What question did this study set out to answer?

To establish a new proof that the theta operator coincides with the Fontaine operator for overconvergent modular eigenforms.

February 5, 2026Open Access

Theta operator equals Fontaine operator on modular curves

Key Points

To establish a new proof that the theta operator coincides with the Fontaine operator for overconvergent modular eigenforms.
Analyzed overconvergent modular eigenforms of weight 1+k with irreducible Galois representations.
Investigated conditions under which the associated Galois representation is de Rham at p.
Demonstrated the equivalence of the theta operator and Fontaine operator mathematically.
Proved that the theta operator equals the Fontaine operator in the specified context.
Established that overconvergent modular eigenform f is classical if its Galois representation is de Rham at p.

Abstract

Inspired by Pan26, we give a new proof that for an overconvergent modular eigenform f of weight 1 + k with k ∈ ℤ ≥ 1 , assuming that its associated Galois representation ρ f : Gal ℚ → GL 2 ( ℚ ¯ p ) is irreducible, then f is classical if and only if the associated Galois representation ρ f is de Rham at p . For the proof, we prove that theta operator θ k coincides with Fontaine operator in a suitable sense.

Read Full Paperexternally

Mark Helpful

Bookmark

Relay

View Full Paper