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Difference of the modular function ω2(τ), revisited

  • Wei-Lun Tsai
  • Dongxi Ye
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Abstract

Adapting the analytic method of Gross and Zagier, Roskam proved a prime-factorization formula for the norm of the difference of two level-two Weber singular moduli. Independently, Yang and Yin obtained an equivalent formula using Borcherds lifts. More precisely, the formula concerns the norm of ω2((−1+√(d1))/2) − ω2((−1+√(d2))/2) for coprime negative fundamental quadratic discriminants d1,d2≡1pmod 8, where ω2(τ) = 212η(2τ)24/η(τ)24, and η(τ) denotes the Dedekind eta function. In this work, we revisit this formula from the perspective of arithmetic intersection theory and give a new proof.

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Cite

BibTeXRIS
BibTeX
@misc{tsai2026difference,
  title = {{Difference of the modular function $ω_2(τ)$, revisited}},
  author = {Wei-Lun Tsai and Dongxi Ye},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29788},
  eprint = {2608.29788},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29788},
}
RIS
TY  - UNPB
TI  - Difference of the modular function ω_2(τ), revisited
AU  - Wei-Lun Tsai
AU  - Dongxi Ye
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2608.29788
UR  - https://arxiv.org/abs/2608.29788
AB  - Adapting the analytic method of Gross and Zagier, Roskam proved a prime-factorization formula for the norm of the difference of two level-two Weber singular moduli. Independently, Yang and Yin obtained an equivalent formula using Borcherds lifts. More precisely, the formula concerns the norm of ω_2((−1+√(d_1))/2) − ω_2((−1+√(d_2))/2) for coprime negative fundamental quadratic discriminants d_1,d_2≡1pmod 8, where ω_2(τ) = 2^12η(2τ)^24/η(τ)^24, and η(τ) denotes the Dedekind eta function. In this work, we revisit this formula from the perspective of arithmetic intersection theory and give a new proof.
ER  -