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)

• When: Mondays 1440-1545
• Where: Math 622
• Organizer: Xiaorun Wu
• References:

  Main References:

  1. Congruences of cusp forms and special values of their zeta functions, Inventiones Math. 63 (1981), 225–261. Download paper here Inventiones Mathematicae

  2. Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms, Inventiones Math. 85 (1986), 545–613. Download paper here Inventiones Mathematicae

  3. Iwasawa modules attached to congruences of cusp forms, Ann. Scient. Ec. Norm. Sup. 4th series 19 (1986), 231–273. Download paper here Annales scientifiques de l'École Normale Supérieure
  

  4. Modules of congruence of Hecke algebras and L–functions associated with cusp forms, Amer. J. Math. 110 (1988), 323–382.Download paper here American Journal of Mathematics

  5. On p–adic Hecke algebras for GL2 over totally real fields, Ann. of Math. 128 (1988), 295–384.Download paper here Annals of Mathematics

  6. On nearly ordinary Hecke algebras for GL(2) over totally real fields, Advanced Studies in Pure Math. 17 (1989), 139–169. Download paper here Advanced STudies in Pure Mathematics

  7. p–adic L–functions for base change lifts of GL2 to GL3, in Proc. of Conference on “Automorphic forms, Shimura varieties, and L–functions”, Perspectives in Math. 11 (1990) 93–142. (digital version available in Columbia Library upon request)

  8. p-Ordinary cohomology groups for SL(2) over number fields, Duke Math. J. 69 (1993), 259–314.Download paper hereDuke Mathe Journal

  9. p–Adic ordinary Hecke algebras for GL(2), Ann. l’insitut Fourier 44 (1994), 1289–1322. Download paper hereAnnals l'institut Fourier

  10. On \(\Lambda\)-adic forms of half integral weight for \(SL(2)/_{\mathbb Q}\), in Number Theory, Paris, Lecture notes series of LMS 215 (1995), 139–166 Download paper hereLecture notes series of LMS

  11. Adjoint modular Galois representations and their Selmer groups, joint work with J. Tilouine and E. Urban, Proc. Natl. Acad. Sci. USA 94 (1997), 11121–11124. Download paper hereLecture notes series of LMS

  12. Adjoint Selmer groups as Iwasawa modules, Israel J. Math. 120 (2000), 361–427.Download paper here Israel Journal of Mathematics

  13. Control theorems for coherent sheaves on Shimura varieties of PEL–type, Journal of the Inst. of Math. Jussieu 1 (2002), 1–76. Download paper here Israel Journal of Mathematics

  14. Higher Hida and Coleman Theories on the Modular Curve (G. Boxer & V. Pilloni)Download paper here

   

  Books on Hida Theory:

  [HIDA] Elementary Theory of L–functions and Eisenstein series, 1993, Cambridge University Press, Book

  [HIDA2]H. Hida, p-Adic Automorphic Forms on Shimura Varieties, Springer Monographs in Mathematics, 2004, Springer

   

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

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

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