Period Relations for Theta Products and Elliptic Integral Moments
Abstract
We construct period polynomial relations for finite systems of theta products stable under modular transformations and use them to derive identities among critical L-values. Our main example is a three-component theta system of weight $5$, whose coupled period polynomials are determined explicitly and yield new cross-form relations among critical values of the associated eta products. These relations are not consequences of the functional equations of the individual forms. We also develop a weight-$4$ system arising from a quadratic twist of conductor $3$ and obtain relations linking critical values of the original and twisted modular forms. Through modular parametrizations by complete elliptic integrals, these L-value identities yield corresponding moment identities. The method gives a unified modular-symbol explanation of several previously known elliptic integral moment relations while producing new critical-value relations between distinct modular forms.
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BibTeX
@misc{bao2026period,
title = {{Period Relations for Theta Products and Elliptic Integral Moments}},
author = {Dianbin Bao},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.26514},
eprint = {2608.26514},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.26514},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Period Relations for Theta Products and Elliptic Integral Moments
AU - Dianbin Bao
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.26514
UR - https://arxiv.org/abs/2608.26514
AB - We construct period polynomial relations for finite systems of theta products stable under modular transformations and use them to derive identities among critical L-values. Our main example is a three-component theta system of weight $5$, whose coupled period polynomials are determined explicitly and yield new cross-form relations among critical values of the associated eta products. These relations are not consequences of the functional equations of the individual forms. We also develop a weight-$4$ system arising from a quadratic twist of conductor $3$ and obtain relations linking critical values of the original and twisted modular forms. Through modular parametrizations by complete elliptic integrals, these L-value identities yield corresponding moment identities. The method gives a unified modular-symbol explanation of several previously known elliptic integral moment relations while producing new critical-value relations between distinct modular forms.
ER -