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Rencontres de théorie analytique des nombres

Diagonal quadric surfaces with a rational point

Salle Grisvard, IHP, Paris

I will discuss the family of diagonal quadric surfaces parameterised by $\left\{Y: wx=yz\right\}$. This family was first studied by Browning, Lyczak, Sarapin who discovered that an unexpectedly large number of these diagonal quadric surfaces have a rational point and attributed this odd behaviour to the presence of thin sets in $Y(\mathbb{Q})$ whose corresponding varieties have obvious rational points. In recent work, I have shown that additional unusual behaviour is revealed once this thin set is removed from $Y$ by providing an asymptotic for the corresponding counting problem. This talk will focus on the character sum methods used to prove this result, focussing in particular on necessary modifications to the large sieve for quadratic characters.