[Paper Review] Formes modulaires modulo 2 : structure de l'algèbre de Hecke

Jean-Louis Nicolas, Jean-Pierre Serre|arXiv (Cornell University)|Apr 4, 2012

Advanced Mathematical Identities1 references4 citations

TL;DR

This paper establishes that the Hecke algebra for modular forms modulo 2 of level 1 is isomorphic to the power series ring $\mathbb{F}_2[[x,y]]$, with $x = T_3$ and $y = T_5$. Using a projective limit construction on finite-dimensional Hecke modules and duality arguments, the authors prove that the action of Hecke operators on the space of odd powers of the cusp form $\Delta$ generates a free module, leading to an isomorphism that reveals the algebra's structure as a regular local ring of dimension 2.

ABSTRACT

Modular forms mod 2 : structure of the Hecke ring We show that the Hecke algebra for modular forms mod 2 of level 1 is isomorphic to the power series ring F2[[x, y]], where x = T3 and y = T5.

Motivation & Objective

• To determine the algebraic structure of the Hecke algebra acting on modular forms modulo 2 of level 1.
• To understand the nilpotent and local properties of the Hecke algebra in characteristic 2.
• To establish a canonical isomorphism between the Hecke algebra and a formal power series ring.
• To characterize the action of Hecke operators on the space of odd powers of the cusp form $\Delta$.
• To provide a complete description of the Hecke algebra as a regular local ring of dimension 2 over $\mathbb{F}_2$.

Proposed method

• Define $\mathcal{F}(n)$ as the $\mathbb{F}_2$-span of $\Delta^{2i-1}$ for $1 \leq i \leq n$, and $A(n)$ as the $\mathbb{F}_2$-algebra generated by Hecke operators $T_p$ on $\mathcal{F}(n)$.
• Use duality to identify $\mathcal{F}(n)^*$ as a free $A(n)$-module generated by the functional $e_n$ that evaluates the coefficient of $q^1$.
• Prove that $A(n)$ is generated by $T_3$ and $T_5$ via contradiction using Nakayama's lemma and the non-vanishing of $T_p$ on certain forms.
• Construct the projective limit $A = \varprojlim A(n)$, which inherits a natural action on $\mathcal{F}$, and define a homomorphism $\psi: \mathbb{F}_2[[x,y]] \to A$ by $x \mapsto T_3$, $y \mapsto T_5$.
• Establish injectivity of $\psi$ by showing that for any nonzero $u \in \mathbb{F}_2[[x,y]]$, there exists $f \in \mathcal{F}$ such that $\psi(u)f = \Delta$, using the structure of $\theta$-series and their $T_p$-actions.
• Use the $\theta_{t,n}$ and $\theta'_{t,n}$ series to analyze the action of Hecke operators and verify the isomorphism via explicit computation of $T_p$-orbits.

Experimental results

Research questions

• RQ1 What is the structure of the Hecke algebra acting on modular forms modulo 2 of level 1?
• RQ2 Can the Hecke algebra be realized as a power series ring, and if so, which generators suffice?
• RQ3 How do the Hecke operators $T_p$ act on the space of odd powers of the cusp form $\Delta$?
• RQ4 What is the role of duality and projective limits in reconstructing the full Hecke algebra from finite-dimensional quotients?
• RQ5 Are there canonical bases or parametrizations of eigenforms in this setting, and how do they relate to quadratic forms?

Key findings

• The Hecke algebra for modular forms modulo 2 of level 1 is isomorphic to the power series ring $\mathbb{F}_2[[x,y]]$, with $x = T_3$ and $y = T_5$.
• The algebra $A(n)$ of Hecke operators on $\mathcal{F}(n)$ is a local ring of dimension $n$ over $\mathbb{F}_2$, with a unique maximal ideal.
• The dual space $\mathcal{F}(n)^*$ is a free $A(n)$-module of rank 1, generated by the functional $e_n$ that extracts the $q^1$-coefficient.
• The map $\psi: \mathbb{F}_2[[x,y]] \to A$ defined by $x \mapsto T_3$, $y \mapsto T_5$ is an isomorphism, establishing the full structure of the Hecke algebra.
• The $\theta_{t,n}$ and $\theta'_{t,n}$ series form bases for the kernels of $T_5$ and $T_3$, respectively, and their $T_p$-actions are explicitly described via a non-standard group law on $\mathbb{Z}/2^n\mathbb{Z}$.
• The algebra $A$ is a regular local ring of dimension 2, hence an integral domain, and is isomorphic to $\mathbb{F}_2[[x,y]]$ as a topological $\mathbb{F}_2$-algebra.