Modular forms and modular curves.

*(English)*Zbl 0853.11032
Murty, V. Kumar (ed.), Seminar on Fermat’s last theorem. The Fields Institute for Research in Mathematical Sciences, 1993-1994, Toronto, Ontario, Canada. Proceedings. Providence, RI: American Mathematical Society (publ. for the Canadian Mathematical Society). CMS Conf. Proc. 17, 39-133 (1995).

This is an extensive survey introduction to the theory of modular forms and modular curves, with a special focus on those parts of the theory that are relevant to the recent work of Wiles, Taylor, and Ribet on the modularity conjecture and Fermat’s last theorem.

The paper is divided into three parts. The first surveys the classical theory of modular forms in one variable, including the Hecke and \(W\) operators, the \(L\)-function, the functional equation, the theory of newforms, and the multiplicity one theorem. The second part introduces modular curves first as Riemann surfaces and then as moduli spaces, discussing canonical models, Hecke correspondences, compactification of modular curves, and the geometry and arithmetic of the Jacobians of modular curves. The final section returns to modular forms, reinterpreting them first as automorphic representations and then as sections of invertible sheaves on modular curves. A final section discusses the modularity conjecture for elliptic curves.

While essentially no proofs are included, there are abundant references to the literature. A substantial portion of this information had not been gathered in one place before, so this proves to be a very useful survey and reference.

For the entire collection see [Zbl 0828.00033].

The paper is divided into three parts. The first surveys the classical theory of modular forms in one variable, including the Hecke and \(W\) operators, the \(L\)-function, the functional equation, the theory of newforms, and the multiplicity one theorem. The second part introduces modular curves first as Riemann surfaces and then as moduli spaces, discussing canonical models, Hecke correspondences, compactification of modular curves, and the geometry and arithmetic of the Jacobians of modular curves. The final section returns to modular forms, reinterpreting them first as automorphic representations and then as sections of invertible sheaves on modular curves. A final section discusses the modularity conjecture for elliptic curves.

While essentially no proofs are included, there are abundant references to the literature. A substantial portion of this information had not been gathered in one place before, so this proves to be a very useful survey and reference.

For the entire collection see [Zbl 0828.00033].

Reviewer: F.Gouvêa (Waterville)

##### MSC:

11F11 | Holomorphic modular forms of integral weight |

11G18 | Arithmetic aspects of modular and Shimura varieties |

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

14G35 | Modular and Shimura varieties |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

##### Keywords:

Hecke operators; survey; modular forms; modular curves; functional equation; newforms; multiplicity one theorem; Hecke correspondences; automorphic representations; modularity conjecture for elliptic curves
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\textit{F. Diamond} and \textit{J. Im}, in: Seminar on Fermat's last theorem. The Fields Institute for Research in Mathematical Sciences, 1993-1994, Toronto, Ontario, Canada. Proceedings. Providence, RI: American Mathematical Society (publ. for the Canadian Mathematical Society). 39--133 (1995; Zbl 0853.11032)