Poincaré institute

Reinventing rational points

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Squares represented by a product of three ternary quadratic forms, and a homogeneous variant of a method of Swinnerton-Dyer

Yonatan Harpaz

Let \(k\) be a number field. In this talk we will consider K3 surfaces over k which admit a degree 2 map to the projective plain, ramified over a union of three conics. Such surfaces always admit a fibration into curves of genus 1, which one can try to exploit in order to study their rational points. When all three conics are simultaneously diagonalizable the associated Jacobian fibration admits a particularly nice form, rendering it amenable to the descent-fibration method of Swinnerton-Dyer. Assuming finiteness of the relevant Tate-Shafarevich groups, we show that when the coefficients of the diagonal forms are sufficiently generic, the Brauer-Manin obstruction is the only one for the Hasse principle on \(X\). Using that the singular fibers of the fibration all lie over rational points, the dependence on Schinzel's hypothesis can be removed by adapting the method to run with the homogeneous version of that hypothesis, which is known in the case of linear forms thanks to the seminal work of Green—Tao—Ziegler.