On the Iwasawa $\mu $-invariant and $\lambda $-invariant associated to tensor products of newforms

Delbourgo, Daniel

doi:10.5802/aif.3593

On the Iwasawa

μ

-invariant and

λ

-invariant associated to tensor products of newforms
[Sur les invariants

μ

λ

d’Iwasawa associés aux produits de formes primitives]

Delbourgo, Daniel ¹

¹ Department of Mathematics University of Waikato Hamilton (New Zealand)

Annales de l'Institut Fourier, Online first, 52 p.

Résumé
Abstract

On fixe un nombre premier impair $p$ , et soit ${\bar{ρ}}_{1}, \dots, {\bar{ρ}}_{t}$ une famille de représentations galoisiennes ordinaires à deux dimensions définies sur un corps fini $𝔽_{p^{e}}$ . On suppose données des formes modulaires primitives $f_{1}, \dots, f_{t}$ dont les représentations $p$ -adiques

ρ_{f_{1}} : G_{ℚ} \to {GL}_{2} (𝒦), \dots, ρ_{f_{t}} : G_{ℚ} \to {GL}_{2} (𝒦) avec 𝒪_{𝒦} / π_{𝒦} ≅ 𝔽_{p^{e}}

satisfont $ρ_{f_{1}} \otimes \dots \otimes ρ_{f_{t}} mod π_{𝒦} ≅ {\bar{ρ}}_{1} \otimes \dots \otimes {\bar{ρ}}_{t}$ , et quelques autres hypothèses supplémentaires. On détermine l’invariant $λ$ cyclotomique pour le groupe de Selmer associé au produit $f_{1} \otimes \dots \otimes f_{t}$ sous l’hypothèse que l’invariant $μ$ est nul. Si $t = 2$ (c’est-à-dire dans le cas du double produit), cela nous permet de déduire la conjecture principale d’Iwasawa pour $f_{1} \otimes f_{2}$ si elle est déjà connue pour une paire congruente $f_{1}^{'} \otimes f_{2}^{'}$ , tout comme Greenberg et Vatsal [19] l’ont fait pour $t = 1$ (i.e. pour les formes modulaires paraboliques).

Fix an odd prime number $p$ , and let ${\bar{ρ}}_{1}, \dots, {\bar{ρ}}_{t}$ be a collection of two-dimensional ordinary Galois representations defined over a finite field $𝔽_{p^{e}}$ . Suppose that we are given newforms $f_{1}, \dots, f_{t}$ whose $p$ -adic representations

ρ_{f_{1}} : G_{ℚ} \to {GL}_{2} (𝒦), \dots, ρ_{f_{t}} : G_{ℚ} \to {GL}_{2} (𝒦) with 𝒪_{𝒦} / π_{𝒦} ≅ 𝔽_{p^{e}}

satisfy $ρ_{f_{1}} \otimes \dots \otimes ρ_{f_{t}} mod π_{𝒦} ≅ {\bar{ρ}}_{1} \otimes \dots \otimes {\bar{ρ}}_{t}$ , and some other extra hypotheses. We shall determine the cyclotomic $λ$ -invariant for the Selmer group attached to the product $f_{1} \otimes \dots \otimes f_{t}$ under the assumption that the $μ$ -invariant is zero. If $t = 2$ (i.e. the double product case) this allows us to deduce the Iwasawa Main Conjecture for $f_{1} \otimes f_{2}$ if it is already known for a congruent pair $f_{1}^{'} \otimes f_{2}^{'}$ , much as Greenberg and Vatsal [19] did for $t = 1$ (i.e. for elliptic cusp forms).

Reçu le : 2020-12-09
Révisé le : 2021-07-01
Accepté le : 2021-11-04
Première publication : 2023-07-03

DOI : 10.5802/aif.3593

Classification : 11F33, 11F80, 11G40, 11R23
Keywords: Galois representations, Iwasawa theory, $p$-adic $L$-functions.
Mot clés : Représentations galoisiennes, théorie d’Iwasawa, fonctions $L$ $p$-adiques.

Affiliations des auteurs :

Delbourgo, Daniel ¹

¹ Department of Mathematics University of Waikato Hamilton (New Zealand)

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Delbourgo, Daniel. On the Iwasawa $\mu $-invariant and $\lambda $-invariant associated to tensor products of newforms. Annales de l'Institut Fourier, Online first, 52 p.

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