[L’opérateur thêta et l’opérateur de Fontaine sur les courbes modulaires]
Inspired by [Pan26], we give a new proof that for an overconvergent modular eigenform $f$ of weight $1+k$ with $k\in \mathbb{Z}_{\ge 1}$, assuming that its associated Galois representation $\rho _{f}:\mathrm{Gal}_{\mathbb{Q}}\rightarrow \mathrm{GL}_{2}(\overline{\mathbb{Q}}_{p})$ is irreducible, then $f$ is classical if and only if the associated Galois representation $\rho _{f}$ is de Rham at $p$. For the proof, we prove that theta operator $\theta ^{k}$ coincides with Fontaine operator in a suitable sense.
En nous inspirant de [Pan26], nous donnons une nouvelle preuve du fait que pour une forme modulaire surconvergente $f$ de poids $1+k$ avec $k\in \mathbb{Z}_{\ge 1}$, si sa représentation galoisienne associée $\rho _{f}:\mathrm{Gal}_{\mathbb{Q}}\rightarrow \mathrm{GL}_{2}(\overline{\mathbb{Q}}_{p})$ est irréductible, alors $f$ est classique si et seulement si $\rho _{f}$ est de Rham en $p$. Pour ce faire, nous démontrons que l’opérateur thêta $\theta ^{k}$ coïncide avec l’opérateur de Fontaine en un sens convenable.
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Keywords: Perfectoid modular curve, Fontaine operator, classicality
Mots-clés : Courbe modulaire perfectoïde, opérateur de Fontaine, classicité
Yuanyang Jiang 1
CC-BY 4.0
@article{JEP_2026__13__349_0,
author = {Yuanyang Jiang},
title = {Theta operator equals {Fontaine} operator on modular curves},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {349--398},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.329},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.329/}
}
TY - JOUR AU - Yuanyang Jiang TI - Theta operator equals Fontaine operator on modular curves JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 349 EP - 398 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.329/ DO - 10.5802/jep.329 LA - en ID - JEP_2026__13__349_0 ER -
%0 Journal Article %A Yuanyang Jiang %T Theta operator equals Fontaine operator on modular curves %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 349-398 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.329/ %R 10.5802/jep.329 %G en %F JEP_2026__13__349_0
Yuanyang Jiang. Theta operator equals Fontaine operator on modular curves. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 349-398. doi: 10.5802/jep.329
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