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 cover basics of Hida Theory, in light of recent developments and advances in this subject.

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$.
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    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.
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    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.
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    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.
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    Week 5 (02/23)
    Xiaorun Wu
    \(p\)-adic Hecke Algebras for \(GL_1\) and \(GL_2\)
    This seminar talk delves into Hida’s influential 1986 paper, which offers a comparative study of \(p\)-adic Hecke algebras for \(GL_1\) and \(GL_2\). By contrasting the classical case of \(GL_1\)—deeply rooted in class field theory and Dirichlet characters—with the more complex world of modular forms, Hida develops the seminal concept of the ordinary Hecke algebra. The presentation highlights his control theorem, which proves that these algebras are finite and free modules over the Iwasawa algebra \(\Lambda = \mathbb{Z}_p[[T]]\), effectively allowing modular forms of varying weights to be interpolated into continuous \(p\)-adic families. This work serves as a cornerstone for the modern arithmetic of automorphic forms, providing the essential framework for understanding the deformation of Galois representations and the evolution of the \(p\)-adic Langlands program.
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    Week 6 (03/02 & 03/03)
    Vivian (Qiyao) Yu
    Modules of congruence of Hecke algebras and L–functions associated with cusp forms
    This seminar talk examines Haruzo Hida’s landmark 1988 paper, which establishes a profound link between the algebraic properties of Hecke algebras and the analytic values of \(L\)-functions. Hida investigates the modules of congruence—which measure the extent to which a specific cusp form is congruent to others modulo a prime—and proves that the "size" of these modules is precisely governed by the special values of the adjoint \(L\)-function associated with the form. A central theme of the discussion is the construction of the ordinary Hecke algebra, demonstrating how these congruences allow for the \(p\)-adic interpolation of modular forms into continuous families. By bridging the gap between the ring-theoretic structure of Hecke algebras and the arithmetic of \(L\)-values, this work provides a foundational piece of the evidence for the Bloch-Kato conjecture and remains a cornerstone of modern \(p\)-adic Langlands theory. For one time only, we will continue our discussion on 03/03, as many of important things were not being discussed due to time limit
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    Week 7 (03/09)
    Xiaorun Wu
    On p–adic Hecke algebras for GL2 over totally real fields
    In this landmark 1988 paper, Haruzo Hida generalizes his theory of \(p\)-adic ordinary Hecke algebras from the setting of classical modular forms over \(\mathbb{Q}\) to the more complex arithmetic of Hilbert modular forms over a totally real field \(F\). The core of the work is the construction of a "big" Hecke algebra \(\mathbf{h}_{\infty}\)—a universal object that $p$-adically interpolates the ordinary parts of Hecke algebras across varying weights. Hida demonstrates that this big Hecke algebra is a finite and free module over the appropriate Iwasawa algebra (the weight space), effectively proving that ordinary Hilbert modular forms live in $p$-adic analytic families.A critical technical pivot in the paper is the proof of the Control Theorem, which establishes that the specialization of the universal algebra at a specific weight \(k\) is isomorphic to the classical ordinary Hecke algebra of that weight, provided the weights satisfy certain regularity conditions (\(k_i \geq 2\)). By establishing this interpolation, Hida provides the necessary machinery to study the variation of Galois representations and \(L\)-values within these families, extending the reach of the Langlands program into the $p$-adic deformation theory of automorphic forms for totally real fields.
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    Week 8 (03/23)
    Julius Zhang
    p-Ordinary Cohomology Groups for \(SL(2)\) over Number Fields
    In his 1993 paper "p-Ordinary Cohomology Groups for \(SL(2)\) over Number Fields," Haruzo Hida generalizes his earlier work on ordinary \(\Lambda\)-adic forms from the rational field \(\mathbb{Q}\) to general number fields \(F\). By focusing on the group \(SL(2)_{/F}\), Hida constructs a "nearly ordinary" Hecke algebra that is proven to be a finite module over a multi-variable Iwasawa algebra \(\Lambda\). The technical core of the paper is the Control Theorem, which establishes that the \(\Lambda\)-adic cohomology—when specialized at specific arithmetic weights—recovers the classical ordinary cohomology groups of the corresponding weight and level. This result is foundational for the \(p\)-adic deformation theory of automorphic forms over number fields and provides a crucial link between the geometry of Hilbert modular varieties (or their analogs) and the construction of \(p\)-adic \(L\)-functions and Galois representations.
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    Week 9 (03/30)
    Ethan Bottomley-Mason
    p–Adic ordinary Hecke algebras for \(GL(2)\)
    In this landmark 1994 paper, Haruzo Hida provides the rigorous foundation and full proof of the Control Theorem for universal nearly ordinary Hecke algebras over an arbitrary number field \(F\). While his 1986 work established the theory for \(GL(2)\) over \(\mathbb{Q}\), this paper generalizes those results to the restriction of scalars \(G = \text{Res}_{F/\mathbb{Q}} GL(2)\), constructing a "big" Hecke algebra that acts as a module over a multi-variable Iwasawa algebra \(\Lambda\). A defining contribution of this work is Hida's discovery of a sharp structural divide based on the field’s signature: he proves that while the Hecke algebra is a finite module in the totally real case, it becomes a torsion module over \(\Lambda\) whenever the base field \(F\) possesses at least one complex place. By linking the annihilator of these CM components to the Leopoldt conjecture, Hida demonstrates that these algebras are not just tools for \(p\)-adic interpolation, but are fundamental arithmetic objects that encode the non-abelian class field theory of the underlying number field.
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    Week 10 (04/06)
    TBD
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