University of Calgary

On the areas of rational triangles - PIMS Number Theory CRG Distinguished Lecture

Submitted by ccameron on Mon, 01/10/2011 - 3:06pm.
Jan 17 2011 - 3:00pm
Jan 17 2011 - 4:00pm

Dr. Noam Elkies, Harvard University

ICT 114


 By a "rational triangle" we mean a plane triangle whose sides are  rational numbers.  By Heron's formula, there exists such a triangle  of area sqrt(a) if and only if  a > 0  and  x y z (x + y + z) = a  for some rationals x, y, z.  In a 1749 letter to Goldbach,  Euler constructed infinitely many such  (x, y, z)  for any rational  $a$ (positive or not), remarking that it cost him much effort, but not  explaining his method.  We suggest one approach, using only tools  available to Euler, that he might have taken, and use this approach  to construct several other infinite families of solutions.

 We then reconsider the problem as a question in arithmetic geometry:

 xyz(x+y+z) = a  gives a K3 surface, and each family of solutions is  a singular rational curve on that surface defined over Q.

 The structure of the Neron-Severi group of that K3 surface  explains why the problem is unusually hard.  Along the way  we also encounter the Niemeier lattices (the even unimodular  lattices in R^24) and the non-Hamiltonian Petersen graph.