##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Jennifer Balakrishnan

#### Boston University

## Databases of elliptic curves ordered by height

##### Abstract:

Elliptic curves defined over the rational numbers are of great interest in modern number theory. The rank of an elliptic curve is a crucial invariant, with many open questions about its behavior. In particular, there is great interest in the ``average'' rank of an elliptic curve. The minimalist conjecture is that the average rank should be 1/2. In 2007, Bektemirov, Mazur, Stein, and Watkins [BMSW], using well-known databases of elliptic curves, set out to numerically compute the average rank of elliptic curves, ordered by conductor. They found that ``there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either.'' In joint work with Ho, Kaplan, Spicer, Stein, and Weigandt, we have assembled a new database of elliptic curves ordered by height. I will describe the database and examine some of the questions posed by [BMSW]. I will also discuss ongoing work by a team of undergraduates at Oxford on similar questions about families of elliptic curves.

Host: Kiran Kedlaya

### February 9, 2017

### 1:00 PM

### AP&M 7321

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