Arithmetic properties of signed Selmer groups at non-ordinary primes

Hatley, Jeffrey; Lei, Antonio

Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1259-1294.

Nous généralisons de nombreux résultats sur les groupes de Selmer des courbes elliptiques et des formes modulaires dans le cas non-ordinaire. Plus précisément, nous étudions les groupes de Selmer signés définis au moyen de la théorie du module de Wach sur les ${\mathbb{Z}}_p$-extensions cyclotomiques. Nous commençons par donner une définition de groupes de Selmer résiduels et non-primitifs pour les nombres premiers non-ordinaires. Cela nous permet d’étendre les techniques développées par Greenberg (pour les courbes elliptiques ordinaires en $p$) et par Kim (pour les courbes elliptiques supersingulières en $p$) pour démontrer que si deux formes modulaires non-ordinaires $p$ sont congruentes, alors les invariants d’Iwasawa de leurs groupes de Selmer sont reliés de manière explicite. Nos résultats ont plusieurs applications. Dans un premier temps, ils nous permettent de relier la parité des rangs analytiques de telles formes modulaires en généralisant un résultat récent du premier auteur sur les courbes elliptiques supersingulières en $p$. Dans un deuxième temps, nous pouvons démontrer une formule à la Kida pour les groupes de Selmer signés en généralisant les résultats de Pollack et Weston.

We extend many results on Selmer groups for elliptic curves and modular forms to the non-ordinary setting. More precisely, we study the signed Selmer groups defined using the machinery of Wach modules over ${\mathbb{Z}}_p$-cyclotomic extensions. First, we provide a definition of residual and non-primitive Selmer groups at non-ordinary primes. This allows us to extend techniques developed by Greenberg (for $p$-ordinary elliptic curves) and Kim ($p$-supersingular elliptic curves) to show that if two $p$-non-ordinary modular forms are congruent to each other, then the Iwasawa invariants of their signed Selmer groups are related in an explicit manner. Our results have several applications. First of all, this allows us to relate the parity of the analytic ranks of such modular forms generalizing a recent result of the first-named author for $p$-supersingular elliptic curves. Second, we can prove a Kida-type formula for the signed Selmer groups generalizing results of Pollack and Weston.

Classification : 11R18, 11F11, 11R23, 11F85
Keywords: Cyclotomic extensions, Selmer groups, modular forms, non-ordinary primes
Mot clés : Extensions cyclotomiques, groupes de Selmer, formes modulaires, nombres premiers non-ordinaires

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