The number of quartic 𝐷₄-fields with monogenic cubic resolvent ordered by conductor

Tsang, Cindy; Xiao, Stanley

doi:10.1090/tran/8260

The number of quartic $D_4$-fields with monogenic cubic resolvent ordered by conductor
HTML articles powered by AMS MathViewer

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

A. Altug, A. Shankar, I. Varma, and K. Wilson, The number of quartic $D_4$-fields ordered by conductor, arXiv:1704.01729v1 [math.NT].
Andrew Marc Baily, On the density of discriminants of quartic fields, J. Reine Angew. Math. 315 (1980), 190–210. MR 564533, DOI 10.1515/crll.1980.315.190
Manjul Bhargava, Higher composition laws. III. The parametrization of quartic rings, Ann. of Math. (2) 159 (2004), no. 3, 1329–1360. MR 2113024, DOI 10.4007/annals.2004.159.1329
Manjul Bhargava, The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162 (2005), no. 2, 1031–1063. MR 2183288, DOI 10.4007/annals.2005.162.1031
Manjul Bhargava and Arul Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Ann. of Math. (2) 181 (2015), no. 1, 191–242. MR 3272925, DOI 10.4007/annals.2015.181.1.3
B. J. Birch and J. R. Merriman, Finiteness theorems for binary forms with given discriminant, Proc. London Math. Soc. (3) 24 (1972), 385–394. MR 306119, DOI 10.1112/plms/s3-24.3.385
Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and scientists, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised. MR 0277773, DOI 10.1007/978-3-642-65138-0
Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier, Enumerating quartic dihedral extensions of $\Bbb Q$, Compositio Math. 133 (2002), no. 1, 65–93. MR 1918290, DOI 10.1023/A:1016310902973
K. Conrad, Galois groups of cubics and quartics (not in characteristic 2), Online notes, retrieved 24 Nov, 2017. http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/cubicquartic.pdf http://www.math.uconn.edu/$\sim $kconrad/blurbs/galoistheory/cubicquartic.pdf
J. E. Cremona, Reduction of binary cubic and quartic forms, LMS J. Comput. Math. 2 (1999), 64–94. MR 1693411, DOI 10.1112/S1461157000000073
H. Davenport, On a principle of Lipschitz, J. London Math. Soc. 26 (1951), 179–183. MR 43821, DOI 10.1112/jlms/s1-26.3.179
B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, R.I., 1964. MR 0160744
J. B. Friedlander and H. Iwaniec, Square-free values of quadratic polynomials, Proc. Edinb. Math. Soc. (2) 53 (2010), no. 2, 385–392. MR 2653239, DOI 10.1017/S0013091508000989
M. N. Huxley, Exponential sums and lattice points. III, Proc. London Math. Soc. (3) 87 (2003), no. 3, 591–609. MR 2005876, DOI 10.1112/S0024611503014485
Zev Klagsbrun and Robert J. Lemke Oliver, The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion point, Res. Math. Sci. 1 (2014), Art. 15, 10. MR 3375649, DOI 10.1186/s40687-014-0015-4
Kostadinka Lapkova, On the average number of divisors of reducible quadratic polynomials, J. Number Theory 180 (2017), 710–729. MR 3679824, DOI 10.1016/j.jnt.2017.05.002
Kurt Mahler, Zur Approximation algebraischer Zahlen. III, Acta Math. 62 (1933), no. 1, 91–166 (German). Über die mittlere Anzahl der Darstellungen grosser Zahlen durch binäre Formen. MR 1555382, DOI 10.1007/BF02393603
Jin Nakagawa, Binary forms and orders of algebraic number fields, Invent. Math. 97 (1989), no. 2, 219–235. MR 1001839, DOI 10.1007/BF01389040
SageMath, the Sage Mathematics Software System (Version 9.0), The Sage Developers, 2020, https://www.sagemath.org.
C. L. Stewart and S. Y. Xiao, On the representation of $k$-free integers by binary forms, to appear in Rev. Mat. Iberoamericana, DOI 10.4171/rmi/1213.
Cindy Tsang and Stanley Yao Xiao, Binary quartic forms with bounded invariants and small Galois groups, Pacific J. Math. 302 (2019), no. 1, 249–291. MR 4028773, DOI 10.2140/pjm.2019.302.249
Melanie Matchett Wood, Quartic rings associated to binary quartic forms, Int. Math. Res. Not. IMRN 6 (2012), 1300–1320. MR 2899953, DOI 10.1093/imrn/rnr070
Melanie Eggers Matchett Wood, Moduli spaces for rings and ideals, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–Princeton University. MR 2713054