gpt 6 astra pro attempt on Math-69

Leopoldt's Conjecture

Definitions and mathematical statement

Let $\mathcal O_K^\times$ be the units of a number field $K$, of rank $r_1+r_2-1$. For a prime $p$, use the local $p$-adic logarithms to define \[ \ell_p:\mathcal O_K^\times\otimes_{{\mathbb{Z}}}{\mathbb{Q}}_p\longrightarrow\prod_{v\mid p}K_v, \qquad u\longmapsto(\log_v u)_v. \] The logarithm on units is extended from principal units and kills torsion. The Leopoldt defect is $\delta_{K,p}=r_1+r_2-1-\dim_{{\mathbb{Q}}_p}\operatorname{im}\ell_p$. Leopoldt's conjecture asserts $\delta_{K,p}=0$ for every $K,p$, equivalently injectivity of $\ell_p$. This is the full-rank assertion behind the $p$-adic regulator, not a statement that the ordinary real regulator vanishes or that local logarithms are algebraically independent.

Formulation and concept references: [S1].

Short English statement

Do independent global units remain linearly independent when measured by their logarithms in all completions above any chosen prime?

Research attempt

Current frontier.

Ferri and Johnston's 2026 paper (version details in [E1]) proves representation-theoretic restrictions on defects and constructs infinite nonabelian families satisfying Leopoldt at prescribed finite sets of primes. Of particular use below is their Corollary 1.11: for an $S_3$-extension $L/{\mathbb{Q}}$ and a cubic subfield $K$, $\delta_{L,p}=2\delta_{K,p}$ [S1]. These results do not establish injectivity for every number field and every prime. The attempt below gives an explicit congruence family and a complete finite-residue calculation at $p=3$. It is a special-case lemma, not a claimed improvement over all known families.

Attack route.

Representation descent can transfer a regulator result from a cubic field to its normal closure, but does not eliminate the cubic input. A transcendence route must prove independence over ${\mathbb{Q}}_p$, not merely rule out rational relations among logarithms. We instead look for two explicit units whose suitably scaled logarithms have independent reductions modulo $3$. A nonzero residue minor proves exact $3$-adic independence: numerical precision is then a certificate rather than a heuristic. The congruence transfer idea is related to the explicit-unit methods reviewed in [S1], Theorem 6.1 and the section on infinite families. All calculations for the following polynomial are supplied here.

Attempt: an explicit family.

Lemma candidate. For every integer $a\ge5$ with $a\equiv5\pmod9$, let $\theta$ be a root of \begin{equation}\label{wsref-0-65-71-31-35-36-2d-66-61-6d-69-6c-79} f_a(T)=T^3-aT^2+(a-1)T+1, \end{equation} let $K_a={\mathbb{Q}}(\theta)$, and let $L_a$ be its Galois closure. Then $K_a$ is totally real of degree three, $\operatorname{Gal}(L_a/{\mathbb{Q}})\cong S_3$, and \[ \delta_{K_a,3}=\delta_{L_a,3}=0. \] Moreover, this parameter set contains infinitely many isomorphism classes of fields $K_a$, and hence infinitely many normal closures $L_a$.

Algebraic preliminaries. A rational root of the monic cubic must be $1$ or $-1$. But \[ f_a(1)=1,\quad f_a(-1)=1-2a,\quad f_a(0)=1,\quad f_a(2)=7-2a,\quad f_a(a)=a^2-a+1. \] There are therefore three distinct real roots, one in each of $(-1,0)$, $(1,2)$, and $(2,a)$, and no rational root. In particular the unit rank is two. The polynomial discriminant is \begin{equation}\label{wsref-0-65-71-31-35-36-2d-64-69-73-63-72-69-6d-69-6e-61-6e-74} D(a)=a^4-2a^3-5a^2+6a-23=(a^2-a-3)^2-32. \end{equation} Write $B=a^2-a-3$. For $a\ge5$ we have $B\ge17$ and \[ (B-1)^2<D(a)<B^2, \] since $D(a)-(B-1)^2=2B-33>0$. Hence $D(a)$ is not a square. The irreducible cubic has Galois group $S_3$. Both \[ u=\theta,\qquad v=\theta-1 \] are units, because $N_{K_a/{\mathbb{Q}}}(u)=N_{K_a/{\mathbb{Q}}}(v)=-1$.

