Contact: Pieter Moree (moree @ mpim-bonn.mpg.de)
A theorem of Serre states that almost all plane conics over Q have no rational point. We prove an analogue of this for a family of conics parametrised by certain elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve. Another way to think about this result is: we show that 0% of points on elliptic curves have a denominator which is a sum of two squares.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/246