Abstract
We discuss how to compute the Brauer group of the product of two elliptic curves over a finite field. Specifically, we apply the Artin–Tate formula for abelian surfaces to give a simple formula for computing the order of the Brauer group of the product of two elliptic curves. Our formula enables to compute the order of the Brauer group using traces of Frobenius maps on elliptic curves and the discriminant of the group of homomorphisms between two elliptic curves. In addition, for the product of non-isogenous elliptic curves, we give an algorithm for computing torsion subgroups of a certain Galois cohomology that can be embedded as subgroups of the Brauer group.
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Acknowledgements
We would like to thank Professor Kazuhiro Yokoyama for his useful comments. This work was supported by JST CREST Grant Number JPMJCR2113 and JSPS KAKENHI Grant Number JP23K18469.
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Katayama, A., Yasuda, M. Computing the Brauer group of the product of two elliptic curves over a finite field. Japan J. Indust. Appl. Math. 41, 919–943 (2024). https://doi.org/10.1007/s13160-023-00638-y
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DOI: https://doi.org/10.1007/s13160-023-00638-y