}) = 3

with

…”

Section: Introductionmentioning

confidence: 99%

Variation of the analytic λ‐invariant over a solvable extension

Delbourgo

2019

Proc. London Math. Soc.

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“…There seem to be two approaches to (2), either using congruences modulo trace ideals [1,17,20,21,23], or instead by deriving p-adic congruences [10,11,12,16,18,19]. Naturally both approaches should be equivalent to one another.…”

Section: Introductionmentioning

confidence: 99%

$$\hbox {K}_{1}$$ K 1 -congruences for three-dimensional Lie groups

Delbourgo

Chao

2018

Ann. Math. Québec

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We completely describe K 1 (Z p [[G ∞ ]]) and its localisations by using an infinite family of p-adic congruences, where G ∞ is any solvable p-adic Lie group of dimension 3. This builds on earlier work of Kato when dim(G ∞ ) = 2, and of the first named author and Lloyd Peters when G ∞ ∼ = Z × p Z d p with a scalar action of Z × p . The method exploits the classification of 3-dimensional p-adic Lie groups due to González-Sánchez and Klopsch, as well as the fundamental ideas of Kakde, Burns, etc. in non-commutative Iwasawa theory.

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