$\ell$-Adic Abel-Jacobi map for abelian $t$-modules
HTML articles powered by AMS MathViewer
- by Yujia Qiu;
- Proc. Amer. Math. Soc.
- DOI: https://doi.org/10.1090/proc/17201
- Published electronically: April 17, 2025
- HTML | PDF | Request permission
Abstract:
We study the function field analogue of the $\ell$-adic Abel-Jacobi map for an abelian $t$-module $E$ defined over a function field $L$ of positive characteristic. When the Mordell-Weil theorem (in the sense of Poonen) holds for the $L$-valued points of $E$, we show that our Abel-Jacobi map is injective and even an isomorphism when $L$ is finite.References
- Emiliano Ambrosi, Perfect points of abelian varieties, Compos. Math. 159 (2023), no. 11, 2261–2278. MR 4638607, DOI 10.1112/s0010437x23007467
- Greg W. Anderson, $t$-motives, Duke Math. J. 53 (1986), no. 2, 457–502. MR 850546, DOI 10.1215/S0012-7094-86-05328-7
- W. Dale Brownawell and Matthew A. Papanikolas, A rapid introduction to Drinfeld modules, $t$-modules, and $t$-motives, $t$-motives: Hodge structures, transcendence and other motivic aspects, EMS Ser. Congr. Rep., EMS Publ. House, Berlin, [2020] ©2020, pp. 3–30. MR 4321964
- David Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Springer-Verlag, Berlin, 1996. MR 1423131, DOI 10.1007/978-3-642-61480-4
- Urs Hartl and Ann-Kristin Juschka, Pink’s theory of Hodge structures and the Hodge conjecture over function fields, $t$-motives: Hodge structures, transcendence and other motivic aspects, EMS Ser. Congr. Rep., EMS Publ. House, Berlin, [2020] ©2020, pp. 31–182. MR 4321965
- Uwe Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988), no. 2, 207–245. MR 929536, DOI 10.1007/BF01456052
- Peter Jossen, On the arithmetic of 1-motives, Ph.D. Thesis, Central European University, 2009 https://www.jossenpeter.ch/PdfDvi/Dissertation.pdf.
- Yen-Liang Kuan, The Mordell-Weil theorem for certain $t$-modules, Proc. Amer. Math. Soc. 151 (2023), no. 3, 989–999. MR 4531633, DOI 10.1090/proc/16172
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, NJ, 1980. MR 559531
- Richard Pink, Kummer theory for Drinfeld modules, Algebra Number Theory 10 (2016), no. 2, 215–234. MR 3477742, DOI 10.2140/ant.2016.10.215
- Bjorn Poonen, Local height functions and the Mordell-Weil theorem for Drinfel′d modules, Compositio Math. 97 (1995), no. 3, 349–368. MR 1353279
- Wayne Raskind, Higher $l$-adic Abel-Jacobi mappings and filtrations on Chow groups, Duke Math. J. 78 (1995), no. 1, 33–57. MR 1328751, DOI 10.1215/S0012-7094-95-07803-X
- The Stacks Project Authors, The stacks project, 2024 https://stacks.math.columbia.edu.
- Lenny Taelman, 1-$t$-motifs, $t$-motives: Hodge structures, transcendence and other motivic aspects, EMS Ser. Congr. Rep., EMS Publ. House, Berlin, [2020] ©2020, pp. 417–439. MR 4321971
- John Tate, Relations between $K_{2}$ and Galois cohomology, Invent. Math. 36 (1976), 257–274. MR 429837, DOI 10.1007/BF01390012
Bibliographic Information
- Yujia Qiu
- Affiliation: Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, People’s Republic of China
- ORCID: 0009-0002-8285-0799
- Email: woodchiu@gmail.com
- Received by editor(s): July 31, 2024
- Received by editor(s) in revised form: December 18, 2024, December 28, 2024, and December 29, 2024
- Published electronically: April 17, 2025
- Additional Notes: The author was supported by the Fundamental Research Foundation (8402/217045519) for Universities of Heilongjiang University.
- Communicated by: Rachel Pries
- © Copyright 2025 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
- MSC (2020): Primary 11G09; Secondary 11R58
- DOI: https://doi.org/10.1090/proc/17201