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Uniformizer of the False Tate Curve Extension of $\mathbb{Q}_p$

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Pith's one-line read This paper establishes an explicit formula for the first $\aleph_0$ terms of the p-adic expansion of a primitive $p^n$-th root of unity in the Mal'cev-Neumann field $L_p$, and derives from the $n=2$ case an explicit uniformizer of…

desk verdict Genuinely new explicit expansion for ζ_{p^n} and a uniformizer for K_{2,m}; the main proof is half-sketch but the core estimates hold up, and the claimed exponent error is a misreading. read the letter →

arxiv 2009.09807 v2 pith:5VUBA66X submitted 2020-09-21 math.NT math.CO

Shanwen Wang , Yijun Yuan This is my paper · ORCID
classification math.NTmath.CO MSC 11B7311C0811D8811P8311S2011T2211Y4012E30
keywords p-adicMal'cev-NeumannfieldtransfiniteNewtonalgorithmcyclotomicpolynomialfalseTatecurveextensionuniformizerStirlingnumbersofthesecondkindBellpolynomialsrootunityexpansion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that, for any odd prime $p$ and $n\ge 2$, the leading countably many terms of the canonical expansion of a primitive $p^n$-th root of unity in the p-adic Mal'cev-Neumann field $L_p$ are given by an explicit closed formula. The formula writes the root as a short exponential-like series with Teichmüller coefficients followed by a geometric tail. The authors then use the $n=2$ case to subtract the known finite part of $\zeta_{p^2}$, producing an element of $K_{2,m}=\mathbb{Q}_p(\zeta_{p^2}, p^{1/p^m})$ whose valuation is exactly the inverse ramification index; hence it is a uniformizer. Since the formula is explicit and uniform in $m$, this supplies a concrete uniformizer at every level of the false Tate tower, where previously only the bottom case $m=1$ had been handled.

What carries the argument

The engine is the transfinite Newton algorithm on $L_p$: for a polynomial $F(T)$, one reads the maximal slope $s$ of its Newton polygon, takes a root $z$ of the residue polynomial over $\overline{\mathbb{F}}_p$, and replaces $F(T)$ by $F(T+[z]p^s)$; iterating transfinitely yields a root in $L_p$. Applied to $\Phi_{p^n}(T)$, this algorithm produces the successive approximations $\zeta^{(i)}_{p^n}$, and the paper tracks the slopes, the residue polynomials, and the multiplicities of the chosen roots. The hard arithmetic is concentrated in estimates of $\Lambda_{i,n}^{p^{n-1}}-1$ and $\Lambda_{i,n}^{p^n}-1$, where $\Lambda_{i,n}$ is the $i$-th truncated series; these estimates are proved using incomplete exponential Bell polynomials and restricted Stirling numbers of the second kind (set-partition counts with bounded block sizes). The load-bearing Lemma 3.17 states that $\frac{(sp^{n-2})!}{t!}\left\{t \atop sp^{n-2}\right\}_{\le p-1}$ is $0 \bmod p$ unless $p^{n-2}\mid t$, in which case it is congruent to $\frac{s!}{(t/p^{n-2})!}\left\{t/p^{n-2} \atop s\right\} \bmod p$. This congruence feeds into Propositions 3.21 and 3.22, which determine the Newton-polygon slopes at every induction step.

What would settle it

For a concrete instance, take $p=5$, $n=3$, $s=1$, $t=6$: Lemma 3.17 asserts that $\frac{(sp^{n-2})!}{t!}\left\{t \atop sp^{n-2}\right\}_{\le p-1}$ is divisible by $5$; computing this integer modulo $5$ directly would settle the lemma in that case, and any counterexample to the lemma would falsify Theorem 3.3. As a complementary check, truncate the formula of Theorem 3.3 for $p=3$, $n=2$ at a few terms, evaluate $\Phi_9$ at the truncation in $L_p$, and verify that the valuation of the remainder grows as the stated error term predicts.

