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The Gross--Kuz'min conjecture

Let HH be a CM field and H+H^+ its maximal totally real subfield. Define

Up−={u∈H∗:∣u∣w=1 for all w∤p}.U_p^- = \{u\in H^*:|u|_w=1\text{ for all }w\nmid p\}.

Let rr be the number of primes of H+H^+ above pp that split completely in HH. Let XpX_p be the Cp\mathbf C_p-vector space with basis indexed by the places of HH above pp, and let Xp−X_p^- be its largest quotient on which complex conjugation acts by −1-1. Define ℓp,op:Up−→Xp−\ell_p,o_p:U_p^-\to X_p^- by the local pp-adic logarithm and valuation coordinates, respectively, and extend them Cp\mathbf C_p-linearly. Since opo_p is an isomorphism, set

Rp−(H)=det⁡(ℓp∘op−1).R_p^-(H)=\det(\ell_p\circ o_p^{-1}).

Gross--Kuz'min conjecture.

Rp−(H)≠0.R_p^-(H)\neq0.

This is the pp-adic analogue of a class-number-formula assertion at s=0s=0 and is presented as another consequence of the pp-adic Structural Rank Conjecture. Its status is not specified in the source.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Gross–Kuz'min conjecture

    Let KK be a number field, let K∞cyclK_\infty^{\mathrm{cycl}} be its cyclotomic Zp\mathbb{Z}_p-extension, and let Tpcycl(K)T_p^{\mathrm{cycl}}(K) be the associated Kuz'min–Tate module. Write Γ=Gal⁡(K∞cycl/K)\Gamma=\operatorname{Gal}(K_\infty^{\mathrm{cycl}}/K) and let Tpcycl(K)ΓT_p^{\mathrm{cycl}}(K)_\Gamma denote the group of coinvariants. Gross–Kuz'min conjecture. The group

    Tpcycl(K)ΓT_p^{\mathrm{cycl}}(K)_\Gamma

    is finite. The group agrees with the logarithmic class group defined by Jaulent, so the conjecture asserts finiteness of that logarithmic class group. It was formulated by Kuz'min and independently by Jaulent and is widely known as the generalized Gross conjecture.

    source: Julian Feuerpfeil, “A Hilbert 90 Property for S-Class Groups and Applications to the Gross–Kuz'min Conjecture”, arXiv:2509.20144 (2025).

Sources & referencesView supporting material

Primary source

Samit Dasgupta, “Ranks of matrices of logarithms of algebraic numbers I: the theorems of Baker and Waldschmidt-Masser”, arXiv:2303.02037 (2023).

class-groupscm-fieldsgross-kuzmingross-kuzmin-conjectureiwasawa-theoryp-adic-number-theory

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