Abstract
Let d and n be positive integers and let K be a totally real number field of discriminant d and degree n. We construct a theta series \(\theta _{K} \in {\mathcal {M}}_{d,n}\), where \({\mathcal {M}}_{d,n}\) is a space of modular forms defined in terms of n and d. Moreover, if d is square free and n is at most 4 then \(\theta _{K}\) is a complete invariant for K. We also investigate whether or not the collection of \(\theta \)-series, associated to the set of isomorphism classes of quartic number fields of a fixed squarefree discriminant d, is a linearly independent subset of \({\mathcal {M}}_{d,4}\). This is known to be true if the degree of the number field is less than or equal to 3. We give computational and heuristic evidence suggesting that in degree 4 these theta series should be independent as well.
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Acknowledgements
We would like to thank the organizers of the workshop Number Theory in the Americas for providing us the opportunity to start this collaboration and the Casa Matemática Oaxaca (CMO-BIRS) for the hospitality. We also would like to thank an anonymous referee for a careful reading of the paper which allowed us to make it more readable.
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Barquero-Sanchez, A., Mantilla-Soler, G. & Ryan, N.C. Theta series and number fields: theorems and experiments. Ramanujan J 56, 613–630 (2021). https://doi.org/10.1007/s11139-021-00394-y
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DOI: https://doi.org/10.1007/s11139-021-00394-y