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# 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.