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Ju-Feng Wu

Publications and source records attributed to Ju-Feng Wu.

9 recordsLinked to original sources
arXiv2026

Families of symplectic Galois representations over small parabolic eigenvarieties for Siegel cuspforms of genus $2$

We construct small parabolic eigenvarieties for holomorphic Siegel cuspforms of genus $2$ and study families of Galois representations attached to them in the spirit of Bella\"iche--Chenevier. In the course, we introduce the notion of $(\varphi, \Gamma)$-modules with $G$-structures and the notion of refined families of symplectic Galois representations by implementing the theory of symplectic Galois determinant d'apr\`es Moakher--Quast. Such families of symplectic Galois representations provide two applications: In the first application, we show that the small parabolic eigenvarieties are smooth at non-critical points by proving an infinitesimal $R=\mathbb{T}$ theorem. In the second application, we study the relationship between the geometry of the small parabolic eigenvarieties at the Saito--Kurokawa lifts for cuspidal eigenforms (both finite- and infinite-slope) and the Bloch--Kato Selmer groups of those eigenforms.

math.NT↗
arXiv2026

Perfectoid overconvergent Siegel modular forms and the overconvergent Eichler--Shimura morphism

The aim of this paper is twofold. We first present a construction of the overconvergent automorphic sheaves for Siegel modular forms by generalising the perfectoid method, originally introduced by Chojecki--Hansen--Johansson for automorphic forms on compact Shimura curves over $\mathbf{Q}$. The global sections of these automorphic sheaves are precisely the overconvergent Siegel modular forms. In particular, one can compare these automorphic sheaves with the ones constructed by Andreatta--Iovita--Pilloni. Secondly, we establish an (explicit) overconvergent Eichler--Shimura morphism for Siegel modular forms, generalising the result of Andreatta--Iovita--Stevens for the elliptic modular forms.

math.NT↗
arXiv2025

Eisenstein degeneration of Beilinson--Kato classes and circular units

The aim of this note is to explore the Euler system of Beilinson--Kato elements in families passing through the critical $p$-stabilization of an Eisenstein series attached to two Dirichlet characters $(ψ,τ)$. In this context, we establish an explicit connection with the system of circular units, utilizing suitable factorization formulas in a situation where several of the $p$-adic $L$-functions vanish. In that regard, our main results may be seen as an Euler system incarnation of the factorization formula of Bellaïche and Dasgupta. One of the most significant aspects is that, depending on the parity of $ψ$ and $τ$, different phenomena arise; while some can be addressed with our methods, others pose new questions. Finally, we discuss analogous results in the framework of Beilinson--Flach classes.

math.NT↗
arXiv2025

Overconvergent Eichler-Shimura morphisms for $\mathrm{GSp}_4$

We construct explicit Eichler-Shimura morphisms for families of overconvergent Siegel modular forms of genus two. These can be viewed as $p$-adic interpolations of the Eichler-Shimura decomposition of Faltings-Chai for classical Siegel modular forms. In particular, we are able to $p$-adically interpolate the entire decomposition, extending our previous work on the $H^0$-part. The key new inputs are the higher Coleman theory of Boxer-Pilloni and a theory of pro-Kummer étale cohomology with supports.

math.NT↗
arXiv2024

On $p$-adic adjoint $L$-functions for Bianchi cuspforms: the $p$-split case

We construct a Hecke-equivariant pairing on the overconvergent cohomology of Bianchi threefolds. Applying the strategy of Kim and Bellaïche, we use this pairing to construct $p$-adic adjoint $L$-functions for Bianchi cuspforms and show that it detects the ramification locus of the cuspidal Bianchi eigenvariety over the weight space. Combining results of Barrera Salazar--Williams, we show a non-vanishing result of this $p$-adic adjoint $L$-function at certain points. Finally, we obtain a formula relating this pairing with the adjoint $L$-values of the corresponding cuspidal Bianchi eigenforms (of level 1).

math.NT↗
arXiv2024

Finite polynomial cohomology with coefficients

We introduce a theory of finite polynomial cohomology with coefficients in this paper. We prove several basic properties and introduce an Abel-Jacobi map with coefficients. As applications, we use such a cohomology theory to study arithmetics of compact Shimura curves over $\mathbb{Q}$, and simplify proofs of the works of Darmon-Rotger and Bertolini-Darmon-Prasanna.

math.NT↗
arXiv2022

On adjoint Bloch--Kato Selmer groups for $\mathrm{GSp}_{2g}$

We study the adjoint Bloch--Kato Selmer groups attached to a classical point in the cuspidal eigenvariety associated with $\mathrm{GSp}_{2g}$. Our strategy is based on the study of families of Galois representations on the eigenvariety, which is inspired by the book of J. Bellaiche and G. Chenevier.

math.NT↗
arXiv2021

A pairing on the cuspidal eigenvariety for $\mathrm{GSp}_{2g}$ and the ramification locus

In the present paper, we first construct a pairing on the space of analytic distributions associated with $\mathrm{GSp}_{2g}$. By considering the overconvergent parabolic cohomology groups and following the work of Johansson--Newton, we construct the cuspidal eigenvariety for $\mathrm{GSp}_{2g}$. The pairing on the analytic distributions then induces a pairing on some coherent sheaves of the cuspidal eigenvariety. As an application, we follow the strategy of Bellaïche to study the ramification locus of the cuspidal eigenvariety over the corresponding weight space.

math.NT↗