Triangulation of refined families

Paraboline variation of $p$-adic families of $(φ,Γ)$-modules

We study the $p$-adic variation of triangulations over $p$-adic families of $(φ,Γ)$-modules. In particular, we study certain canonical sub-filtrations of the pointwise triangulations and show that they extend to affinoid neighborhoods of crystalline points. This generalizes results of Kedlaya, Pottharst and Xiao and (independently) Liu in the case where one expects the entire triangulation to extend. As an application, we study the ramification of weight parameters over natural $p$-adic families.

Wach modules, regulator maps, and epsilon-isomorphisms in families

We prove the local epsilon-isomorphism conjecture of Fukaya and Kato [FK06] for certain crystalline families of G_Qp-representations. This conjecture can be regarded as a local analogue of the Iwasawa main conjecture for families. Our work extends earlier work of Kato for rank-1 modules (cf. [Ven13]), of Benois and Berger for crystalline G_Qp-representations with respect to the cyclotomic extension (cf. [BB08]), as well as of Loeffler, Venjakob, and Zerbes (cf. [LVZ13]) for crystalline G_Qp- representations with respect to abelian p-adic Lie extensions of Qp. Nakamura [Nak13, Nak14] has also f…

Refined $q$-Trinomial Coefficients and Two Infinite Hierarchies of $q$-Series Identities

We will prove an identity involving refined $q$-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined $q$-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarchies contains an identity which is equivalent to Capparelli's first Partition Theorem.

刘与钱猜想的一个反例及$\mathbb{Z}_p$中受限和集的改进逆定理

Infinite Families of Counterexamples to a Conjecture of Liu and Qian and a Refined Inverse Theorem for Restricted Sumsets in $\mathbb{Z}_p$

本文针对Liu和Qian关于$\mathbb{Z}_p$中受限和集临界对的猜想，在边界情形给出反例，并在$|A|+|B|\le p-1$假设下证明改进的逆定理。

On Stark elements of arbitrary weight and their $p$-adic families

We develop a detailed arithmetic theory related to special values at arbitrary integers of the Artin $L$-series of linear characters. To do so we define canonical generalized Stark elements of arbitrary `rank' and `weight', thereby extending the classical theory of Rubin-Stark elements. We then formulate an extension to arbitrary weight of the refined version of the Rubin-Stark Conjecture that we studied in an earlier article and also show that generalized Stark elements constitute a $p$-adic family by formulating precise conjectural congruence relations between elements of differing weights. …

关于 $p$-细化 Friedberg-Jacquet 积分与 $\mathrm{GL}_{2n}$ 特征簇中的经典辛轨迹

On $p$-refined Friedberg-Jacquet integrals and the classical symplectic locus in the $\mathrm{GL}_{2n}$ eigenvariety

本文提出关于 $P$-parahoric $p$-细化下 Friedberg-Jacquet 积分非零性与函子转移等价性的猜想，通过证明 $\mathrm{GL}_{2n}$ 特征簇中经典辛族维数的上界，获得了支持该猜想的若干结果。