On the rank of Leopoldt's and Gross's regulator maps

Technical Summary: On the Rank of Leopoldt's and Gross's Regulator Maps

Problem Statement
The paper addresses two central conjectures in Iwasawa theory and $p$-adic transcendence theory: Leopoldt's conjecture and the Gross-Kuz'min conjecture.

• Leopoldt's Conjecture: Posits that the $p$-adic regulator map $\iota_K$ from the global units of a number field $K$ to the product of its local units at $p$-adic places is injective. The defect $\delta^L_K = \dim_{\mathbb{Q}_p} \ker \iota_K$ is conjectured to be zero.
• Gross-Kuz'min Conjecture: Posits that the Gross regulator map $L_K$, mapping $p$-units to a specific hyperplane in the product of local norms, is surjective. The defect $\delta^G_K = \dim_{\mathbb{Q}_p} \operatorname{coker} L_K$ is conjectured to be zero.

While these conjectures are known for abelian extensions of $\mathbb{Q}$ or imaginary quadratic fields, and for fields with at most two $p$-adic primes, they remain open for general number fields. The paper seeks to establish upper bounds for these defects and prove new cases of the conjectures, particularly for non-abelian extensions and non-cyclotomic $\mathbb{Z}_p$-extensions.

Methodology
The author employs a synthesis of $p$-adic transcendence theory and Artin formalism.

1. Artin Formalism: The defects $\delta^L_K$ and $\delta^G_K$ are decomposed into isotypic components associated with irreducible representations $\rho$ of the Galois group. The paper defines defect invariants $\delta^L_k(\rho)$ and $\delta^G_k(\rho)$ which satisfy the standard properties of Artin formalism (induction, inflation, and additivity). This allows the reduction of global problems to questions about specific representations over $\mathbb{Q}$.
2. $p$-adic Transcendence Theory: The core analytical tool is the application of theorems by Waldschmidt, Roy, and the Baker-Brumer theorem regarding the linear independence of $p$-adic logarithms of algebraic numbers. Specifically, the paper utilizes Roy's sharpening of Waldschmidt's theorem (Theorem 2.2.1) to establish lower bounds on the rank of matrices with coefficients in the $\mathbb{Q}$-linear span of $p$-adic logarithms ($\Lambda$).
3. Galois Cohomology and Class Field Theory: The defects are reinterpreted via Galois cohomology and the Poitou-Tate duality theorem. The Gross defect is identified with the dimension of the cokernel of a localization map involving $p$-units and local reciprocity maps. This framework facilitates the analysis of non-cyclotomic $\mathbb{Z}_p$-extensions.

Key Contributions and Results

1. Generalized Bounds on Defects
The paper generalizes Waldschmidt's bound for Leopoldt's defect and establishes a parallel bound for Gross's defect.

• Theorem 1.4: For an extension of number fields $K/k$, the defects satisfy:
  $$\delta^L_K \leq \delta^L_k + \frac{|S_\infty(K)| - |S_\infty(k)|}{2}$$
  $$\delta^G_K \leq \delta^G_k + \frac{|S_p(K)| - |S_p(k)|}{2}$$
  The second bound is strict if $K$ has a real place and the number of $p$-adic primes changes. This implies the Gross-Kuz'min conjecture holds for all cubic number fields.

2. Representation-Theoretic Bounds (Theorem 1.5)
For an irreducible representation $\rho \in \operatorname{Art}_{\mathbb{Q}_p}(G_{\mathbb{Q}})$ with invariants $d = [K:\mathbb{Q}]$, $d_+ = |S_\infty(K)|$, and $f = |S_p(K)|$, the paper proves:

• $\delta^L_{\mathbb{Q}}(\rho) \leq \frac{d_+^2}{d + d_+}$
• $\delta^G_{\mathbb{Q}}(\rho) \leq \frac{f^2}{d_+ + 2f}$ (provided $d_+, f$ are not both zero).
  These bounds are derived using a lemma on local Galois representations and Roy's theorem on matrix ranks.

3. Applications to Conjectures

• Gross's Regulator Rank: Corollary 1.6 demonstrates that for a totally real field $k$ and a totally odd Artin representation $V$, Gross's $p$-adic regulator matrix $R_p(V)$ has a rank of at least half its size. This strengthens Gross's classical result that the rank is positive.
• New Cases of Gross-Kuz'min: Corollary 1.7 and Theorem 4.3.2 prove the Gross-Kuz'min conjecture (i.e., $\delta^G_K = 0$) for:
  • Abelian extensions of imaginary quadratic fields.
  • Abelian extensions of real quadratic fields having at least one real place.
  • Various other cases involving Galois extensions with constraints on the number of $p$-adic primes (e.g., $|S_p(K)| \leq 6$ for real Galois extensions).

4. Non-Cyclotomic Analogues and Algebraic Independence
The paper investigates the vanishing of the defect for non-cyclotomic $\mathbb{Z}_p$-extensions.

• Theorem 1.8: For an imaginary quadratic field $k$ and an abelian extension $K$ where $p$ splits completely, there are at most finitely many $\mathbb{Z}_p$-extensions $k_\infty$ of $k$ for which $\delta^G_{K k_\infty/K} > 0$.
• Conditional Non-Existence: If a specific polynomial $XY Z^2 - (AX - BY)(CX - DY)$ does not vanish on any $\mathbb{Q}$-linearly independent 7-tuple in the space of $p$-adic logarithms (a condition implied by the weak $p$-adic Schanuel conjecture), then no such $\mathbb{Z}_p$-extensions exist.
• Slope Analysis: The paper parameterizes $\mathbb{Z}_p$-extensions by points on a linear subspace of $\mathbb{P}^{n-1}(\mathbb{Q}_p)$. It shows that if a defect is non-zero, the corresponding "slope" of the extension must be transcendental (Theorem 5.1.1).

Significance
The paper claims to provide the best known upper bounds for Leopoldt's and Gross's defects for arbitrary number fields, generalizing previous results by Waldschmidt. By leveraging Artin formalism, it extends the validity of the Gross-Kuz'min conjecture beyond the classical abelian case to specific non-abelian settings. Furthermore, the work establishes a concrete link between the non-vanishing of Gross's defect for non-cyclotomic extensions and the algebraic independence of $p$-adic logarithms, suggesting that the conjecture holds unconditionally under standard transcendence hypotheses. The results also offer a quantitative refinement on the rank of Gross's regulator, showing it is at least half the conjectured rank in broad generality.