Math Conjectures

Stark Conjecture over Q\mathbb Q

OPENLandmarkConjectureProposed 1971–1980 · Standard version

Canonical statement

Let K/kK/k be a finite Galois extension with group GG, and let SS contain the archimedean and ramified places of kk. Let XSX_S be the augmentation-zero Z\mathbb Z-module on the places of KK above SS, let US=OK,S×U_S=\mathcal O_{K,S}^{\times}, and choose a Q[G]\mathbb Q[G]-isomorphism f:Q⊗XS→∼Q⊗USf:\mathbb Q\otimes X_S\xrightarrow{\sim}\mathbb Q\otimes U_S. Define λS(u)=−∑w∣Slog⁡∣u∣w w\lambda_S(u)=-\sum_{w\mid S}\log|u|_w\,w. For every finite-dimensional complex representation VχV_\chi of GG with character χ\chi, let RSf(χ)R_S^f(\chi) be the determinant of the endomorphism induced by λS∘f\lambda_S\circ f on
Hom⁡C[G](Vχ∗,C⊗XS), \operatorname{Hom}_{\mathbb C[G]}(V_\chi^*,\mathbb C\otimes X_S),
and let LS∗(0,χ)L_S^*(0,\chi) be the first nonzero Taylor coefficient at zero. Then
ASf(χ)=RSf(χ)LS∗(0,χ)∈Q(χ), A_S^f(\chi)=\frac{R_S^f(\chi)}{L_S^*(0,\chi)}\in\mathbb Q(\chi),
and τ(ASf(χ))=ASf(χτ)\tau(A_S^f(\chi))=A_S^f(\chi^\tau) for every τ∈Aut⁡(C)\tau\in\operatorname{Aut}(\mathbb C).
View source LaTeX
Let \(K/k\) be a finite Galois extension with group \(G\), and let \(S\) contain the archimedean and ramified places of \(k\). Let \(X_S\) be the augmentation-zero \(\mathbb Z\)-module on the places of \(K\) above \(S\), let \(U_S=\mathcal O_{K,S}^{\times}\), and choose a \(\mathbb Q[G]\)-isomorphism \(f:\mathbb Q\otimes X_S\xrightarrow{\sim}\mathbb Q\otimes U_S\). Define \(\lambda_S(u)=-\sum_{w\mid S}\log|u|_w\,w\). For every finite-dimensional complex representation \(V_\chi\) of \(G\) with character \(\chi\), let \(R_S^f(\chi)\) be the determinant of the endomorphism induced by \(\lambda_S\circ f\) on
\[
  \operatorname{Hom}_{\mathbb C[G]}(V_\chi^*,\mathbb C\otimes X_S),
\]
and let \(L_S^*(0,\chi)\) be the first nonzero Taylor coefficient at zero. Then
\[
 A_S^f(\chi)=\frac{R_S^f(\chi)}{L_S^*(0,\chi)}\in\mathbb Q(\chi),
\]
and \(\tau(A_S^f(\chi))=A_S^f(\chi^\tau)\) for every \(\tau\in\operatorname{Aut}(\mathbb C)\).
NT-LFUNC-042:[Stark1975Artin][Stark1980Derivatives][Tate1984Stark][BurnsMaciasSeo2024RefinedStark]
Number theoryAlgebra and representation theory

Notes

Stark's conjectures compare the first nonzero Taylor coefficient of an Artin LL-function at s=0s=0 with a determinant formed from logarithms of SS-units. Stark developed the leading-term program across a sequence of papers, including the rational-character case [Stark1975Artin] and the first-derivative setting that predicts distinguished units [Stark1980Derivatives]. The regulator removes the transcendental logarithmic contribution, leaving a quantity expected to be algebraic.

Tate's formulation “over Q\mathbb Q” asserts both that this normalized leading term lies in the character field and that it transforms correctly under every automorphism of C\mathbb C [Tate1984Stark]. The assertion is known for rational-valued characters and many abelian or rank-one settings, while refined and integral forms continue to yield progress in special cases [BurnsMaciasSeo2024RefinedStark].

For arbitrary non-rational Artin characters and general Galois extensions, the algebraicity and Galois-equivariance statement remains open. “Over Q\mathbb Q” names the rationality version and does not require the base field kk to equal Q\mathbb Q; explicit-unit, Brumer–Stark, and integral ETNC refinements are stronger claims outside this card.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.