Theta series and number fields: theorems and experiments

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 θ K ∈ M d , n , where 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 θ K is a complete invar...

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Bibliographic Details
Published in:The Ramanujan journal 2021-11, Vol.56 (2), p.613-630
Main Authors: Barquero-Sanchez, Adrian, Mantilla-Soler, Guillermo, Ryan, Nathan C.
Format: Article
Language:English
Subjects:
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Isomorphism
Mathematics
Mathematics and Statistics
Number Theory
Numbers
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Summary: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 θ K ∈ M d , n , where 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 θ K is a complete invariant for K . We also investigate whether or not the collection of θ -series, associated to the set of isomorphism classes of quartic number fields of a fixed squarefree discriminant d , is a linearly independent subset of 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.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-021-00394-y