##
**Arithmetic of p-adic modular forms.**
*(English)*
Zbl 0641.10024

Lecture Notes in Mathematics, 1304. Berlin etc.: Springer-Verlag. VIII, 121 p.; DM 23.00 (1988).

The first chapter of this monograph gives the basic definitions and properties of p-adic modular forms, mostly following work of Serre and Katz. Particular attention is given to the notion of overconvergence and congruences of classical modular forms. This is a useful outline but it could not be used as a primer, owing to its compression and omission of proofs and examples. Precise references are given for the theorems quoted both here and in the rest of the book.

The second chapter defines the Hecke algebra acting on p-adic modular forms. It then concentrates on the U operator, a sort of Hecke operator at p. An important feature of a p-adic Hecke-eigenform is the p-adic valuation of the eigenvalue \(\lambda\) of U, and this is studied using spectral theory on p-adic Banach spaces. The congruences and overconvergence properties come into play here. A short appendix explains the connection of this study to Hida’s work on the case when \(\lambda\) is a p-adic unit. The main new theorem in this chapter (according to the author) is that the space of overconvergent eigenforms for U with fixed integral weight and fixed valuation for \(\lambda\) is finite dimensional.

The last chapter deals with representations of the Galois group of the algebraic closure of \({\mathbb{Q}}\), some of which are “attached” to modular forms in a standard way. Using the fact that modular forms can be interpreted as certain functions on the Hecke algebra (“duality theory”), the author constructs a “universal modular deformation” of a given absolutely irreducible Galois representation attached to a modular form. He then sees how it sits in the “universal deformation” constructed by Mazur. It is unknown when the two coincide, i.e. whether every deformation of a representation attached to a modular form is itself attached to a modular form. The author shows that the Krull dimension of the modular deformation ring is at least 3, using twists by wild characters of \({\mathbb{Z}}^{\times}_ p\).

The second chapter defines the Hecke algebra acting on p-adic modular forms. It then concentrates on the U operator, a sort of Hecke operator at p. An important feature of a p-adic Hecke-eigenform is the p-adic valuation of the eigenvalue \(\lambda\) of U, and this is studied using spectral theory on p-adic Banach spaces. The congruences and overconvergence properties come into play here. A short appendix explains the connection of this study to Hida’s work on the case when \(\lambda\) is a p-adic unit. The main new theorem in this chapter (according to the author) is that the space of overconvergent eigenforms for U with fixed integral weight and fixed valuation for \(\lambda\) is finite dimensional.

The last chapter deals with representations of the Galois group of the algebraic closure of \({\mathbb{Q}}\), some of which are “attached” to modular forms in a standard way. Using the fact that modular forms can be interpreted as certain functions on the Hecke algebra (“duality theory”), the author constructs a “universal modular deformation” of a given absolutely irreducible Galois representation attached to a modular form. He then sees how it sits in the “universal deformation” constructed by Mazur. It is unknown when the two coincide, i.e. whether every deformation of a representation attached to a modular form is itself attached to a modular form. The author shows that the Krull dimension of the modular deformation ring is at least 3, using twists by wild characters of \({\mathbb{Z}}^{\times}_ p\).

Reviewer: A.Ash

### MSC:

11F33 | Congruences for modular and \(p\)-adic modular forms |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11F80 | Galois representations |

14H52 | Elliptic curves |

11F11 | Holomorphic modular forms of integral weight |

11R39 | Langlands-Weil conjectures, nonabelian class field theory |