A proof of a conjecture on trace-zero forms and shapes of number fields
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RESEARCH
A proof of a conjecture on trace-zero forms and shapes of number fields Guillermo Mantilla-Soler2,3* * Correspondence:
[email protected] Department of Mathematics, Universidad Konrad Lorenz, Bogotá, Colombia Full list of author information is available at the end of the article 2
and Carlos Rivera-Guaca1 Abstract In 2012 the first named author conjectured that totally real quartic fields of fundamental discriminant are determined by the isometry class of the integral trace zero form; such conjecture was based on computational evidence and the analog statement for cubic fields which was proved using Bhargava’s higher composition laws on cubes. Here, using Bhargava’s parametrization of quartic fields we prove the conjecture by generalizing the ideas used in the cubic case. Since at the moment, for arbitrary degrees, there is nothing like Bhargava’s parametrizations we cannot deal with degrees n > 5 in a similar fashion. Nevertheless, using some of our previous work on trace forms we generalize this result to higher degrees. We show that if n is an integer bigger than 2 such that (Z/nZ)∗ is a cyclic group, the shape is a complete invariant for degree n number fields that are totally real and have fundamental discriminant.
1 Introduction Let K be a number field of degree n := [K : Q] and let OK be its maximal order. The trace zero module of OK is the Z-submodule of OK given by the kernel of the trace map, i.e., OK0 = K 0 ∩ OK where K 0 := {x ∈ K : trK /Q (x) = 0}. The integral trace-zero form of K is the isometry class of the rank n−1 quadratic Z-module (OK0 , trK /Q ) given by restricting the trace pairing from OK × OK to OK0 × OK0 . It is clear that the isometry class of the quadratic module (OK0 , trK /Q ) determines the field K for n = 1, 2. This is not the case for cubic fields, see [11, Sect. 3]. However, it can be shown, using Delone-Faddeev-Gross parametrization of cubic rings and Bhargava’s higher composition laws on cubes, that for totally real cubic fields of fundamental discriminant (OK0 , trK /Q ) determines the field (see [11, Theorem 6.5]). In [10] the first named author conjectured that the above property of the trace zero form is not only particular of degrees less than 4 but also works for quartic fields (see [10, Conjecture 2.10]). In Sect. 4 we prove such conjecture via Bhargava’s parametrization of quartic rings: Theorem (cf. Sect. 4). Let K be a totally real quartic number field with fundamental discriminant. If L is a tamely ramified number field such that an isomorphism of quadratic modules (OK0 , trK /Q ) ∼ = (OL0 , trL/Q )
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G. Mantilla-Soler, C. Rivera-Guaca Res. Number Theory (2020)6:35
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exists, then K ∼ = L. Another quadratic invariant, with a more geometric interpretation and closely related to the trace zero form, that has been studied by several authors is the shape of K . Endow K with the real-valued Q-bilinear form bK whose associated quadratic form is given by bK (x, x) := |σ (x)|2 . σ :K →C
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