**SMS scnews item created by Leo Tzou at Mon 20 Oct 2014 1825**

Type: Seminar

Distribution: World

Expiry: 2 Oct 2015

**Calendar1: 14 Nov 2014 1430-1530**

**CalLoc1: Carslaw 275**

CalTitle1: Joint Colloquium: Dokchitser -- Average ranks of curves

Auth: leo@dyn-129-78-251-29.wirelessguest.usyd.edu.au (ltzo2369) in SMS-WASM

### Joint Colloquium

# Average ranks of curves

### Tim Dokchitser

Location:

** (Carslaw Lect. 275, Sydney Uni) **

Speaker:

** Tim Dokchitser, Bristol**

Title:

**Average ranks of curves **

Abstract:

Rational points and ranks of elliptic curves are subjects of many important conjectures, such as the Birch-Swinnerton-Dyer conjecture and conjectures on `typical’ and `maximal’ ranks. In a recent series of papers, Manjul Bhargava and his collaborators made several fundamental breakthroughs on average ranks and Selmer ranks of elliptic curves over the rationals. In particular, they prove that the average rank of all elliptic curves over Q is less than 1 (this average was not even known to be bounded), and deduce that a positive proportion of elliptic curves satisfy the Birch-Swinnerton-Dyer conjecture. This beautiful work combines techniques from invariant theory, Selmer groups, geometry and analytic number theory. In this lecture I will give a brief and elementary overview of their approach and explain some related results.

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