[Paper Review] Model Complete Expansions of the Real Field by Modular Functions and Forms

Ricardo Bianconi|arXiv (Cornell University)|Jun 27, 2014

Numerical methods for differential equations5 references3 citations

TL;DR

This paper establishes strong model completeness for expansions of the real field by modular functions (including the j-invariant), modular forms $E_4$ and $E_6$, quasimodular form $E_2$, and the unrestricted exponential and restricted sine functions. Using techniques from Weierstrass systems and results on definability in $\mathbb{R}_{\mathit{an,exp}}$, the author shows that every definable set is existentially definable with a unique witness, extending model completeness results to non-analytic modular objects via Eisenstein series and exponential function integration.

ABSTRACT

We prove a strong form of model completenes for expansions of the field of real numbers by (the real and imaginary parts of) the modular function J, by the modular forms $E_4$ and $E_6$ and quasimodular form $E_2$ defined in the usual fundamental domain, and the restricted sine function and the (unrestricted) exponential function. This is done using ideas of Peterzil and Starchenko's paper \cite{peterzil-starchenko-wp2004} on the uniform definability of $\wp$ function in $\mathbb{R}_{\mathit{an}}$ (and of the modular function $J$). In the conclusion we pose some open problems related to this work.

Motivation & Objective

• To establish strong model completeness for expansions of the real field by modular functions and forms, including the j-invariant, $E_4$, $E_6$, $E_2$, and the unrestricted exponential and restricted sine functions.
• To extend model completeness results beyond analytic functions by incorporating modular objects that are not analytic at infinity, using the unrestricted exponential function as a key tool.
• To bridge the gap between o-minimal structures and modular functions by showing that these functions are definable in $\mathbb{R}_{\mathit{an,exp}}$ via Eisenstein series and Weierstrass systems.
• To address the lack of effectiveness in existing proofs by posing open problems on effective model completeness and decidability of such structures.

Proposed method

• Leverages Peterzil and Starchenko’s result that the modular function $J$ is definable in $\mathbb{R}_{\mathit{an}}$ and extends it to include modular forms and the exponential function.
• Uses Eisenstein series to transform the problem into a reduct of $\mathbb{R}_{\mathit{an}}$, enabling the application of known model completeness techniques.
• Applies the theory of Weierstrass systems to handle convergence and division properties of power series in the context of modular functions.
• Integrates the unrestricted exponential function via techniques from van den Dries, Macintyre, and Marker, building on Wilkie’s work on $\mathbb{R}_{\mathit{exp}}$.
• Employs a strong form of model completeness where each formula is equivalent to an existential formula with a unique witness, ensuring definable sets are strongly definable.
• Relies on topological compactness of intervals and polydisks, though this introduces non-effectiveness in the proofs.

Experimental results

Research questions

• RQ1 Can effective model completeness proofs be developed for expansions of the real field by modular functions and forms, avoiding reliance on compactness arguments?
• RQ2 Is the unrestricted exponential function necessary for model completeness in these structures, or can it be defined from modular functions alone?
• RQ3 Can the model completeness results be extended to $p$-adic fields, generalizing Wilkie’s $p$-adic exponential work?
• RQ4 Can effective bounds on the number of connected components of zero sets of noetherian functions be established, especially when restricted to compact domains?
• RQ5 Can the reliance on deep conjectures like Schanuel’s Conjecture be circumvented in model completeness proofs for modular and exponential structures?

Key findings

• The theory of the structure $<\mathbb{R}, \mathit{constants}, +, -, \cdot, <, (E_{k,\mathit{re}}, E_{k,\mathit{im}})_{k=2,4,6}, \exp, \sin|_{[-\pi,\pi]}\rangle$ is strongly model complete.
• The modular function $J$, along with $E_4$, $E_6$, and $E_2$, are definable in $\mathbb{R}_{\mathit{an,exp}}$ via Eisenstein series and Weierstrass systems.
• Model completeness holds even though the modular functions and forms are not analytic at infinity, overcoming a major obstacle in previous approaches.
• The proof relies on the topological compactness of $[-1,1]$ and closed polydisks, which introduces non-effectiveness and remains a key limitation.
• The results suggest that the unrestricted exponential function may not be definable from modular functions alone, indicating its foundational role in the structure.
• The work opens pathways toward proving decidability of the theory of such expansions, particularly if effective versions of the proofs can be developed.