Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion
Chan S, Hanselman J, Li W. 2019. Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion. The Open Book Series. 2, 173–189.
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Author
Chan, StephanieISTA
;
Hanselman, Jeroen;
Li, Wanlin

Abstract
In 2016, Balakrishnan, Ho, Kaplan, Spicer, Stein and Weigandt produced a database of elliptic curves over Q ordered by height in which they computed the rank, the size of the 2-Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over
Q whose rational torsion subgroup is isomorphic to Z∕2Z×Z∕8Z. Conditional on GRH and BSD, we compute the rank of 92% of the 202,461 curves with parameter height less than 103. We also compute the size of the 2-Selmer group and the Tamagawa product, and prove that their averages tend to infinity for this family.
Publishing Year
Date Published
2019-02-13
Journal Title
The Open Book Series
Publisher
Mathematical Sciences Publishers
Volume
2
Page
173-189
ISSN
eISSN
IST-REx-ID
Cite this
Chan S, Hanselman J, Li W. Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion. The Open Book Series. 2019;2:173-189. doi:10.2140/obs.2019.2.173
Chan, S., Hanselman, J., & Li, W. (2019). Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion. The Open Book Series. Mathematical Sciences Publishers. https://doi.org/10.2140/obs.2019.2.173
Chan, Stephanie, Jeroen Hanselman, and Wanlin Li. “Ranks, 2-Selmer Groups, and Tamagawa Numbers of Elliptic Curves with ℤ∕2ℤ × ℤ∕8ℤ-Torsion.” The Open Book Series. Mathematical Sciences Publishers, 2019. https://doi.org/10.2140/obs.2019.2.173.
S. Chan, J. Hanselman, and W. Li, “Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion,” The Open Book Series, vol. 2. Mathematical Sciences Publishers, pp. 173–189, 2019.
Chan S, Hanselman J, Li W. 2019. Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion. The Open Book Series. 2, 173–189.
Chan, Stephanie, et al. “Ranks, 2-Selmer Groups, and Tamagawa Numbers of Elliptic Curves with ℤ∕2ℤ × ℤ∕8ℤ-Torsion.” The Open Book Series, vol. 2, Mathematical Sciences Publishers, 2019, pp. 173–89, doi:10.2140/obs.2019.2.173.
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