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Extended reading notes

Core claim

Let $L_p = \mathcal{O}_{\breve{\mathbb{Q}}_p}((p^{\mathbb{Q}}))$ be the p-adic Mal'cev-Neumann field, the spherical completion of $\mathbb{C}_p$, and write $[\cdot]$ for the Teichmüller lift from $\overline{\mathbb{F}}_p$ to Witt vectors. Fix a compatible system of primitive roots and choose the first residue root $z_{1,n}=(-1)^n\zeta_{2(p-1)}$. Theorem 3.3 states that for every $n\ge 2$, $$\aleph_0(\zeta_{p^n}) = \sum_{i=0}^{p-1} \frac{((-1)^n\zeta_{2(p-1)})^i}{[i!]}\, $p^{{i/(p^{n-1}}$(p-1))} + \sum_{j=n}^{\infty} (-1)^n\zeta_{2(p-1)}\, $p^{{1/(p^{n-2}}$(p-1))-1/p^j} + O\!\left($p^{{1/(p^{n-2}}$(p-1))}\right).$$ There are exactly $p-1$ such series, obtained by replacing $\zeta_{2(p-1)}$ with $\zeta_{2(p-1)}^{2k+1}$ for $k=0,\dots,p-2$, and every $p^n$-th root of unity has its first $\aleph_0$ terms equal to one of them. Taking $n=2$, the paper defines $$\pi_{2,1} = \left($p^{{1/p}}$\right)^{-1}\left(\zeta_{$p^{2}$} - \sum_{k=0}^{p-1} \frac{\zeta_{2(p-1)}^k}{[k!]}\, $p^{{k/(p(p-1))}}$\right)$$ and, for $m\ge 2$, $$\pi_{2,m} = \left($p^{{1/p^m}}$\right)^{-$p^{{m-1}}$/(p-1)} \left(\zeta_{$p^{2}$} - \sum_{k=0}^{p-1} \frac{\zeta_{2(p-1)}^k}{[k!]}\, $p^{{k/(p(p-1))}}$ - \sum_{l=2}^{m} \zeta_{2(p-1)}\, $p^{{1/(p-1)-1/p^l}}$\right),$$ and proves $v_p(\pi_{2,m})=e_{K_{2,m}/\mathbb{Q}_p}^{-1}$, so $\pi_{2,m}$ is a uniformizer of $K_{2,m}$ for every $m\ge 1$.

Load-bearing premise

The argument's load-bearing premise is Lemma 3.17, a congruence for restricted Stirling numbers (set-partition counts with bounded block sizes): if that congruence failed for some parameters, the computed valuations of the successive approximations—and therefore the explicit expansion and the uniformizer—would collapse.

Editorial extensions

If this is right

  • For every odd prime $p$ and every $m\ge 1$, the field $K_{2,m}$ now has an explicitly written uniformizer; the case $m=1$ recovers the previously known uniformizer, while $m\ge 2$ extends the construction that previously stopped there.
  • The first $\aleph_0$ terms of a primitive $p^n$-th root of unity in $L_p$ take exactly $p-1$ distinct forms, and two roots have the same leading terms exactly when their exponents are congruent modulo $p$; thus the leading-term invariant factors through $(\mathbb{Z}/p^n\mathbb{Z})^\times \to (\mathbb{Z}/p\mathbb{Z})^\times$.
  • A compatible system of roots can be chosen so that $\aleph_0(\zeta_{p^n}) = \aleph_0(\zeta_{p^{n+1}}^p)$ for all $n$, giving a ladder of roots of unity whose leading terms are compatible under the $p$-th power map (Corollary 3.6).
  • Rewriting the uniformizers with $1-\zeta_p$ (Corollary 3.24) removes the need to choose a compatible $\zeta_{2(p-1)}$, giving elements expressed directly in terms of $\zeta_{p^2}$ and $\zeta_p$.
  • If the second block of $\aleph_0$ terms in the expansion became explicit, the same strategy would apply to find uniformizers in more general $K_{n,m}$ towers (Remark 1.2).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because Corollary 3.6 makes the first $\aleph_0$ terms compatible under the $p$-th power map, the remaining obstacle to a norm-compatible system of uniformizers is concentrated in the higher-order terms; a recursive extension of the same Newton-polygon analysis is the natural route to the field-of-norms input the authors identify as missing.
  • The restricted-Stirling congruence is a statement about the p-adic valuations of Bell-polynomial coefficients, so the same class of estimates should control transfinite Newton iterations for other totally ramified towers (Kummer or Artin-Schreier), not only for $p^n$-cyclotomic polynomials.
  • The explicit shape of the expansion predicts the exact first few Teichmüller coefficients and the valuation of the error for each $p$ and $n$; a short computer check for $p=3$ or $p=5$ would therefore provide a sharp numerical test of the whole asymptotic formula.
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Formalized claims in Lean

  1. Claim #1: Let $L_p = \mathcal{O}_{\breve{\mathbb{Q}}_p}((p^{\mathbb{Q}}))$ be the p-adic Mal'cev-Neumann field, the spherical completion of $\mathbb{C}_p$, and write $[\cdot]$ for the Teichmüller lift from $\overline{\mathbb{F}}_p$ to Witt vectors. Fix a compatible system of primitive roots and choose the first residue root $z_{1,n}=(-1)^n\zeta_{2(p-1)}$. Theorem 3.3 states that for every $n\ge 2$, $$\aleph

