The Abelian Rank-One Stark Conjecture

The Abelian Rank-One Stark Conjecture

Definitions and mathematical statement

For an abelian extension $K/F$ and $S$ as specified, the partial zeta function $\zeta_{K/F,S}(\sigma,s)$ sums $(N\mathfrak a)^{-s}$ over ideals of $F$ prime to $S$ having Artin symbol $\sigma$, continued to $s=0$. The group $U_{v,S}$ of permitted Stark units, its normalized local absolute values, and the Artin-symbol action convention are those of the cited notes; these conventions affect the displayed identity. If $w\mid v$ and $e=|\mu(K)|$, the requested unit satisfies \[ \zeta'_{K/F,S}(\sigma,0)=-e^{-1}\log|u^\sigma|_w \quad(\sigma\in G),\qquad K(u^{1/e})/F\text{ abelian}. \] The last condition is part of the conjecture, not a consequence to be dropped from a regulator-only reformulation. The source definition of $U_{v,S}$ is retained rather than replacing it with unrestricted units.

Formulation and concept references: [S1].

Short English statement

Can all the relevant partial-zeta derivatives be realized by logarithms of conjugates of a single permitted unit whose root generates an abelian extension?

Research attempt

Current frontier.

The formulation is Conjecture 1.1 of the source, including the permitted-unit condition and the abelian radical extension. Its Section 1.4 separates archimedean and finite-place cases. Dasgupta--Kakde prove the Brumer--Stark conjecture away from 2; the subsequent Dasgupta--Kakde--Silliman--Wang manuscript gives a proof over ${\mathbb{Z}}$ in the finite-place, totally-real/CM setting. The latter manuscript's statements were inspected here, not its entire proof independently audited. Neither statement should be relabelled as a proof for every archimedean split place in the present question. [S1], [E1], [E2]. Honnor's work on root-of-unity ambiguity in a Brumer--Stark formula also underscores that obtaining logarithms and obtaining the specified radical extension are different tasks. [E3].

Attack route.

Separate three possible losses: existence in a real regulator space, integrality in the unit lattice, and the commutator obstruction for the radical extension. A regulator argument controls the first, rational Stark statements can control a rational lattice, and Kummer theory controls the last; none automatically supplies the missing integral point. The route pursued here makes these obstructions independently testable. The Kummer criterion below is proved under an explicit full-degree hypothesis; it is not asserted to cover the lower-degree cases without modification.

Attempt.

1. The exact lattice, not unrestricted $S$-units. The source defines $U=U_{v,S}$ as follows. If $|S|\ge3$, require $|u|_{w'}=1$ at every place not above $v$. If $S=\{v,v'\}$, require the absolute values at places above $v'$ to be equal to one another, and $|u|_{w'}=1$ outside $S$. The values are the source's product-formula normalized ones. [S1], Section 1.2, printed page 9. Set \begin{equation}\label{wsref-0-65-71-3a-33-30-30-30-31-38-34-36-3a-6c-61-74-74-69-63-65} \lambda(u)=(\log|u^\sigma|_w)_{\sigma\in G},\qquad b=(-e\zeta'_{K/F,S}(\sigma,0))_{\sigma\in G}. \end{equation} Then $\ker\lambda=\mu(K)$. Indeed, vanishing of these coordinates gives vanishing at every place above $v$, because $v$ splits completely. For $|S|\ge3$ all other logarithms vanish by definition. For $|S|=2$ the product formula forces the common logarithm above $v'$ to vanish as well. An algebraic number with absolute value one at every place is a root of unity: it is an algebraic integer unit, and all conjugates have modulus one, so the bounded-coefficient argument applied to its powers proves Kronecker's assertion.

More precisely, the monic polynomials whose roots are the conjugates of $u^m$ have integral coefficients bounded independently of $m$. There are only finitely many such polynomials and hence finitely many possible values of $u^m$. Two powers coincide, proving that $u$ is torsion. Thus the kernel assertion does not use Stark's conjecture.

The full $S$-unit logarithms form a lattice modulo torsion by the $S$-unit theorem. On the real subspace specified by the preceding zero and equal-coordinate conditions, projection to the coordinates above $v$ has a continuous linear inverse onto its image: the omitted common coordinate, when present, is recovered by the product formula. Consequently $\lambda(U)$ is discrete in its real span. The logarithmic part of the target is the exact membership test \begin{equation}\label{wsref-0-65-71-3a-33-30-30-30-31-38-34-36-3a-6d-65-6d-62-65-72-73-68-69-70} b\in\lambda(U). \end{equation} Showing $b\in{\mathbb{R}}\lambda(U)$ is much weaker. Even showing $b\in{\mathbb{Q}}\lambda(U)$ proves only $db=\lambda(u_0)$ for some integer $d>0$; taking a $d$th root of $u_0$ may leave $K$. All solutions of (eq:30001846:membership), when any exists, are $u\xi$ with $\xi\in\mu(K)$. These assertions concern the original unit group, not an enlarged real or rational group.

2. An explicit abelianity certificate. Suppose $u\in U$, $\beta^e=u$, and $[K(\beta):K]=e$. Let $c_\sigma\in{\mathbb{Z}}$ represent the action of $\sigma$ on $\mu_e$: $\sigma(\zeta)=\zeta^{c_\sigma}$. The following two conditions are equivalent to $K(\beta)/F$ being abelian: \begin{align} \sigma(u)&=\gamma_\sigma^e u^{c_\sigma} &&\text{for some }\gamma_\sigma\in K^*,\quad\sigma\in G, \label{wsref-0-65-71-3a-33-30-30-30-31-38-34-36-3a-70-6f-77-65-72}\\ \sigma(\gamma_\tau)\gamma_\sigma^{c_\tau} &=\tau(\gamma_\sigma)\gamma_\tau^{c_\sigma} &&\text{for every }\sigma,\tau\in G. \label{wsref-0-65-71-3a-33-30-30-30-31-38-34-36-3a-63-6f-6d-6d-75-74-61-74-6f-72} \end{align} Here the representatives $c_\sigma$ are fixed before choosing the $\gamma_\sigma$.

