A proof of a conjecture on trace-zero forms and shapes of number fields


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 \((\mathbb {Z}/n\mathbb {Z})^{*}\) is a cyclic group, the shape is a complete invariant for degree n number fields that are totally real and have fundamental discriminant.

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Correspondence to Guillermo Mantilla-Soler.

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We would like to thank the referee for the careful reading of the paper, and for their helpful comments.

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Mantilla-Soler, G., Rivera-Guaca, C. A proof of a conjecture on trace-zero forms and shapes of number fields. Res. number theory 6, 35 (2020). https://doi.org/10.1007/s40993-020-00210-4

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