We obtain upper and lower bounds, of the expected order of magnitude, regarding
Manin's conjecture for any quartic del Pezzo surface over Q that contains
a rational conic defined over Q.
We use fibrations to translate the problem into an analytic problem regarding
divisor sums. These sums cannot be directly evaluated and we use algebraic
arguments to convert them into averages of certain arithmetic functions in
number fields. These averages are then bounded by adopting an important technique
of Nair and Tenenbaum in the new setting.
This is joint work with T. Browning.