##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Jason Colwell

#### Univ. of Southern California

## The Conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication by a nonmaximal order

##### Abstract:

Gross has refined the Birch Swinnerton Dyer Conjecture in the case of an elliptic curve with complex multiplication by a nonmaximal order. Gross Conjecture has been reformulated in the language of derived categories and determinants of perfect complexes. Burns and Flach have realized that this immediately leads to a refinement of Gross Conjecture. The conjecture is now expressed as a statement concerning a generator of the image of a map of 1-dimensional modules. This conjecture is proved by a construction which shows it to follow from the Explicit Reciprocity Law and Rubin`s Main Conjecture.

Host: Cristian Popescu

### January 29, 2004

### 2:00 PM

### AP&M 6438

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