arXiv subjects

Luca Mastella

Publications and source records attributed to Luca Mastella.

4 recordsLinked to original sources

arXiv2026

Vanishing of $\mathrm{Sha}(A/K)[\mathfrak{P}^\infty]$ and its consequences for the anticyclotomic Iwasawa theory of $\mathrm{GL}_2$-abelian varieties

In this article we generalise a classical Kolyvagin's result on the vanishing of the $p$-part of the Shafarevich--Tate group of an elliptic curve (defined over $\mathbb{Q}$) over an imaginary quadratic field $K$ to modular abelian varieties of $\mathrm{GL}_2$-type (defined over a totally real number field $F$) and a CM field $K/F$. Combining this result with the abstract Iwasawa-theoretical argument of [MN19], we show that a similar vanishing holds over the layers of a suitably defined anticyclotomic multi-$\mathbb{Z}_p$-extension of $K$.

math.NT↗

arXiv2025

On anticyclotomic Euler and Kolyvagin systems

We introduce an axiomatization of the notion of ( $p$-complete) anticyclotomic Euler system for a wide class of Galois representations, including those attached to a cuspidal eigenform and to a Hida family of modular forms. Under a minimal set of assumptions, we show how to build from these data a universal Kolyvagin system for the representation and for its anticyclotomic twist. Eventually, we recover some applications to the structure of Selmer groups and Iwasawa main conjectures and we review a few concrete examples of these abstract notions that can be found in the literature.

math.NT↗

arXiv2024

On the structure of the Bloch--Kato Selmer groups of modular forms over anticyclotomic $\mathbf{Z}_p$-towers

Let $p$ be an odd prime number and let $K$ be an imaginary quadratic field in which $p$ is split. Let $f$ be a modular form with good reduction at $p$. We study the variation of the Bloch--Kato Selmer groups and the Bloch--Kato--Shafarevich--Tate groups of $f$ over the anticyclotomic $\mathbf{Z}_p$-extension $K_\infty$ of $K$. In particular, we show that under the generalized Heegner hypothesis, if the $p$-localization of the generalized Heegner cycle attached to $f$ is primitive and certain local conditions hold, then the Pontryagin dual of the Selmer group of $f$ over $K_\infty$ is free over the Iwasawa algebra. Consequently, the Bloch--Kato--Shafarevich--Tate groups of $f$ vanish. This generalizes earlier works of Matar and Matar--Nekov\'a\v{r} on elliptic curves. Furthermore, our proof applies uniformly to the ordinary and non-ordinary settings.

math.NT↗

arXiv2023

Vanishing of the p-part of the Shafarevich-Tate group of a modular form and its consequences for Anticyclotomic Iwasawa Theory

In this article we prove a refinement of a theorem of Longo and Vigni in the anticyclotomic Iwasawa theory for modular forms. More precisely we give a definition for the ($\mathfrak{p}$-part of the) Shafarevich-Tate groups $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K)$ and $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K_\infty)$ of a modular form $f$ of weight $k >2$, 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 $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K) = \widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K_\infty) = 0$.

math.NT↗