##### Department of Mathematics,

University of California San Diego

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### Special Colloquium

## Romyar Sharifi

#### University of Arizona

## Modular symbols and arithmetic

##### Abstract:

In 1844, Kummer showed that cyclotomic integer rings can fail to be principal ideal domains. In 1976 and 1984, Ribet and Mazur-Wiles used Galois representations attached to modular forms to partially describe class groups that measure the extent of this failure. The exact structure of these class groups remains a mystery to this day. I will explain how to attach ideal classes to geodesics in the complex upper half-plane. A conjecture of mine states these two constructions are inverse to each other in an appropriate sense. I hope to motivate a broader philosophy, developed jointly with Takako Fukaya and Kazuya Kato, that certain arithmetic objects attached to Galois representations of global fields can be described using higher-dimensional modular symbols.

Host: Cristian Popescu

### December 3, 2015

### 3:00 PM

### AP&M 6402

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