For sufficiency, (eq:30001846:power) defines an extension $S_\sigma$ of $\sigma$ by $S_\sigma(\beta)=\gamma_\sigma\beta^{c_\sigma}$. Irreducibility of $T^e-u$ and the displayed $e$th-power identity make this an $F$-automorphism of $K(\beta)$. Such lifts for every $\sigma$, together with $\beta\mapsto\zeta\beta$, make the extension Galois. Each lift commutes with this cyclic kernel, since both orders of composition send $\beta$ to $\gamma_\sigma\zeta^{c_\sigma}\beta^{c_\sigma}$. The two compositions $S_\sigma S_\tau$ and $S_\tau S_\sigma$ agree on $K$ and send $\beta$ respectively to \[ \sigma(\gamma_\tau)\gamma_\sigma^{c_\tau} \beta^{c_\sigma c_\tau},\qquad \tau(\gamma_\sigma)\gamma_\tau^{c_\sigma} \beta^{c_\sigma c_\tau}. \] Equation (eq:30001846:commutator) therefore makes all the generators commute.

Conversely, if the extension is abelian, choose any lift $S_\sigma$. Commutation with every $\beta\mapsto\zeta\beta$ says that $S_\sigma(\beta)/\beta^{c_\sigma}$ is fixed by $\operatorname{Gal}(K(\beta)/K)$, hence is some $\gamma_\sigma\in K^*$. This gives (eq:30001846:power); commutation of the lifts gives (eq:30001846:commutator). Moreover, under (eq:30001846:power) the ratio of the two sides of (eq:30001846:commutator) belongs to $\mu_e$. Replacing a $\gamma_\sigma$ by a root-of-unity multiple does not change that ratio, because $\sigma(\zeta)=\zeta^{c_\sigma}$. The obstruction is therefore not removed merely by changing the chosen lifts.

3. Normalization test. For $K=F={\mathbb{Q}}$, $S=\{\infty,p\}$, one has $e=2$ and \[ \zeta_S(s)=(1-p^{-s})\zeta(s),\qquad \zeta'_S(0)=-\tfrac12\log p. \] With $v=\infty$ take $u=p$; with $v=p$ take $u=p^{-1}$, since $|p^{-1}|_p=p$. Both choices satisfy the source's $U_{v,S}$ condition, and their square roots generate quadratic, hence abelian, extensions. The computation checks both the sign and the factor $e$; it does not produce units over general $F$.

Stress test.

A central extension of an abelian group need not be abelian: the quaternion group maps onto the Klein four group with central kernel. Thus the power congruences alone cannot justify deleting (eq:30001846:commutator). This group example is an obstruction to an inference, not a counterexample to Stark's statement. Changing $u$ by $\xi\in\mu(K)$ preserves every logarithm but may change its Kummer class. One must find a suitable representative, not choose one arbitrarily. Honnor's result concerns this ambiguity in its stated finite-place formula, not all instances of (eq:30001846:membership). [E3].

The full-degree assumption in the certificate is essential to the proof as written: it identifies the entire cyclic kernel with $\mu_e$. No claim about all lower-degree radicals has been smuggled into the criterion. Most importantly, no argument above proves the lattice membership (eq:30001846:membership), even before abelianity is imposed.

Outcome.

Outcome: Conditional reduction. The original logarithmic requirement is isolated as an integral lattice problem with exactly root-of-unity ambiguity. In the full-degree Kummer case, equations (eq:30001846:power)--(eq:30001846:commutator) give a proved certificate for the second requirement. A resolution still needs the exact lattice point, a representative with the required abelianity, and treatment of the remaining Kummer degrees. The rational leading-term problem in Section 166 is not a substitute for these integral conditions.

Sources

• S1 S. Dasgupta and M. Greenberg, The Rank One Abelian Stark Conjecture, Arizona Winter School 2011 course notes, March 11, 2011. https://swc-math.github.io/aws/2011/2011DasguptaGreenbergNotes.pdf. Catalogue formulation source.
• E1 Samit Dasgupta and Mahesh Kakde, On the Brumer--Stark conjecture, Annals of Mathematics 197 $2023$, 289--388; arXiv:2010.00657. Main theorem over ${\mathbb{Z}}[1/2]$ and its precise totally-real/CM setting. https://arxiv.org/abs/2010.00657. Research reference; consulted 27 September 2026.
• E2 Samit Dasgupta, Mahesh Kakde, Jesse Silliman, and Jiuya Wang, The Brumer--Stark Conjecture over ${\mathbb{Z}}$, authors' manuscript, 56 pages. Section 1.1, Theorem 1.1 and Section 1.2, Theorem 1.2. https://sites.math.duke.edu/~dasgupta/papers/Brumer-Stark-Z.pdf. Research reference; statement and scope inspected 27 September 2026; the complete proof was not independently audited here.
• E3 Matthew H. Honnor, On the Root of Unity Ambiguity in a Formula for the Brumer--Stark Units, arXiv:2302.12332v3, Introduction and the stated root-of-unity refinement. https://arxiv.org/html/2302.12332v3. Research reference; consulted 27 September 2026.