##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Dino Lorenzini

#### Univ. of Georgia, Athens

## Torsion and Tamagawa Numbers

##### Abstract:

Let $A/K$ be an abelian variety over a global field $K$. For each place $v$ of $K$, one associates an integer $c(v)$ called the Tamagawa number of the place, using the reduction of the abelian variety at $v$. Let $c$ denote the product of the $c(v)'s$. Let $t$ denote the order of the torsion subgroup of Mordell-Weil group $A(K)$. The ratio $c/t$ is a factor in the leading term of the L-function of $A/K$ at $s=1$ predicted by the conjecture of Birch and Swinnerton-Dyer. We investigate in this talk possible cancellations in the ratio $c/t$. For elliptic curves over $Q$. the smallest ratio $c/t$ is $1/5$, obtained only by the modular curve $X_1(11)$.

Host: Cristian Popescu

### January 20, 2011

### 2:00 PM

### AP&M 7321

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