Seminar: Topics in Number Theory: Hida Theory (Spring 2026)

Topic:s Classical Hida Theory, Control Theorems of Hida, Higher Hida Theory

In this seminar, we will be mainly discussing 3 topics: (basic) Hida theory, Hida control theory, and if time permits, higher Hida Theory (especially those pertaining to Shimura varieties). It is roughtly divided into three parts: in the first part, we will be reviewing a sequence of Hida's papers that eventually built up into (the most basic version of) Hida's theory. Then after a brief discussion on basics of Galois deformation, we will be reviewing Hida's work on control theorems, and discuss how its relation with Hida theory. If time permits, we will look at higher cohomology class and discussing some of the basics of higher Hida theory. Mostly, it will

Logistics (To be continuously updated thorughout the semester)

Plan(This is NOT the abstract: please go to the bottom for weekly summary):

Module 1 (8-9 Weeks): Basic Hida Theory

  • Congruences of Cusp Forms and Special Values of Zeta functions
  • Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms
  • Iwasawa modules attached to congruences of cusp forms

    • Module 2 (2-3 Weeks): Hida Control Theory

    Module 3 (1-2 Weeks): Higher Hida Theory

    Tentative Syllabus:

    See here

    Schedule & Weekly Summary:

    Week 1 (01/29)
    Xiaorun Wu
    Logistics & Introduction
    We will first go through the general logistics, assign speakers for seminars the next few weeks. We will also update our syllabus with some new papers being proposed. Following that, we will talk about Hida's 1981 paper. "Congruences of cusp forms and special values of their zeta functions". In the paper "Congruences of cusp forms and special values of their zeta functions," Hida establishes a connection between congruences of cusp forms and the special values of their adjoint $L$-functions. The work shows that congruences between a primitive cusp form and others are related to the Petersson inner product, which links to the critical value of the adjoint $L$-function, and the existence of non-trivial congruences modulo a prime $p$ is connected to the divisibility of this $L$-value by $p$.
    notes here
    Week 2 (02/02)
    Xiaorun Wu
    Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms
    We will again go through the general logistics, assign speakers for seminars the next few weeks. Following that, we will talk about Hida's 1986 paper. "Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms". In his 1986 paper, "Galois representations into \(GL_{2}(\mathbb{Z}_{p}[[X]])\) attached to ordinary cusp forms," Haruzo Hida introduced the concept of \(p\)-adic analytic families of ordinary modular forms (Hida families). He demonstrated that these families are associated with a \(p\)-adic Galois representation interpolating classical Deligne representations. This work established (the most basic version of) Hida theory, offering tools for studying \(p\)-adic \(L\)-functions and advancing research in areas like the Iwasawa main conjecture and Galois representation deformation theory.
    notes here
    Week 3 (02/09)
    Ethan Bottomley-Mason
    Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms (Cont'd)
    Picking up from where we left last week, we continue on Hida's 1986 paper. "Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms". In his 1986 paper, "Galois representations into \(GL_{2}(\mathbb{Z}_{p}[[X]])\) attached to ordinary cusp forms," He introduced the concept of \(p\)-adic analytic families of ordinary modular forms (Hida families). He demonstrated that these families are associated with a \(p\)-adic Galois representation interpolating classical Deligne representations. This work established (the most basic version of) Hida theory, offering tools for studying \(p\)-adic \(L\)-functions and advancing research in areas like the Iwasawa main conjecture and Galois representation deformation theory. If time permits, we will briefly talk about the next paper: Iwasawa modules attached to congruences of cusp forms, which complements with the one we've discussed.
    notes here
    Week 4 (02/16)
    Rafah Hajjar Muñoz
    Iwasawa modules attached to congruences of cusp forms
    This week we are going to talk about the paper "Iwasawa modules attached to congruences of cusp forms". Haruzo Hida’s seminal 1986 paper establishes the theory of p-adic ordinary families of modular forms. Hida constructs a "big" Hecke algebra \(\mathfrak{h}_\infty\) that acts as a finite, free module over the Iwasawa algebra \(\Lambda = \mathbb{Z}_p[[T]]\). By introducing an idempotent ordinary projector, he demonstrates that modular forms of varying weights \(k\ge2\) belong to continuous \(p\)-adic families. The work's crowning achievement is a Control Theorem, showing that specializing these families at specific weights recovers classical space. This provides a bridge between Iwasawa theory and arithmetic geometry, offering a framework for studying \(p\)-adic \(L\)-functions and the Galois representations attached to cusp forms.
    notes here