## The number of quartic $D_4$-fields with monogenic cubic resolvent ordered by conductor

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- by Cindy (Sin Yi) Tsang and Stanley Yao Xiao PDF
- Trans. Amer. Math. Soc.
**374**(2021), 1987-2033

## Abstract:

In this paper, we consider maximal and irreducible quartic orders which arise from integral binary quartic forms, via the construction of Birch and Merriman, and whose field of fractions is a quartic $D_4$-field. By a theorem of Wood, such quartic orders may be regarded as quartic $D_4$-fields whose ring of integers has a monogenic cubic resolvent. We shall determine the asymptotic number of such objects when ordered by conductor. We shall also give a lower bound, which we suspect has the correct order of magnitude, and a slightly larger upper bound for the number of such objects when ordered by discriminant. A simplified version of the techniques used allows us to give a count for those elliptic curves with a marked rational 2-torsion point when ordered by discriminant.## References

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## Additional Information

**Cindy (Sin Yi) Tsang**- Affiliation: School of Mathematics (Zhuhai), Sun Yat-Sen University, Tangjiawan, Zhuhai, Guangdong, 519082, China
- MR Author ID: 1136383
- ORCID: 0000-0003-1240-8102
- Email: zengshy26@mail.sysu.edu.cn
**Stanley Yao Xiao**- Affiliation: Department of Mathematics, University of Toronto Bahen Centre, Toronto, Ontario, Canada M5S 2E4
- MR Author ID: 1224579
- ORCID: 0000-0003-2352-3472
- Email: syxiao@math.toronto.edu
- Received by editor(s): November 13, 2018
- Received by editor(s) in revised form: March 31, 2020, and July 19, 2020
- Published electronically: January 12, 2021
- Additional Notes: Part of this research was done while the first author was visiting the second author at the Mathematical Institute at the University of Oxford. She would like to thank the institute for its hospitality during her stay. The visit was supported by the China Postdoctoral Science Foundation Special Financial Grant (Award No.: 2017T100060).
- © Copyright 2021 Cindy (Sin Yi) Tsang, Stanley Yao Xiao
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 1987-2033 - MSC (2020): Primary 11E76, 11R04, 11R45; Secondary 11G05
- DOI: https://doi.org/10.1090/tran/8260
- MathSciNet review: 4216730