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Brauer-Manin obstructions on Kummer surfaces of products of elliptic curves

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Mohamed Tawfik
King's College London
Fri, 15/09/2023 - 12:05 - 12:25
MPIM Lecture Hall

We discuss Brauer-Manin obstructions to weak approximation on Kummer surfaces of products of CM elliptic curves. In particular, we are interested in the Kummer surfaces $Kum(E^c\times E^d)$ where $E^c$ and $E^d$ are elliptic curves with CM by $\mathbb{Z}[\zeta_3]$, and $\zeta_3$ is a primitive cubic root of unity. First, we use a theorem by Skorobogatov and Zarhin to show that the transcendental Brauer group is finite. Then, we show that the possibly non-trivial p-primary parts of the transcendental Brauer group of the $Kum(E^c\times E^d)$ are for p one of the primes 2,3,5, or 7. Then, we put necessary and sufficient conditions on $c$ and $d$ such that the transcendental Brauer group of $Kum(E^c\times E^d)$ is non-trivial. Moreover, we find generators to the transcendental Brauer groups of orders 5 and 7. And finally, we calculate the Manin pairing and prove that a transcendental element of the Brauer group of the Kummer surface gives rise to Brauer-Manin obstruction to weak approximation on the Kummer surface.

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