[Paper Review] A Variation on Leopoldt's Conjecture: Some Local Units instead of All Local Units

Dawn Nelson|arXiv (Cornell University)|Aug 21, 2013

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper investigates a variation of Leopoldt’s Conjecture by replacing the full product of all local units with a subset of local units at primes above a fixed prime $p$. Using the $p$-adic Schanuel Conjecture, it establishes that the $\mathbb{Z}_p$-rank of the diagonal embedding of global units into a restricted product of local units equals the $\mathbb{Z}$-rank of global units minus a correction term $t$ plus one, with equality in real or CM fields and an inequality in general complex non-CM fields.

ABSTRACT

Leopoldt's Conjecture is a statement about the relationship between the global and local units of a number field. Approximately the conjecture states that the Z_p-rank of the diagonal embedding of the global units into the product of all local units equals the Z-rank of the global units. The variation we consider asks: Can we say anything about the Z_p-rank of the diagonal embedding of the global units into the product of some local units? We use the p-adic Schanuel Conjecture to answer the question in the affirmative and moreover we give a value for the Z_p-rank (of the diagonal embedding of the global units into the product of some local units) in terms of the Z-rank of the global units and a property of the the local units included in the product.

Motivation & Objective

• To investigate whether the $\mathbb{Z}_p$-rank of the diagonal embedding of global units into a restricted product of local units (only over a subset $\Gamma$ of primes above $p$) can be determined.
• To determine how the rank of the image of global units in a partial product of local units depends on the choice of local primes in $\Gamma$.
• To establish a formula for the $\mathbb{Z}_p$-rank of the image under the diagonal map into a subset of local units, using transcendence theory.
• To extend Leopoldt’s Conjecture beyond the full product of local units by introducing a correction term $t$ dependent on the selected local units.

Proposed method

• The method uses $p$-adic logarithms to linearize the multiplicative structure of units, transforming the rank problem into a linear algebra problem over $\mathbb{Q}_p$.
• A matrix of $p$-adic logarithms of global units under various embeddings into $\mathbb{C}_p$ is constructed to analyze the rank of the image in the restricted product of local units.
• Transcendence theory, particularly the $p$-adic Schanuel Conjecture, is applied to bound the $\mathbb{Z}_p$-rank of the image of the diagonal map into the product over $\Gamma$.
• The correction term $t$ is defined based on the structure of the local units included in $\Gamma$, particularly their intersection with the principal units $\mathcal{O}^*_{\mathfrak{p},1}$.
• The analysis distinguishes between real/CM fields (where equality holds) and complex non-CM fields (where only a lower bound is obtained).
• Topological closure and the use of principal local units ensure compatibility with the $p$-adic topology in the formulation of the rank.

Experimental results

Research questions

• RQ1 Can the $\mathbb{Z}_p$-rank of the diagonal embedding of global units into a restricted product of local units (over a subset $\Gamma$ of primes above $p$) be determined?
• RQ2 How does the $\mathbb{Z}_p$-rank of the image depend on the choice of local primes in $\Gamma$?
• RQ3 Is there a formula for the $\mathbb{Z}_p$-rank in terms of the $\mathbb{Z}$-rank of global units and a structural invariant of the selected local units?
• RQ4 What role does the $p$-adic Schanuel Conjecture play in determining the rank of the image in a partial product of local units?

Key findings

• For real or CM number fields $M$, the $\mathbb{Z}_p$-rank of the diagonal embedding of global units into the product over $\Gamma$ is exactly $\operatorname{rank}_{\mathbb{Z}}\mathcal{O}_M^* - t + 1$, where $t$ is a constant depending on the selected primes in $\Gamma$.
• In the case of complex, non-CM extensions, the $\mathbb{Z}_p$-rank satisfies $\operatorname{rank}_{\mathbb{Z}_p}(\Delta_\Gamma \mathcal{O}_M^*) \geq \operatorname{rank}_{\mathbb{Z}}\mathcal{O}_M^* - t + 1$.
• The correction term $t$ is determined by the structure of the local units in $\Gamma$, particularly their intersection with the principal units $\mathcal{O}^*_{\mathfrak{p},1}$.
• The results are conditional on the $p$-adic Schanuel Conjecture, which is used to control the linear independence of $p$-adic logarithms of units.
• The proof relies on constructing a matrix of $p$-adic logarithms and applying transcendence-theoretic bounds via Schanuel’s Conjecture to determine its rank.
• The work provides a precise formula for the rank in the CM and real case, generalizing Leopoldt’s Conjecture to partial products of local units.