[Déformations formelles de formes modulaires et valeurs multiples de fonctions $\mathrm{L}$]
We relate analytically defined deformations of modular curves and modular forms from the literature to motivic periods via cohomological descriptions of deformation theory. Leveraging cohomological vanishing results, we prove the existence and essential uniqueness of deformations, which we make constructive via established Lie algebraic arguments and a notion of formal Lie deformations. Further, we construct a canonical and a totally holomorphic universal family of deformations of modular forms of all weights, which we obtain from the canonical cocycle associated with periods on the moduli space $\mathcal{M}_{1,1}$. Our uniqueness statement shows that non-critical multiple $\mathrm{L}$-values, which appear in our deformations but are a priori non-geometric, are genuinely linked to deformations. Our work thus suggests a new geometric perspective on them.
Nous établissons un lien entre les déformations de courbes modulaires et de formes modulaires définies analytiquement, telles qu’elles apparaissent dans la littérature, et les périodes motiviques, par le biais de descriptions cohomologiques issues de la théorie des déformations. En nous appuyant sur des résultats d’annulation cohomologiques, nous prouvons l’existence et l’unicité essentielle des déformations, que nous rendons constructives à l’aide d’arguments bien établis d’algèbres de Lie et d’une notion de déformation formelle de Lie. De plus, nous construisons une famille universelle canonique et totalement holomorphe de déformations de formes modulaires de tous poids, que nous obtenons à partir du cocycle canonique associé aux périodes sur l’espace de modules $\mathcal{M}_{1,1}$. Notre énoncé d’unicité montre que les valeurs multiples non critiques de fonctions $\mathrm{L}$, qui apparaissent dans nos déformations mais sont a priori non géométriques, sont véritablement liées aux déformations. Nos travaux suggèrent ainsi une nouvelle perspective géométrique sur celles-ci.
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Keywords: Non-critical multiple $\mathrm{L}$-values, cohomological Lie formal deformations
Mots-clés : Valeurs multiples non critiques de fonctions $\mathrm{L}$, déformations formelles de Lie cohomologiques
CC-BY 4.0
Adam Keilthy; Martin Raum. Formal deformations of modular forms and multiple L-values. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 853-890. doi: 10.5802/jep.338
@article{JEP_2026__13__853_0,
author = {Adam Keilthy and Martin Raum},
title = {Formal deformations of modular forms and multiple {L-values}},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {853--890},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.338},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.338/}
}
TY - JOUR AU - Adam Keilthy AU - Martin Raum TI - Formal deformations of modular forms and multiple L-values JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 853 EP - 890 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.338/ DO - 10.5802/jep.338 LA - en ID - JEP_2026__13__853_0 ER -
%0 Journal Article %A Adam Keilthy %A Martin Raum %T Formal deformations of modular forms and multiple L-values %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 853-890 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.338/ %R 10.5802/jep.338 %G en %F JEP_2026__13__853_0
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