The number of $S_4$ fields with given discriminant

Juergen Klueners

The number of $S_4$ fields with given discriminant

Number Theory 2007-05-23 v3

Authors: Juergen Klueners

Abstract

We prove that the number of quartic $S_4$ --extensions of the rationals of given discriminant $d$ is $O_\eps(d^{1/2+\eps})$ for all $\eps>0$ . For a prime number $p$ we derive that the dimension of the space of octahedral modular forms of weight 1 and conductor $p$ or $p^2$ is bounded above by $O(p^{1/2}\log(p)^2)$ .

Keywords

class number of quadratic fields prime number theory quadratic forms

Cite

@article{arxiv.math/0411484,
  title  = {The number of $S_4$ fields with given discriminant},
  author = {Juergen Klueners},
  journal= {arXiv preprint arXiv:math/0411484},
  year   = {2007}
}

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