Date | 2022-03-24 |
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Speaker | Preston Wake |

Dept. | Michigan State University |

Room | 선택 |

Time | 22:00-23:00 |

**일시: 3월 21일, 22일, 24일 10:00PM**

**Zoom meeting: 220 305 8101 (passcode: 196884)**

**Abstract** : In his landmark 1977 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and then Eisenstein series of weight 2 and prime level N. He proved many deep results describing the structure of these congruences, and noted, based on computational evidence, that there is usually only one such cusp form, but sometimes there are several. He asked if there any arithmetic significance to this number of cusp forms. In this series of three lectures, we will address this question and see that this number is significant both algebraically (in terms of Galois cohomology) and analytically (in terms of L-functions). The first lecture will be an introduction to the subject with many examples. The second lecture will focus on the analytic aspects (Eisenstein series and L-functions) and will include generalizations to higher weight. The third lecture will focus on algebraic aspects (Galois representations and Galois cohomology) and will include generalizations to non-prime level. This is joint work with Carl Wang-Erickson.

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