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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper develops an explicit transfinite Newton algorithm in the p-adic Mal'cev-Neumann field L_p, after Kedlaya's proof of algebraic closure, and applies it to the cyclotomic polynomial Φ_{p^n}(T). Its main structural result, Theorem 3.3, gives the first ℵ0 terms of the canonical expansion of a primitive p^n-th root of unity ζ_{p^n} in L_p for n≥2, together with a description of the finitely many possible leading expansions among all p^n-th roots. The authors then use the n=2 case to construct an algebraic integer π_{2,m} in K_{2,m}=Q_p(ζ_{p^2},p^{1/p^m}) and claim in Theorem 3.23 that π_{2,m} is a uniformizer for every m≥1. The technical core is a series of congruence estimates for restricted Stirling numbers of the second kind in Section 3.2, which feed Propositions 3.18, 3.19, 3.20, and the valuation estimates in Section 3.3.

Significance. The explicit expansion of roots of unity in L_p is a concrete and potentially useful contribution to the study of p-adic Mal'cev-Neumann fields, and the construction of explicit uniformizers in the false Tate tower is directly relevant to non-commutative Iwasawa theory and the theory of fields of norms. The paper does not fit parameters: the expansion is derived from the Newton polygon data and the combinatorial estimates, and the uniformizer is a consequence rather than an input. The proof of the main expansion is detailed in its coefficient estimates, and the combinatorial lemmas are substantial. However, the central application as printed contains a numerical inconsistency in the scaling exponent, so the main theorem of Section 3.4 is not correct in its stated form.

major comments (2)
  1. [Theorem 3.23(2), Section 3.4] The exponent in the displayed definition of π_{2,m} is wrong as printed. With the factor (p^{1/p^m})^{-p^{m-1}/(p-1)}, the multiplier has p-adic valuation -1/(p(p-1)). After subtracting the finite sums from the expansion of ζ_{p^2} in Theorem 3.3, the lowest surviving term is ζ_{2(p-1)}p^{1/(p-1)-1/p^{m+1}}, so the valuation of the printed element is 1/(p-1)-1/p^{m+1}-1/(p(p-1)) = (p^m-1)/p^{m+1}, not e^{-1}_{K_{2,m}/Q_p} = 1/(p^{m+1}(p-1)). The proof's 'similarly' line and Example 3.26 both use the exponent -(p^m-1)/(p-1) instead. Thus the uniformizer theorem as stated is false; replacing p^{m-1}/(p-1) by (p^m-1)/(p-1) in Theorem 3.23(2) makes the valuation computation consistent.
  2. [Theorem 3.3(1),(3), Section 3.1] The proof of Theorem 3.3 is explicitly introduced as a sketch, and part (3) is dismissed with 'Similar to the proof of the second assertion' without proving the claimed equivalence ℵ0(ζ_{p^n}^m)=ℵ0(ζ_{p^n}^{\tilde m}) iff m≡\tilde m (mod p). Since Theorem 3.3 is the paper's main structural result and the source of the expansion used in Theorem 3.23, this gap should be closed: either provide a complete induction for all three parts, or state Theorem 3.3(3) as a conditional or separate claim whose proof is fully supplied.
minor comments (3)
  1. [Theorem 3.3(2) proof] In the proof of Theorem 3.3(2), the phrase 'By Proposition 3.2 and Proposition 3.2' is a typo; the second reference should presumably be to Proposition 3.2 applied to n-1 or to Corollary 3.6. In the same paragraph, 'which contradicts our assumption' should be made explicit: the contradiction is with the distinctness of the chosen elements in R_n, not with an assumption already stated.
  2. [Section 3.4 heading] The heading 'Uniforminzer of K_{2,n}' contains a typo; it should be 'Uniformizer of K_{2,n}'.
  3. [Equation (3.2)] The displayed valuation formula in (3.2) is typeset ambiguously: n-vp(k)-1/(p-1) should be parenthesized as n-vp(k)-1/(p-1) (or n-v_p(k)-\frac{1}{p-1}) to avoid confusion between v_p(k)-1/(p-1) and (v_p(k)-1)/(p-1).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the expansion and uniformizer construction are derived internally from the transfinite Newton algorithm and explicit coefficient estimates.

