Effective Hasse principle for the intersection of two quadrics
  We consider a smooth system of two homogeneous quadratic equations over the rationals in at least 13 variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over the reals. In this paper, we give an explicit algorithm to decide whether a nonzero rational solution exists, and if so, to compute one. 
Tony Quertier
math.NT