We prove that a pair of integral quadratic forms in 5 or more variables will
simultaneously represent "almost all" pairs of integers that satisfy the
necessary local conditions, provided that the forms satisfy a suitable
nonsingularity condition. In particular such forms simultaneously attain prime
values if the obvious local conditions hold. The proof uses the circle method,
and in particular pioneers a two-dimensional version of a Kloosterman
refinement.Comment: 63 page