Constructing $2$-dimensional Lubin-Tate formal groups over $\mathbb{Z}_{p}$ (I)

Technical Summary of "Constructing 2-Dimensional Lubin-Tate Formal Groups over $\mathbb{Z}_p$ (I)"

Problem Statement
The paper addresses a significant gap in the generalization of Lubin-Tate theory from dimension one to higher dimensions. While Lubin and Tate (1964) successfully constructed one-dimensional formal groups over $\mathbb{Z}_p$ whose torsion points generate maximal totally ramified abelian extensions of $\mathbb{Q}_p$, higher-dimensional analogues have remained poorly understood. Previous attempts by Honda, Hazewinkel, and others either failed to produce non-trivial higher-dimensional groups or resulted in direct sums of one-dimensional groups, which do not capture the intended complexity. The authors aim to construct a genuine class of 2-dimensional Lubin-Tate formal groups over $\mathbb{Z}_p$ (non-isomorphic to direct sums of 1-dimensional groups) and initiate the study of the field extensions generated by their $p^\infty$-torsion points.

Methodology
The authors employ a functional equation approach rooted in the work of Hazewinkel and Lubin, adapted for the 2-dimensional case.

1. Logarithm Construction: Instead of defining the formal group law directly, the authors construct the group via its logarithm. They define a specific family of 2-dimensional power series $L_g(X)$ satisfying a functional equation involving a sequence of matrices $(s_i)$ and a Frobenius-like endomorphism.
2. Exponentiation: Using the Functional Equation Lemma (Lemma 3.2), they invert the logarithm $L_g$ to define the formal group law $F(X, Y) = L_g^{-1}(L_g(X) + L_g(Y))$.
3. Dynamical Approximation: To analyze the torsion points, the authors introduce a dynamical system $D_{\vec{h}}$ which serves as a $p$-adic approximation of the multiplication-by-$p$ endomorphism $[p]_F$. This system is defined by $D_{\vec{h}}(x_1, x_2) = (px_1 + x_2^{h_1}, px_2 + x_1^{h_2})$.
4. Newton Copolygons: A novel geometric tool, the "Newton copolygon," is introduced for $p$-adic power series in two variables. This generalizes the classical Newton polygon to 3D space, allowing the authors to relate the valuations of the coefficients of a series to the valuations of its zeros.
5. Ramification Analysis: The paper analyzes the ramification properties of the extensions generated by torsion points by calculating the $p$-adic valuations of the coordinates of these points using the dynamical system and the Newton copolygon.

Key Contributions and Results

• Construction of 2-Dimensional Groups: The authors define a family of 2-dimensional Lubin-Tate formal groups $F$ over $\mathbb{Z}_p$ associated with a pair of relatively prime positive integers $\vec{h} = (h_1, h_2)$. They prove that these groups have an endomorphism ring larger than $\mathbb{Z}_p$ when considered over certain unramified extensions, distinguishing them from direct sums of 1-dimensional groups.
• Height and Endomorphisms: The constructed groups are shown to be of finite height $h = h_1 + h_2$. The multiplication-by-$p$ endomorphism $[p]_F$ satisfies specific congruence conditions modulo $p$ and degree 2, placing it within a specific subset of power series $C_{LT}$.
• Galois Structure of Torsion Extensions:
  • Theorem 5.1: The field extension $\mathbb{Q}_p(F[p^n])$ generated by the coordinates of the $p^n$-torsion points is a Galois extension of $\mathbb{Q}_p$.
  • Theorem 5.2: The extension generated by the full $p^\infty$-torsion points over the unramified extension $U_h$ (of degree $h=h_1+h_2$) is an abelian extension. This is a central result, establishing that while the extension over $\mathbb{Q}_p$ is likely non-abelian (similar to the 1-dimensional case over unramified extensions), the extension over the appropriate unramified base is abelian.
• Valuation of Torsion Points: Theorem 5.5 provides explicit formulae for the $p$-adic valuations of the coordinates of non-trivial $p^n$-torsion points for odd $p$. The valuations depend on the parity of $n$ and the parameters $h_1, h_2$.
• Ramification Properties:
  • Theorem 5.6: In the specific case where the sum of the heights $h = h_1 + h_2$ is odd, the extension generated by the $p$-torsion points over $U_h$ is totally ramified of degree $(p^h - 1)/2$. The coordinates of the $p$-torsion points generate the same field extension.

Significance and Scope
The paper claims to partially fill the gap in the theory of higher-dimensional Lubin-Tate groups by providing the first explicit construction of non-trivial 2-dimensional groups over $\mathbb{Z}_p$ and establishing their fundamental arithmetic properties. The introduction of the Newton copolygon offers a new geometric perspective for studying $p$-adic power series in multiple variables.

The authors are modest regarding the scope of their results. They explicitly state that this paper only "initiates" the study of ramification properties. While they prove total ramification for the $p$-torsion case (when $h$ is odd), they note that a comprehensive analysis of ramification for $p^n$-torsion points with $n > 1$ is reserved for forthcoming work. Furthermore, they acknowledge that not every element of the candidate set $C_{LT}$ yields a valid Lubin-Tate formal group, highlighting the constraints specific to the higher-dimensional setting compared to the 1-dimensional case. The paper does not claim to fully characterize the maximal abelian extension of $\mathbb{Q}_p$ via these groups but rather identifies the specific unramified extension $U_h$ over which the torsion points generate an abelian tower.