On fine Selmer groups and the greatest common divisor of signed and chromatic $p$-adic $L$-functions

On fine Selmer groups and signed Selmer groups of elliptic modular forms

Let $f$ be an elliptic modular form and $p$ an odd prime that is coprime to the level of $f$. We study the link between divisors of the characteristic ideal of the $p$-primary fine Selmer group of $f$ over the cyclotomic $\mathbb{Z}_p$ extension of $\mathbb{Q}$ and the greatest common divisor of signed Selmer groups attached to $f$ defined using the theory of Wach modules. One of the key ingredients of our proof is a generalization of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction at $p$ to the context of modular forms.

Functional equations for supersingular abelian varieties over $\mathbf{Z}_p^2$-extensions

Let $K$ be an imaginary quadratic field and $K_\infty$ be the $\mathbf{Z}_p^2$-extension of $K$. Answering a question of Ahmed and Lim, we show that the Pontryagin dual of the Selmer group associated to a supersingular polarized abelian variety admits an algebraic functional equation. The proof uses the theory of $Γ$-system developed by Lai, Longhi, Tan and Trihan. We also show the algebraic functional equation holds for Sprung's chromatic Selmer groups of supersingular elliptic curves along $K_\infty$.

塞尔默调和三项式的单模零点

Unimodular Zeros of Selmer's Harmonic Trinomials

本文对塞尔默调和三项式H_{ε,δ}的单模零点进行分类，通过引入n、k的最大公约数g及相关参数，给出其存在单模零点的充要同余条件，明确了零点数量并完成证明。

精细Mordell–Weil群与正负Mordell–Weil群的代数结构和特征理想

Algebraic structure and characteristic ideals of fine Mordell--Weil groups and plus/minus Mordell--Weil groups

该研究推广了Lee关于普通Mordell–Weil群的结果，证明了精细Mordell–Weil群在$\boldsymbol{Z}_p$-扩张上的控制定理，验证了其特征理想与相关预测的一致性，并定义正负Mordell–Weil群验证了其对偶特征理想最大公因子的对应预测。