Abstract
We compute the Grothendieck group of the category of abelian varieties over an algebraically closed field k. We also compute the Grothendieck group of the category of A-isotypic abelian varieties, for any simple abelian variety A, assuming k has characteristic 0, and for any elliptic curve A in any characteristic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cox, D.A.: Primes of the Form \(x^2 + ny^2\). A Wiley-Interscience Publication. Wiley, New York (1989)
Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Semin. Univ. Hamburg 14(1), 197–272 (1941)
Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73(3), 349–366 (1983)
Gille, P., Szamuely, T.: Central Simple Algebras and Galois Cohomology. Cambridge Studies in Advanced Mathematics, vol. 101. Cambridge University Press, Cambridge (2006)
Jordan, B.W., Keeton, A.G., Poonen, B., Rains, E.M., Shepherd-Barron, N., Tate, J.T.: Abelian varieties isogenous to a power of an elliptic curve (2016). arXiv:1602.06237
Kani, E.: Products of CM elliptic curves. Collect. Math. 62(3), 297–339 (2011)
Kohel, D.: Endomorphism Rings of Elliptic Curves over Finite Fields. PhD thesis, University of California at Berkeley, Berkeley (1996)
Milne, J.S.: Extensions of abelian varieties defined over a finite field. Invent. Math. 5(1), 63–84 (1968)
Mumford, D.: Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Oxford University Press, Oxford (1970)
Oort, F.: Commutative Group Schemes. Lecture Notes in Mathematics, vol. 15. Springer, Berlin (1966)
Reiner, I.: Maximal Orders. London Mathematical Society Monographs, vol. 5. Academic Press, London (1975)
Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106, 2nd edn. Springer, Dordrecht (2009)
Acknowledgements
The author thanks Julian Rosen for suggesting the proof of Theorem 1.1 and for other conversations on this topic. He also thanks Taylor Dupuy, Yuri Zarhin, and the anonymous referee for their helpful feedback.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shnidman, A. Grothendieck groups of categories of abelian varieties. European Journal of Mathematics 3, 507–519 (2017). https://doi.org/10.1007/s40879-017-0159-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40879-017-0159-z