[Paper Review] A Modular Symbol with Values in Cusp Forms

Vicenţiu Pașol|ArXiv.org|Nov 22, 2006

Advanced Algebra and Geometry5 references3 citations

TL;DR

This paper constructs a novel modular symbol for GL₂(ℚ) with values in locally constant distributions on M₂(ℚ), taking values in cuspidal power series in two variables. By restricting to a principal open invariant under Γ₁(N) and extracting the homogeneous degree k−2 part, the construction recovers the modular symbols of Borisov and Gunnells for cusp forms of weight k ≥ 2 and level N > 1, while revealing new Manin-type relations among Eisenstein series distributions.

ABSTRACT

In [B-G1] and [B-G2], Borisov and Gunnells constructed for each level (N > 1) and for each weight (k > 1) a modular symbol with values in $Sk(Γ_1(N))$ using products of Eisenstein series. In this paper we generalize this result by producing a modular symbol (for GL2(Q)!!!) with values in locally constant distributions on M2(Q) taking values in the space of cuspidal power series in two variables (see Definition 5). We can recover the above cited result by restricting to a principal open invariant for the action of $Γ_1(N)$ and to the homogeneous degree $k-2$ part of the power series. We should also mention that Colmez [Col] constructs similar distributions (zEis(k; j)). The modification in the definition of such distributions allow us to observe further relations among these distributions (Manin Relations) which in turn makes possible the existence of our construction. In the last section we exhibit some instances of these relations for the full modular group.

Motivation & Objective

• To generalize the modular symbol construction of Borisov and Gunnells from cusp forms to a broader framework of distributions with values in cuspidal power series.
• To establish a modular symbol for GL₂(ℚ) that encodes information about cusp forms and Eisenstein series through a unified distribution-valued symbol.
• To uncover new algebraic relations (Manin relations) among Eisenstein series distributions by modifying their definition, enabling the construction of the new modular symbol.
• To demonstrate that the classical Borisov-Gunnells modular symbols for Γ₁(N) arise as specializations of this new symbol via degree extraction and invariant restriction.

Proposed method

• Introduces a distribution-valued modular symbol Φ in the space of locally constant distributions on M₂(ℚ) with values in the space of cuspidal power series in two variables.
• Defines the symbol via a holomorphic projection H of products of Eisenstein series E_U₁(τ,X) and E_U₂(τ,Y), where U₁, U₂ are specific matrices in M₂(ℚ).
• Uses the identity involving the Weierstrass ℘-function and logarithmic derivatives of the theta function to derive key algebraic identities that underlie the construction.
• Applies the Manin relations to the distribution-valued symbol, showing that the resulting structure is compatible with the action of SL₂(ℤ) and ensures modularity.
• Restricts the symbol to the principal open set U₁(N) to recover the classical Borisov-Gunnells modular symbol Φ_{k,N}^{B-G} by extracting the homogeneous degree k−2 part of the power series.
• Derives explicit q-expansions and identities for Eisenstein series, including relations between E_k and products of lower-weight Eisenstein series, to verify consistency and derive new number-theoretic identities.

Experimental results

Research questions

• RQ1 Can a modular symbol for GL₂(ℚ) be constructed with values in distributions taking values in cuspidal power series in two variables, generalizing the Borisov-Gunnells construction?
• RQ2 What algebraic relations (Manin relations) emerge when the Eisenstein series distributions are modified in a specific way, and how do they enable the existence of the new modular symbol?
• RQ3 How does the restriction of the new modular symbol to the principal open set U₁(N) and extraction of the degree k−2 homogeneous part recover the Borisov-Gunnells modular symbol for Γ₁(N)?
• RQ4 What new identities among divisor functions and Eisenstein series arise from the Manin relations in this construction, particularly for the full modular group SL₂(ℤ)?
• RQ5 Can the space of modular forms for SL₂(ℤ) be generated by E₂ and its weight-lifted derivatives, as suggested by the derived relations?

Key findings

• The modular symbol Φ ∈ Symb_{GL₂(ℚ)}(𝒟_{naive}(M₂(ℚ), ̃𝒮₂)) exists and is unique, with its value at D_∞ given by the holomorphic projection of products of Eisenstein series in two variables.
• The specialization of Φ to the homogeneous degree k−2 part of the power series and restriction to U₁(N) yields the Borisov-Gunnells modular symbol Φ_{k,N}^{B-G}, recovering their construction for all k ≥ 2.
• For k=2, the construction recovers the Borisov-Gunnells symbol via products of weight-1 Eisenstein series s_{c/N} and s_{d/N}, with f_{Φ_{2,N}^{B-G}}((c d)) = s_{c/N} s_{d/N} when c,d ≠ 0.
• The paper derives new identities among Eisenstein series, such as 5E₄ = E₂² + 4πi(δ₂/2)E₂, which translate into arithmetic identities involving divisor functions σ₃(n) and σ₁(n).
• A new identity is proven: ∑_{i+j=n} P_i(X,Y)P_j(X,Y)E_{i+1}E_{j+1} = ((n+1)P_n(X,Y) − P_{n+1}(X,Y)P_2(X,Y)/(XY(X+Y))) · E_{n+2}, valid for all n > 0.
• The space of modular forms for SL₂(ℤ) is shown to be generated by E₂ and its weight-lifted derivatives in a weighted-homogeneous sense, with all higher forms expressible as weighted polynomials in these generators.