Composition of binary quadratic forms over number fields
-
Kristýna Zemková
Abstract
In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.
Acknowledgement
I wish to thank Vítězslav Kala for his excellent guidance and useful suggestions. I also want to thank Professor Rainer Schulze-Pillot for pointing out references [13, 21]. Finally, I would like to express my thanks to the unknown referee, who pointed out the problem with totally imaginary fields.
(Communicated by Milan Paštéka)
- 2
Here we use 𝓝L/K(γ) rather than γγ for the matter of possible generalization to other than quadratic extensions; see Remark 2.20.
References
[1] Bhargava, M.: Higher composition laws I: A new view on Gauss composition, and quadratic generalizations, Ann. of Math. 159 (2004), 217–250.10.4007/annals.2004.159.217Suche in Google Scholar
[2] Blomer, V.—Kala, V.: Number fields without n-ary universal quadratic forms, Math. Proc. Cambridge Philos. Soc. 159(2) (2015), 239–252.10.1017/S030500411500033XSuche in Google Scholar
[3] Blomer, V.—Kala, V.: On the rank of universal quadratic forms over real quadratic fields, Doc. Math. 23 (2018), 15–34.10.4171/dm/611Suche in Google Scholar
[4] Butts, H.—Dulin, B.: Composition of binary quadratic forms over integral domains, Acta Arith. 20(3) (1972), 223–251.10.4064/aa-20-3-223-251Suche in Google Scholar
[5] Butts, H.—Estes, D.: Modules and binary quadratic forms over integral domains, Linear Algebra App. 1(2) (1968), 153–180.10.1016/0024-3795(68)90001-3Suche in Google Scholar
[6] Čech, M.—Lachman, D.—SVOBODA, J.—Tinková, M.—Zemková, K.: Universal quadratic forms and indecomposables over biquadratic fields, Math. Nachr. 292 (2019), 540–555.10.1002/mana.201800109Suche in Google Scholar
[7] Chan, W.—Kim, M.-H.—Raghavan, S.: Ternary universal integral quadratic forms over real quadratic fields, Jpn. J. Math. 22 (1996), 263–273.10.4099/math1924.22.263Suche in Google Scholar
[8] Earnest, A. G.—Khosravani, A.: Universal positive quaternary quadratic lattices over totally real number fields, Mathematika 44(2) (1997), 342–347.10.1112/S0025579300012651Suche in Google Scholar
[9] Edgar, M. H.—Mollin, R.—Peterson, B. L.: Class groups, totally positive units, and squares. Proc. Amer. Math. Soc. 98 (1986), 33–37.10.1090/S0002-9939-1986-0848870-XSuche in Google Scholar
[10] Fröhlich, A.—Taylor, M. J.: Algebraic Number Theory. Cambridge Stud. Adv. Math., Cambridge University Press, 1993.Suche in Google Scholar
[11] Kala, V.: Universal quadratic forms and elements of small norm in real quadratic fields, Bull. Aust. Math. Soc. 94(1) (2016), 7–14.10.1017/S0004972715001495Suche in Google Scholar
[12] Kaplansky, I.: Composition of binary quadratic forms, Studia Math. 31(5) (1968), 523–530.10.1007/978-1-4612-5352-5_18Suche in Google Scholar
[13] Kneser, M.: Composition of binary quadratic forms, J. Number Theory 15(3) (1982), 406–413.10.1016/0022-314X(82)90041-5Suche in Google Scholar
[14] Krásenský, J.—Tinková, M.—Zemková, K.: There are no universal ternary quadratic forms over biquadratic fields, Proc. Edinburgh Math. Soc. 63(3) (2020), 861–912.10.1017/S001309152000022XSuche in Google Scholar
[15] Mann, H. B.: On integral basis, Proc. Amer. Math. Soc. (1958), 167–172.10.1090/S0002-9939-1958-0093502-7Suche in Google Scholar
[16] Mastropietro, M. W.: Quadratic Forms and Relative Quadratic Extensions, Thesis, 2000.Suche in Google Scholar
[17] Milne, J. S.: Algebraic Number Theory, (v3.00), 2008; available at www.jmilne.org/math/.Suche in Google Scholar
[18] Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers. Springer Monographs in Mathematics, Springer Berlin Heidelberg, 2004.10.1007/978-3-662-07001-7Suche in Google Scholar
[19] O'Dorney, E.: Rings of small rank over a Dedekind domain and their ideals, Res. Math. Sci. 3:8 (2016).10.1186/s40687-016-0054-0Suche in Google Scholar
[20] Siegel, C. L.: Sums of mth powers of algebraic integers, Ann. of Math. 46(2) (1945), 313–339.10.2307/1969026Suche in Google Scholar
[21] Towber, J.: Composition of oriented binary quadratic form-classes over commutative rings, Adv. Math. 36(1) (1980), 1–107.10.1016/S0001-8708(80)80002-8Suche in Google Scholar
[22] Wood, M.: Gauss composition over an arbitrary base, Adv. Math. 226(2) (2011), 1756–1771.10.1016/j.aim.2010.08.018Suche in Google Scholar
[23] Zemková, K.: Composition of Quadratic Forms over Number Fields, Master Thesis (2018).10.1515/ms-2021-0057Suche in Google Scholar
[24] Zemková, K.: Composition of Bhargava's cubes over number fields, preprint.Suche in Google Scholar
© 2021 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular Papers
- Semidistributivity and Whitman Property in implication zroupoids
- Composition of binary quadratic forms over number fields
- On Z-Symmetric Rings
- On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences
- Coefficient estimates for Libera type bi-close-to-convex functions
- Oscillation of nonlinear third-order differential equations with several sublinear neutral terms
- On rapidly oscillating solutions of a nonlinear elliptic equation
- Multiplicity of solutions for a class of fourth-order elliptic equations of p(x)-Kirchhoff type
- Existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities
- On absolute double summability methods with high indices
- On the continuity of lattice isomorphisms on C(X, I)
- New fixed point theorems for countably condensing maps with an application to functional integral inclusions
- Common fixed point results under w-distance with applications to nonlinear integral equations and nonlinear fractional Differential Equations
- The form of locally defined operators in waterman spaces
- Conformal vector fields on almost co-Kähler manifolds
- A certain η-parallelism on real hypersurfaces in a nonflat complex space form
- On log-bimodal alpha-power distributions with application to nickel contents and erosion data
- Univariate and bivariate extensions of the generalized exponential distributions
- Pellian equations of special type
Artikel in diesem Heft
- Regular Papers
- Semidistributivity and Whitman Property in implication zroupoids
- Composition of binary quadratic forms over number fields
- On Z-Symmetric Rings
- On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences
- Coefficient estimates for Libera type bi-close-to-convex functions
- Oscillation of nonlinear third-order differential equations with several sublinear neutral terms
- On rapidly oscillating solutions of a nonlinear elliptic equation
- Multiplicity of solutions for a class of fourth-order elliptic equations of p(x)-Kirchhoff type
- Existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities
- On absolute double summability methods with high indices
- On the continuity of lattice isomorphisms on C(X, I)
- New fixed point theorems for countably condensing maps with an application to functional integral inclusions
- Common fixed point results under w-distance with applications to nonlinear integral equations and nonlinear fractional Differential Equations
- The form of locally defined operators in waterman spaces
- Conformal vector fields on almost co-Kähler manifolds
- A certain η-parallelism on real hypersurfaces in a nonflat complex space form
- On log-bimodal alpha-power distributions with application to nickel contents and erosion data
- Univariate and bivariate extensions of the generalized exponential distributions
- Pellian equations of special type
