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PreprintCC BY 4.0arXiv

Escape and Non-Escape of Mass for Monogenic Binary Cubic Forms

  • Petros Ploumidis
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Abstract

We study the distribution of monogenic integral binary cubic forms f(x,y) with bounded discriminant and bounded monogenizer inside a fundamental domain for the action of GL2(ℤ) on the space of real binary cubic forms. We show that when the monogenizer for f is [±1:0] (i.e., f(±1,0)=1), the corresponding set is fully concentrated in the cusp. We also prove a complementary result, namely, that when the monogenizers [±1:0] (with height bounded in terms of the discriminant bound) are excluded, the resulting set has no escape of mass to the cusp. However, we surprisingly show that despite this non-escape of mass, the points in the fundamental domain are not equidistributed.

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Cite

BibTeXRIS
BibTeX
@misc{ploumidis2026escape,
  title = {{Escape and Non-Escape of Mass for Monogenic Binary Cubic Forms}},
  author = {Petros Ploumidis},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33027},
  eprint = {2609.33027},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33027},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Escape and Non-Escape of Mass for Monogenic Binary Cubic Forms
AU  - Petros Ploumidis
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.33027
UR  - https://arxiv.org/abs/2609.33027
AB  - We study the distribution of monogenic integral binary cubic forms f(x,y) with bounded discriminant and bounded monogenizer inside a fundamental domain for the action of GL_2(ℤ) on the space of real binary cubic forms. We show that when the monogenizer for f is [±1:0] (i.e., f(±1,0)=1), the corresponding set is fully concentrated in the cusp. We also prove a complementary result, namely, that when the monogenizers [±1:0] (with height bounded in terms of the discriminant bound) are excluded, the resulting set has no escape of mass to the cusp. However, we surprisingly show that despite this non-escape of mass, the points in the fundamental domain are not equidistributed.
ER  -