Counterexamples to a Conjecture of Ahmadi and Shparlinski

Abstract

Ahmadi-Shparlinski conjectured that every ordinary, geometrically simple Jacobian over a finite field has maximal angle rank. Using the L-Functions and Modular Forms Database, we provide two counterexamples to this conjecture in dimension 4.

Keywords:

• Angle rank
• abelian varieties
• finite fields
• arithmetic geometry

Acknowledgments

We thank the referee for helpful suggestions, including pointing out the reference [8].

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors began this project during the semester program “Computational aspects of the Langlands program” held at ICERM in fall 2015. Subsequent workshops in support of the project were sponsored by the American Institute of Mathematics and the Simons Foundation. In addition Dupuy was partially supported by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 291111/MODAG while working on this project. Kedlaya was supported by NSF (grants DMS-1501214, DMS-1802161), IAS (visiting professorship 2018–2019), UCSD (Warschawski Professorship), and a Guggenheim Fellowship (fall 2015). Roe was supported by Simons Foundation grant 550033. Vincent was partially supported by NSF (grant DMS-1802323).