The local algebra. For $a\equiv5\pmod9$, reduction modulo three gives \[ f_a(T)\equiv(T+1)(T^2+1),\qquad D(a)\equiv2\pmod3. \] Thus $3$ does not divide the index $[\mathcal O_{K_a}:{\mathbb{Z}}[\theta]]$, and \[ A_a:={\mathbb{Z}}_3[T]/(f_a)\cong\mathcal O_{K_a}\otimes{\mathbb{Z}}_3 \cong\prod_{w\mid3}\mathcal O_{K_a,w}. \] In particular $1,\theta,\theta^2$ are an integral ${\mathbb{Z}}_3$-basis of this product algebra. The norm used here is the $3$-adic one, not a single archimedean embedding.

A residue calculation with two units. In the quotient by $f_5$, ordinary polynomial division yields the exact identities \[ T^8\equiv5082T^2-5523T-1298,\qquad (T-1)^8\equiv480T^2-525T-119. \] Since $f_a\equiv f_5\pmod9$, these imply in $A_a/9A_a$ \[ u^8\equiv7+3\theta+6\theta^2, \qquad v^8\equiv7+6\theta+3\theta^2. \] Both eighth powers belong to $1+3A_a$. Define $q_u=(u^8-1)/3$ and $q_v=(v^8-1)/3$ in $A_a$. Then \begin{equation}\label{wsref-0-65-71-31-35-36-2d-72-65-73-69-64-75-65-73} q_u\equiv2+\theta+2\theta^2,\qquad q_v\equiv2+2\theta+\theta^2\pmod{3A_a}. \end{equation} This is a statement for the whole congruence family, not a computation at only one value of $a$.

From residues to exact logarithms. For every $q\in A_a$, \[ \frac13\log(1+3q) =q+\sum_{k\ge2}\frac{(-1)^{k+1}3^{k-1}q^k}{k} \equiv q\pmod{3A_a}. \] The congruence and convergence are justified term by term: for $k\ge2$, $k-1-v_3(k)\ge1$, and these valuations tend to infinity. The local logarithms in the question satisfy $\log(u^8)=8\log u$, and similarly for $v$. Consequently the two integral vectors \[ \frac83\ell_3(u),\qquad\frac83\ell_3(v) \] have, in the displayed integral basis, the reduction matrix \begin{equation}\label{wsref-0-65-71-31-35-36-2d-6d-69-6e-6f-72} \begin{pmatrix}2&2\\1&2\\2&1\end{pmatrix}\quad\text{over }{\mathbb{F}}_3. \end{equation} Its first two rows have determinant $2\ne0$ in ${\mathbb{F}}_3$. The corresponding minor over ${\mathbb{Z}}_3$ is therefore a unit. The two logarithm vectors are linearly independent over ${\mathbb{Q}}_3$.

It follows at once that $u$ and $v$ are multiplicatively independent modulo torsion. Since the global unit rank is two, they span $\mathcal O_{K_a}^{\times}\otimes{\mathbb{Q}}_3$: it is unnecessary to show that they are a fundamental integral system of units. Thus $\ell_3$ is injective on that entire vector space, proving $\delta_{K_a,3}=0$. Finally apply the $S_3$ defect identity from [S1], Corollary 1.11, to obtain $\delta_{L_a,3}=2\delta_{K_a,3}=0$. The transfer is an established theorem; the residue-minor computation above is the deduction made in this attempt.

Attempt: infinitude of distinct fields, not just parameters.

We include this argument because an infinite polynomial family can in principle repeat finitely many number fields. First, any nonconstant integer polynomial $Q$ has infinitely many prime divisors of its nonzero integer values. Here is the elementary proof needed. If those primes were the finite set $\mathcal P$, choose $n_0$ with $Q(n_0)\ne0$, put $P=\prod_{\ell\in\mathcal P}\ell$, and consider \[ \frac{Q(n_0+Q(n_0)Pt)}{Q(n_0)}. \] This is an integer polynomial in $t$, congruent to $1$ modulo every $\ell\in\mathcal P$, and unbounded in absolute value. Its values could have no prime divisors, although some have absolute value greater than one, a contradiction.

