[Paper Review] The Statement of Mochizuki's Corollary 3.12: Initial Theta Data

Taylor Dupuy, Anton Hilado|arXiv (Cornell University)|Apr 28, 2020

Historical, Religious, and Philosophical Studies5 references4 citations

TL;DR

This paper provides a user-friendly, model-theoretic reinterpretation of Mochizuki's Corollary 3.12 from IUT3, focusing on the setup of initial theta data and the first two indeterminacies. It formalizes the structures underlying Mochizuki's inequality using interpretations in infinitary logic, enabling clearer application of his inequality to Diophantine problems without requiring deep anabelian geometry knowledge.

ABSTRACT

We show that Mochizuki's initial theta data is computable from an elliptic curve defined over $\mathbb{Q}$. We work out the case of initial theta data for the elliptic curve with Cremona label 11a1 in detail.

Motivation & Objective

• To clarify the foundational setup of Mochizuki’s Corollary 3.12, particularly the concept of initial theta data, for working mathematicians unfamiliar with the full anabelian framework.
• To resolve ambiguities in Mochizuki’s constructions by introducing a model-theoretic black-boxing approach using interpretations with countable conjunctions and multiple sorts.
• To prepare the groundwork for subsequent papers by isolating and formalizing the structures necessary to apply Corollary 3.12, especially the adelically structured group $\mathbb{L}$ and its interpretations.
• To make Mochizuki’s inequality—originally seen as opaque—transparent and usable in Diophantine contexts by explicitly stating assumptions and definitions.
• To enable researchers to derive new consequences from Mochizuki’s inequality by clearly distinguishing it from Theorem 1.10 of IUT4 through structural contrasts.

Proposed method

• Uses model theory to interpret Mochizuki’s functorial algorithms, treating them as interpretations in a many-sorted, infinitary first-order logic framework.
• Introduces the abelian group $\mathbb{L}$, isomorphic to a structure in Mochizuki’s Proposition 3.9, as a central object in the setup of initial theta data.
• Defines and analyzes four key interpretations of $\mathbb{L}$: holomorphic/frobenius, holomorphic/etale, mono-analytic/frobenius, and mono-analytic/etale, ordered by strength via interpretability.
• Applies the Tate uniformization at $p=11$ to compute $q_{11}$, $s_4(q_{11})$, and $s_6(q_{11})$ to $O(11^{25})$ using Sage, enabling explicit local data for the curve.
• Constructs the number field $F = \mathbb{Q}(\sqrt{-1}, E[30])$ via embeddings into $\overline{\mathbb{Q}}_{11}$, using the Tate uniformization to lift points to $\overline{\mathbb{Q}}$.
• Establishes that $l=13$ satisfies the conditions for initial theta data, including surjectivity of the mod-$l$ Galois representation $\rho_l$ and the existence of a generator $\underline{\epsilon} = q^{1/13}$.

Experimental results

Research questions

• RQ1 How can Mochizuki’s Corollary 3.12 be disentangled from its anabelian geometric context to become accessible to non-specialists?
• RQ2 What model-theoretic framework best captures the functorial algorithms in IUT3, particularly those involving indeterminacies and interpretations?
• RQ3 How can the initial theta data for a specific elliptic curve over $\mathbb{Q}$ be concretely constructed using $p$-adic uniformization?
• RQ4 What conditions ensure that the mod-$l$ Galois representation $\rho_l$ is surjective, and how does this relate to the validity of initial theta data?
• RQ5 In what way do the four interpretations of $\mathbb{L}$—hol/fr, hol/et, ma/fr, ma/et—differ in strength and how do they affect the inequality in Corollary 3.12?

Key findings

• The abelian group $\mathbb{L}$, isomorphic to $\mathcal{I}^{\mathbb{Q}}({}^{\mathbf{S}^{\pm}}\mathcal{F}({}^{n,\circ}\mathscr{D}_{\succ}))_{V_{\mathbb{Q}}})$, is constructed as a foundational structure for Corollary 3.12, with explicit interpretations in the model-theoretic framework.
• The four interpretations of $\mathbb{L}$ are ordered by interpretative strength: hol/fr > hol/et > ma/fr > ma/et, with stronger interpretations able to interpret weaker ones.
• For the elliptic curve with conductor 11, the Tate parameter $q_{11}$ is computed to $O(11^{25})$, yielding $q_{11} = 0.0,0,0,0,10,2,6,6,5,4,4,1,4,1,0,5,9,9,3,3,1,3,4,\ldots$ in $11$-adic digits.
• The field $F = \mathbb{Q}(\sqrt{-1}, E[30])$ is constructed by lifting points from $\overline{\mathbb{Q}}_{11}$ using the Tate uniformization, ensuring compatibility with the global minimal Weierstrass model.
• The prime $l=13$ satisfies the divisibility and surjectivity conditions for initial theta data: $\rho_{13}$ is surjective, and $\operatorname{ord}(q)=5$ at $p=11$, with only one bad place.
• The generator $\underline{\epsilon}$ is realized as the image of $q^{1/13}$ in the $13$-torsion of the Tate uniformization, with $K_{\underline{v}} = \mathbb{Q}_{11}(\zeta_6, \zeta_{13}, q_{11}^{1/13}, q_{11}^{1/6})$ at the local place $v$.