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Ternary quadratic forms with rational zeros

John FriedlanderHenryk Iwaniec — 2010

Journal de Théorie des Nombres de Bordeaux

We consider the Legendre quadratic forms ϕ a b ( x , y , z ) = a x 2 + b y 2 - z 2 and, in particular, a question posed by J–P. Serre, to count the number of pairs of integers 1 a A , 1 b B , for which the form ϕ a b has a non-trivial rational zero. Under certain mild conditions on the integers a , b , we are able to find the asymptotic formula for the number of such forms.

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