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Theta series and number fields: theorems and experiments
The Ramanujan Journal ( IF 0.6 ) Pub Date : 2021-03-01 , DOI: 10.1007/s11139-021-00394-y
Adrian Barquero-Sanchez , Guillermo Mantilla-Soler , Nathan C. Ryan

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.



中文翻译:

Theta系列和数字字段:定理和实验

dn为正整数,令K为判别d和度n的全实数字段。我们在{\ mathcal {M}} _ {d,n} \)中构造一个theta系列\(\ theta _ {K} \,其中\({\ mathcal {M}} _ {d,n} \)是由nd定义的模块化形式的空间。此外,如果d为无平方且n最多为4,则\(\ theta _ {K} \)K的完全不变式。我们还将调查\(\ theta \)的集合与固定平方自由判别式d的四次数域的同构类集相关联的-系列是\({\ mathcal {M}} _ {d,4} \)的线性独立子集。如果数域的次数小于或等于3,这就是事实。我们提供了计算和启发式的证据,表明在4级时,这些theta系列也应该是独立的。

更新日期:2021-03-01
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