In this thesis we prove a generalization, for the self-dual twist $V$ of the representation attached to a modular forms $f$ of even weight $k > 2$, of a recent result of A. Matar, J. Nekovar, “Kolyvagin’s result on the vanishing of $\sha(E/K)[p^\infty]$ and its consequences for anticyclotomic Iwasawa theory”, in: J. Théor. Nombres Bordeaux 31.2 (2019), in the anticyclotomic Iwasawa Theory for elliptic curves. More precisely we give a definition for the ($\mathfrak{p}$-part of the) Shafarevich-Tate groups $\sha_{\mathfrak{p}^\infty}(f/K)$ and $\sha_{\mathfrak{p}^\infty}(f/K_\infty)$ of $f$ over an imaginary quadratic field $K$ satisfying the Heegner hypothesis and over its anticyclotomic $\mathbb{Z}_p$-extension $K_\infty$ and we show that if the basic generalized Heegner cycle $z_{f, K}$ is non-torsion and not divisible by $p$, then $\sha_{\mathfrak{p}^\infty}(f/K) = \sha_{\mathfrak{p}^\infty}(f/K_\infty) = 0$; moreover the Pontryagin dual of the Bloch-Kato Selmer group of the representation $A = V/T$, where $T$ is the $G_\mathbb{Q}$-stable lattice inside $V$ constructed by Nekovar in “Kolyvagin’s method for Chow groups of Kuga-Sato varieties”, in: Invent. Math.n 107.1 (1992), is free of rank $1$ over the Iwasawa algebra.
In questa tesi dimostriamo una generalizzazione, per il twist autoduale $V$ della rappresentazione associata ad una forma modulare $f$ di peso $k > 2$, di un recente risultato A. Matar, J. Nekovar, “Kolyvagin’s result on the vanishing of $\sha(E/K)[p^\infty]$ and its consequences for anticyclotomic Iwasawa theory”, in: J. Théor. Nombres Bordeaux 31.2 (2019), in teoria di Iwasawa anticiclotomica per curve ellittiche. Più precisamente diamo una definizione della ($\mathfrak{p}$-parte dei) gruppi di Shafarevich-Tate $\sha_{\mathfrak{p}^\infty}(f/K)$ e $\sha_{\mathfrak{p}^\infty}(f/K_\infty)$ di $f$ sopra un campo quadratico immaginario $K$ che soddifi l'ipotesi di Heegner e sopra la sua $\mathbb{Z}_p$-estensione anticiclotomica $K_\infty$ e mostriamo che se il ciclo di Heegner generalizzato di base $z_{f, K}$ non è di torsione e non è divisibile per $p$, allora $\sha_{\mathfrak{p}^\infty}(f/K) = \sha_{\mathfrak{p}^\infty}(f/K_\infty) = 0$; inoltre il duale di Pontryagin del gruppo di Selmer di Bloch-Kato della rappresentazione $A = V/T$, dove $T$ è il reticolo $G_\mathbb{Q}$-stabile dentro $V$ costruito da Nekovar in “Kolyvagin’s method for Chow groups of Kuga-Sato varieties”, in: Invent. Math.n 107.1 (1992), è libero di rango $1$ sull'algebra di Iwasawa.
Vanishing of the p-part of the Shafarevich-Tate group of a modular form and its consequences for Anticyclotomic Iwasawa Theory / Mastella, Luca. - (2023 Mar 24).
Vanishing of the p-part of the Shafarevich-Tate group of a modular form and its consequences for Anticyclotomic Iwasawa Theory
MASTELLA, LUCA
2023
Abstract
In this thesis we prove a generalization, for the self-dual twist $V$ of the representation attached to a modular forms $f$ of even weight $k > 2$, of a recent result of A. Matar, J. Nekovar, “Kolyvagin’s result on the vanishing of $\sha(E/K)[p^\infty]$ and its consequences for anticyclotomic Iwasawa theory”, in: J. Théor. Nombres Bordeaux 31.2 (2019), in the anticyclotomic Iwasawa Theory for elliptic curves. More precisely we give a definition for the ($\mathfrak{p}$-part of the) Shafarevich-Tate groups $\sha_{\mathfrak{p}^\infty}(f/K)$ and $\sha_{\mathfrak{p}^\infty}(f/K_\infty)$ of $f$ over an imaginary quadratic field $K$ satisfying the Heegner hypothesis and over its anticyclotomic $\mathbb{Z}_p$-extension $K_\infty$ and we show that if the basic generalized Heegner cycle $z_{f, K}$ is non-torsion and not divisible by $p$, then $\sha_{\mathfrak{p}^\infty}(f/K) = \sha_{\mathfrak{p}^\infty}(f/K_\infty) = 0$; moreover the Pontryagin dual of the Bloch-Kato Selmer group of the representation $A = V/T$, where $T$ is the $G_\mathbb{Q}$-stable lattice inside $V$ constructed by Nekovar in “Kolyvagin’s method for Chow groups of Kuga-Sato varieties”, in: Invent. Math.n 107.1 (1992), is free of rank $1$ over the Iwasawa algebra.File | Dimensione | Formato | |
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