Explicit Tamagawa numbers for certain algebraic tori over number fields
HTML articles powered by AMS MathViewer
- by Thomas Rüd HTML | PDF
- Math. Comp. 91 (2022), 2867-2904 Request permission
Abstract:
Given a number field extension $K/k$ with an intermediate field $K^+$ fixed by a central element of $\operatorname {Gal}(K/k)$ of prime order $p$, there exists an algebraic torus over $k$ whose rational points are elements of $K^\times$ sent to $k^\times$ by the norm map $N_{K/K^+}$. The goal is to compute the Tamagawa number such a torus explicitly via Ono’s formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when $K/k$ is Galois. Partial results including the numerator of Ono’s formula are given when the extension is not Galois, or more generally when the torus is defined by an étale algebra.
We also present tools developed in SageMath for this purpose, allowing us to build and compute the cohomology and explore the local-global principles for such an algebraic torus.
Particular attention is given to the case when $[K:K^+]=2$ and $K$ is a CM-field. This case corresponds to maximal tori in $\mathrm {GSp}_{2n}$, and most examples will be in that setting. This is motivated by the application to abelian varieties over finite fields and the Hasse principle for bilinear forms.
References
- J. Achter, S. A. Altug, L. Garcia, and J. Gordon, Counting abelian varieties over finite fields via Frobenius densities, Preprint, arXiv:1905.11603, 2019, With an appendix by Wen-Wei Li and Thomas Rüd.
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
- Anne Cortella, The Hasse principle for the similarities of bilinear forms, Algebra i Analiz 9 (1997), no. 4, 98–118; English transl., St. Petersburg Math. J. 9 (1998), no. 4, 743–762. MR 1604012
- Jia-Wei Guo, Nai-Heng Sheu, and Chia-Fu Yu, Class numbers of CM algebraic tori, CM abelian varieties and components of unitary Shimura varieties, Nagoya Math. J. 245 (2022), 74–99. MR 4413363, DOI 10.1017/nmj.2020.31
- Everett W. Howe, Constructing distinct curves with isomorphic Jacobians, J. Number Theory 56 (1996), no. 2, 381–390. MR 1373560, DOI 10.1006/jnth.1996.0026
- Alexander Hulpke, Constructing transitive permutation groups, J. Symbolic Comput. 39 (2005), no. 1, 1–30. MR 2168238, DOI 10.1016/j.jsc.2004.08.002
- Akinari Hoshi and Aiichi Yamasaki, Rationality problem for algebraic tori, Mem. Amer. Math. Soc. 248 (2017), no. 1176, v+215. MR 3685951, DOI 10.1090/memo/1176
- Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR 1632779, DOI 10.1090/coll/044
- The LMFDB Collaboration, The L-functions and modular forms database, http://www.lmfdb.org, 2020 [Online; accessed 28 July 2021].
- Martin Lorenz, Multiplicative invariant theory, Encyclopaedia of Mathematical Sciences, vol. 135, Springer-Verlag, Berlin, 2005. Invariant Theory and Algebraic Transformation Groups, VI. MR 2131760
- Roger C. Lyndon, The cohomology theory of group extensions, Duke Math. J. 15 (1948), 271–292. MR 25468
- P.-X. Liang, H.-Y. Yang, and C.-F. Yu, On Tamagawa numbers of $CM$ tori, Preprint, arXiv:2109.04121, 2021.
- Takashi Ono, On the Tamagawa number of algebraic tori, Ann. of Math. (2) 78 (1963), 47–73. MR 156851, DOI 10.2307/1970502
- Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR 1278263
- Joseph J. Rotman, An introduction to the theory of groups, 4th ed., Graduate Texts in Mathematics, vol. 148, Springer-Verlag, New York, 1995. MR 1307623, DOI 10.1007/978-1-4612-4176-8
- T. Rüd, Computations of Tate-Shafarevich for certain maximal symplectic tori, https://toadrush.github.io/tamagawa-cmtori/, 2020 [Online; accessed 02 June 2022].
- V. E. Voskresenskiĭ, Adèle groups and Siegel-Tamagawa formulas, J. Math. Sci. 73 (1995), no. 1, 47–113. Algebra. 1. MR 1317607, DOI 10.1007/BF02366354
- V. E. Voskresenskiĭ, Algebraic groups and their birational invariants, Translations of Mathematical Monographs, vol. 179, American Mathematical Society, Providence, RI, 1998. Translated from the Russian manuscript by Boris Kunyavski [Boris È. Kunyavskiĭ]. MR 1634406, DOI 10.1090/mmono/179
Additional Information
- Thomas Rüd
- Affiliation: Department of Mathematics, University of British Columbia, British Columbia V6T 1Z2, Canada
- ORCID: 0000-0002-7356-1234
- Email: tompa.rud@gmail.com
- Received by editor(s): August 19, 2021
- Received by editor(s) in revised form: April 5, 2022, and June 2, 2022
- Published electronically: August 9, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp. 91 (2022), 2867-2904
- MSC (2020): Primary 11E72
- DOI: https://doi.org/10.1090/mcom/3771
- MathSciNet review: 4473106