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Proceedings of the American Mathematical Society

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Effective bounds on the dimensions of Jacobians covering abelian varieties


Authors: Juliette Bruce and Wanlin Li
Journal: Proc. Amer. Math. Soc. 148 (2020), 535-551
MSC (2010): Primary 11M83, 14G15
DOI: https://doi.org/10.1090/proc/14756
Published electronically: September 20, 2019
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Abstract: We show that any abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective and explicit version of Poonen's Bertini theorem over finite fields, which allows us to show the existence of smooth curves arising as hypersurface sections of bounded degree and genus. Additionally, for simple abelian varieties we prove a better bound. As an application, we show that for any elliptic curve $ E$ over a finite field and any $ n\in \mathbb{N}$, there exist smooth curves of bounded genus whose Jacobians have a factor isogenous to $ E^n$.


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Additional Information

Juliette Bruce
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: juliette.bruce@math.wisc.edu

Wanlin Li
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: wanlin@math.wisc.edu

DOI: https://doi.org/10.1090/proc/14756
Received by editor(s): December 10, 2018
Received by editor(s) in revised form: May 8, 2019, May 22, 2019, and June 14, 2019
Published electronically: September 20, 2019
Additional Notes: The first author was partially supported by the NSF GRFP under grant No. DGE-1256259.
Communicated by: Rachel Pries
Article copyright: © Copyright 2019 American Mathematical Society