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arXiv · 2608.26514

Period Relations for Theta Products and Elliptic Integral Moments

Dianbin Bao
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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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Dianbin Bao. 2026-08-27. Period Relations for Theta Products and Elliptic Integral Moments. https://arxiv.org/abs/2608.26514

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