Apply this fact to $Q(n)=D(9n+5)$. The polynomial $D$ has no repeated roots in characteristic zero: from $D'=2(a^2-a-3)(2a-1)$ a common root would either have $D=-32$, or be $a=1/2$, where $D=-343/16$. Thus, outside finitely many primes, a root of $D$ modulo $\ell$ is simple. Infinitely many primes $\ell\ne3$ therefore divide some $D(a_0)$ with $a_0\equiv5\pmod9$ and $D'(a_0)\not\equiv0\pmod\ell$.

For such a prime replace $a_0$ by $a=a_0+9\ell t$. Taylor expansion in integers gives \[ D(a)\equiv D(a_0)+9\ell tD'(a_0)\pmod{\ell^2}. \] Exactly one residue class of $t$ modulo $\ell$ makes this vanish modulo $\ell^2$. Choose another class and then a sufficiently large representative so that $a\ge5$. It follows that $v_\ell(D(a))=1$. Since \[ D(a)=\operatorname{disc}(K_a) [\mathcal O_{K_a}:{\mathbb{Z}}[\theta]]^2, \] the prime $\ell$ ramifies in $K_a$. A finite list of number fields has only finitely many ramified rational primes. The construction supplies infinitely many possible ramified primes, and hence infinitely many isomorphism classes among the $K_a$. A fixed degree-six normal closure has only finitely many subfields, so the same conclusion holds for the $L_a$. This completes the proof of the stated family lemma. $\square$

Stress test.

The prime $3$ is unramified here; otherwise using $1,\theta,\theta^2$ as a basis of the full local integer algebra would require a new index check. The factor $8/3$ is legitimate because the eighth powers are in $1+3A_a$; we have not applied a principal-unit logarithm series directly to $\theta$. The nonzero minor is an exact residue obstruction to dependence, not a floating-point determinant that might be numerically small. Conversely, a zero minor at this precision would not prove a positive defect.

At $a=4$ the discriminant is $49$, so the $S_3$ assertion would fail; that parameter is excluded. Nothing in the argument handles an arbitrary prime $p\ne3$, or a general number field without these explicit units. The proposed family is not asserted to be novel in the literature, and the existence of infinite nonabelian families for finite prime sets is already part of [S1]. What is recorded as a lemma candidate is the specific polynomial, congruence, and certificate proved above. The checking script repeats the polynomial divisions and residue tests exactly; the symbolic proof, not an enumeration of parameters, supplies the universal quantifier over $a\equiv5\pmod9$.

Outcome.

Outcome: New partial result / lemma candidate.

The attempt obtains an explicit, fully argued family statement: Leopoldt holds at $3$ for the totally real cubic fields (eq156-family) with $a\ge5$, $a\equiv5\pmod9$, and for their $S_3$ normal closures. There are infinitely many distinct such fields. The label records a lemma derived in this notebook, not a verified claim of literature priority.

A complete resolution would require a mechanism forcing full logarithmic rank for every field and every prime, rather than this particular nonzero residue minor. Representation-theoretic transfer does not create the missing independent logarithms. No general defect bound is improved and no counterexample to Leopoldt is produced.

Sources

• S1 Fabio Ferri and Henri Johnston, Research in Number Theory 12 $2026$, article 32. https://link.springer.com/article/10.1007/s40993-026-00717-2. Catalogue formulation source.
• E1 Fabio Ferri and Henri Johnston, Applications of representation theory and of explicit units to Leopoldt's conjecture, arXiv:2301.05700v2, 23 March 2026; Research in Number Theory 12 $2026$, article 32. https://arxiv.org/abs/2301.05700v2. Full-title/version identification of the paper retained as [S1], not an independent additional result. Corollary 1.11 supplies the $S_3$ transfer; Theorem 6.1 explains the finite-residue strategy. Consulted 27 September 2026.