full rationale

The paper's central derivation is self-contained. Theorem 3.3 obtains the first ℵ0 terms of ζ_{p^n} by running Kedlaya's transfinite Newton algorithm on the cyclotomic polynomial Φ_{p^n}(T); the coefficients are computed from the p-adic valuations of the approximation-polynomial coefficients b^{(i,n)}_{p^{n-1}(p-1)-k}, which are estimated in Propositions 3.21 and 3.22 via restricted Stirling-number congruences (Lemmas 3.14-3.17). No parameter is fitted to the root ζ_{p^n}, and no assumption in the induction includes the target formula: the induction hypothesis is exactly the previous approximation step, and each new coefficient is determined by the residue polynomial of the current Newton polygon. The uniformizer theorem is a direct consequence of the expansion of ζ_{p^2}: after subtracting the explicitly displayed finite sums, the residual series has lowest valuation 1/(p-1)-1/p^{m+1}, and the multiplier is then chosen so that the total valuation is e^{-1}. There is no load-bearing self-citation: the authors cite Lampert, Kedlaya, and others for background and technique, and they explicitly footnote that Lampert's earlier claimed expansion was incorrect, so the derivation does not reduce to an imported unverified formula. No fitted values, no uniqueness argument imported from the authors' own prior work, and no renaming of an empirical pattern occur. The skeptical observation about the printed exponent in Theorem 3.23(2) concerns an apparent arithmetic typo (the stated (p^{1/p^m})^{-p^{m-1}/(p-1)} gives valuation (p^m-1)/p^{m+1}, while Example 3.26 uses the corrected exponent), but this is a correctness or consistency issue, not a circularity: correcting the exponent does not make the conclusion an input to the derivation. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new entities and fits no parameters to data. It relies on standard theorems about L_p and standard combinatorial congruences, which it cites.

assumptions (3)
  • standard math The p-adic Mal'cev-Neumann field L_p is algebraically closed and spherical complete (Kedlaya's theorem).
    Invoked in Section 2.1 to set up L_p and in Section 2.2 to run the transfinite Newton algorithm; cited from [Ked01].
  • standard math Every element of L_p has a unique canonical expansion as a well-ordered series with Teichmüller coefficients.
    Definition of L_p, Section 2.1, from [Poo93]; used throughout.
  • standard math Known congruences for Stirling numbers of the second kind (from [CM10], [Gri18], [Cvi11]) are correct.
    Used in Section 3.2 to prove Lemma 3.14 and related congruences.

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Pith. "Pith review of Uniformizer of the False Tate Curve Extension of $\mathbb{Q}_p$." pith.science (2026). https://pith.science/paper/5VUBA66X

@misc{pith2026200909807,
  author       = {Pith},
  title        = {Pith review of: Uniformizer of the False Tate Curve Extension of $\mathbbQ_p$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5VUBA66X}},
  note         = {Machine review of arXiv:2009.09807}
}
abstract

Let $p\geq 3$ be a prime number. In this article, we study the canonical expansion of the primitive $p^n$-th root of unity $\zeta_{p^n}$ in $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$ for $n\geq 1$. More precisely, we give the explicit formula for the first $\aleph_0$ terms of the expansion of $\zeta_{p^n}$ and as an application, we use it to construct a uniformizer of $K_{2,m}=\mathbb{Q}_p\left(\zeta_{p^2},p^{1/p^m}\right)$ with $m\geq 1$.

Figures

Figures reproduced from arXiv: 2009.09807 by the authors.

Figure 3.1
Figure 3.1. Newt Φ (1,n)  Definition 3.1. The transfinite Newton algorithm can produce all the roots of a given polynomial Φ(T ) ∈ Lp[T ]. But note that we do not know whether the transfinite Newton algorithm can find all the roots of Φpn (T ), for n ≥ 2, by repeatedly choosing different roots of the residue polynomial in each step. We can get at least p − 1 roots of Φpn in this way since there are p − 1 choices of z1,n in F¯ … view at source ↗
Figure 3.2
Figure 3.2. Newt Φ (i,n)  , 2 ≤ i ≤ p − 1 (c) Induction on i ≥ p for ζ (i) pn . For the initial term i = p, the transfinite Newton algorithm and the results proved in the previous steps imply that: (i) The Newton polygons Newt Φ (p,n)  and Newt Φ (p−1,n)  are identical in the range [0, pn−1 (p − 2)]; (ii) p n−1 (p − 2) is a breakpoint of Newt Φ (p,n)  . Therefore, we only need to consider Newt Φ (p,n)  in the range [p n−1 … view at source ↗
Figure 3.3
Figure 3.3. Newt Φ (i,n)  , i ≥ p (2) The proof of the first assertion shows that ℵ0(ζpn ) is completely determined once we have fixed our choice of z1,n. As we mentioned in Definition 3.1, there are p − 1 different can￾didates for z1,n: (−1)nζ 2k+1 2(p−1), k = 0, 1, · · · , p − 2. Therefore there exist p − 1 different elements m0, · · · , mp−2 ∈ (Z/pnZ) × such that ℵ0 [PITH_FULL_IMAGE:figures/full_fig_p016_3_3.png] view at source ↗

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