Seminar on Algebraic Geometry : Average rank of elliptic curves and geometry of numbers

2015-9-1 10:00 - 2015-9-2 16:00
Faculty of Science Building #3Room413
Takashi Taniguchi (Kobe Univ.), Yasuhiro. Ishizuka (Kyoto Univ.)
It was proved by Bhargava and Shankar that the average rank of elliptic curves over the rational number field, when ordered by height, is less than 1.

Their main theorem is the determination of the average size of the 2, 3, 4 and 5 Selmer groups (which are respectively 3, 4, 7 and 6), and the result of the average rank mentioned above is one consequence of this theorem. Their approach is to count integral orbits in certain coregular spaces via the geometry of numbers.

In this lecture, we give an outline of their proof.