Wenhan Dai

Hida’s Theory on p-adic Modular Forms

In this mini-course, we give an introduction to Hida’s construction of analytic families of ordinary p-adic modular forms and their associated Galois representations. We will explain Hida’s control theorems for ordinary p-adic modular forms and show how these theorems have been useful in relating certain Hecke algebras with universal Galois deformation rings. We will also explain examples and open problems on these topics.

• Lecturer: Bin Zhao (Capital Normal University).
• Time & Venue:
  • March 2/3/10/16/17, 10:00am-12:00pm, 82J12.
  • (Specially) March 9, 13:30pm-15:00pm, 77201.

Syllabus

The collated notes is now available with gaps and typos fixed.

• (3/2) Lecture 1: Introduction.
• (3/3) Lecture 2: Cohomological approach to modular forms.
• (3/9) Lecture 3: Geometric approach to modular forms.
• (3/10) Lecture 4: “False” modular forms à la Deligne.
• (3/16) Lecture 5: Ordinary family of p-adic modular forms.
• (3/17) Lecture 6: Hecke algebras and universal Galois deformation.

References

1. Haruzo Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. Ec. Norm. Sup. 4th series 19 (1986), 231-273;
2. Haruzo Hida, Galois representations into $\mathrm{GL}_2(\mathbb{Z}_p)$ attached to ordinary cusp forms, Inventiones Math. 85 (1986), 545-613;
3. Haruzo Hida, Elementary Theory of $L$-functions and Eisenstein series, LMSST 26, Cambridge University Press, Cambridge, 1993;
4. Haruzo Hida, Geometric Modular Forms and Elliptic Curves, 2nd edition, World Scientific Publishing Company, Singapore, 2011.
5. Haruzo Hida, Hecke fields of analytic families of modular forms, J. Amer. Math. Soc. 24 (2011), 51-80.