Variation of the analytic λ‐invariant over a solvable extension

Delbourgo, Daniel

doi:10.1112/plms.12306

Cited by 4 publications

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“…The aim here, and in the companion work [6], is to provide a positive answer. In fact we only discuss = alg in this paper; the complementary case = an is more difficult and technical, and so it is dealt with in op.…”

Section: Introductionmentioning

confidence: 94%

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Variation of the algebraic λ-invariant over a solvable extension

Delbourgo¹

2019

Math. Proc. Camb. Phil. Soc.

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Fix an odd prime p. Let $\mathcal{D}_n$ denote a non-abelian extension of a number field K such that $K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $ and whose Galois group has the form $ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $ where g > 0 and $0 \lt n'\leq n$ . Given a modular Galois representation $\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$ which is p-ordinary and also p-distinguished, we shall write $\mathcal{H}(\overline{\rho})$ for the associated Hida family. Using Greenberg’s notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant \begin{equation} \lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of } \text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big) \end{equation} at all $f\in\mathcal{H}(\overline{\rho})$ , under the assumption $\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$ for at least one form f0. We can then easily deduce that $\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$ is constant along branches of $\mathcal{H}(\overline{\rho})$ , generalising a theorem of Emerton, Pollack and Weston for $\lambda^{\text{alg}}_{\mathbb{Q}(\mu_{p})}(f)$ . For example, if $\mathcal{D}_{\infty}=\bigcup_{n\geq 1}\mathcal{D}_n$ has the structure of a p-adic Lie extension then our formulae include the cases where: either (i) $\mathcal{D}_{\infty}/K$ is a g-fold false Tate tower, or (ii) $\text{Gal}\big(\mathcal{D}_{\infty}/K(\mu_p)\big)$ has dimension ≤ 3 and is a pro-p-group.

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Section: Introductionmentioning

confidence: 94%

“…Before proving the four main theorems from Section 1•1, we first briefly recall the important properties of the Hecke algebra, T (ρ), which controls the Hida family. The principal reference is of course [11], although [2,6] are maybe shorter to read.…”

Section: Control Theory For the Algebraic λ-Invariantmentioning

confidence: 99%

Variation of the algebraic λ-invariant over a solvable extension

Delbourgo¹

2019

Math. Proc. Camb. Phil. Soc.

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“…If µ (j) I,II ∈ Z ∪ {+∞} denotes the minimum of the µ-invariants for L p (f ⊗ g (I) , ω j ) and L p (f ⊗ g (II) , ω j ), then by Theorem 3.3 one has −µ Σ , ω j )) = λ(L p (f ⊗ g ( ) , ω j )) + e ( ) l (ω j ). 5 This containment is also true for the missing characters, which can be seen by exploiting the p-adic density of finite order characters χ with χ F × p = ω j inside the parameter space 1 + pZp. g , because here E l (f ⊗ g (I) , ω j ) ≡ E l (f ⊗ g (II) , ω j ) mod ep•ν 2 .…”

Section: Remarksmentioning

confidence: 97%

“…Emerton, Pollack and Weston [9] later generalised this construction to allow f to vary within a Hida family, and showed that the λ-invariant was stable along the branches of a certain Hecke algebra, T Σ (ρ), parameterising the deformation. Recently the theory has been extended to cover anticyclotomic λ-invariants in the work of Castella, Kim and Longo [1], and also to treat non-commutative p-adic Lie extensions (with a meta-abelian structure) by the first-named author in [5,6]. Further generalisations of Vatsal's original ideas can be found in [2,7,8,19,22].…”

Section: Question How Do the Analytic And Algebraic λ-Invariants Appearing In The Main Conjecture Vary As We Switch Between Two P ν -Congmentioning

confidence: 99%

Controlling λ-invariants for the double and triple product p-adic L-functions

Delbourgo¹,

Gilmore²

2022

Journal De Théorie Des Nombres De Bordeaux

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Controlling λ-invariants for the double and triple product p-adic L-functions Tome 33, n o 3.1 (2021), p. 733-778. © Société Arithmétique de Bordeaux, 2021, tous droits réservés.L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedramArticle mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/

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On the Iwasawa main conjecture for the double product

Delbourgo

2024

Trans. Amer. Math. Soc.

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Let σ \sigma and τ \tau denote a pair of absolutely irreducible p p -ordinary and p p -distinguished Galois representations into GL 2 ⁡ ( F ¯ p ) \operatorname {GL}_2(\overline {\mathbb {F}}_p) . Given two primitive forms ( f , g ) (f,g) such that wt ⁡ ( f ) > wt ⁡ ( g ) > 1 \operatorname {wt}(f)>\operatorname {wt}(g)> 1 and where ρ ¯ f ≅ σ \overline {\rho }_f\cong \sigma and ρ ¯ g ≅ τ \overline {\rho }_g\cong \tau , we show that the Iwasawa Main Conjecture for the double product ρ f ⊗ ρ g \rho _f\otimes \rho _g depends only on the residual Galois representation σ ⊗ τ : G Q → GL 4 ⁡ ( F ¯ p ) \sigma \otimes \tau : G_{\mathbb {Q}}\rightarrow \operatorname {GL}_4(\overline {\mathbb {F}}_p) . More precisely, if IMC( f ⊗ g f\otimes g ) is true for one pair ( f , g ) (f,g) with ρ ¯ f ≅ σ \overline {\rho }_f \cong \sigma and ρ ¯ g ≅ τ \overline {\rho }_g\cong \tau and whose μ \mu -invariant equals zero, then it is true for every congruent pair too.

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Variation of the analytic λ‐invariant over a solvable extension

Abstract: Fix an odd prime p, and suppose that D∞ = n Dn is a solvable p-adic Lie extension of the rationals, such that Gal(Dn/Q) ∼ = (Z/p n Z) g (Z/p n Z) × for some g > 0 and n n. In particular, this situation covers the well-known cases where (i) D∞ = Q(μp∞ , Δ

Cited by 4 publications

References 42 publications

Variation of the algebraic λ-invariant over a solvable extension

Variation of the algebraic λ-invariant over a solvable extension

Controlling λ-invariants for the double and triple product p-adic L-functions

On the Iwasawa main conjecture for the double product

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