Serre's problem for diagonal conics

Efthymios Sofos

Assume that \(B\) is a large real number and let \(c_1, c_2, c_3\) be three randomly chosen integers in the box \([-B,B]^3\). Consider the probability that the “random” curve \[c_1 X^2+c_2 Y^2+c_3 Z^2=0\] has a non-zero solution \((X,Y,Z)\) in the integers. Serre showed in the 90\'s that this probability is \(\ll (\log B)^{-3/2}\) while Hooley and Guo later proved that it is \(\gg (\log B)^{-3/2}\). In joint work with Nick Rome we prove an asymptotic \(\sim c (\log B)^{-3/2}\), where \(c\) is a positive absolute constant.