Matching Tag: stark-conjectures
Stark's conjecture. For every automorphism α \alpha α C \mathbb{C} C
Burns's conjecture. One has Θ ( ϕ ) ∈ Z [ G ] \Theta(\phi)\in\mathbb{Z}[G] Θ ( ϕ ) ∈ Z [ G ] Θ ( ϕ ) ∈ Ann Z [ G ] ( C l S ( K ) ) \Theta(\phi)\in\operatorname{Ann}_{\mathbb{Z}[G]}(Cl_S(K)) Θ ( ϕ ) ∈ Ann Z [ G ] ( C l S ( K )) S ′ S' S ′
Stark's conjecture over the rationals. For every χ ∈ G ^ \chi\in\widehat{G} χ ∈ G A ( χ , A ) ∈ Q ‾ A(\chi,\mathcal{A})\in\overline{\mathbb{Q}} A ( χ , A ) ∈ Q
Let E / Q E/\mathbb Q E / Q G ˉ \bar G G ˉ E 0 = { x ∈ E : Tr E / Q ( x ) = 0 } E_0=\{x\in E:\operatorname{Tr}_{E/\mathbb Q}(x)=0\} E 0 = { x ∈ E : Tr E / Q ( x ) = 0 } μ ∞ \mu_\infty μ ∞ μ p \mu_p μ p
Let E E E χ \chi χ A ( E , χ ) A(E,\chi) A ( E , χ )
Let ( κ , λ , μ ) ∈ W f c l (\kappa,\lambda,\mu)\in \mathcal{W}_\textup{\bf f}^{\rm cl} ( κ , λ , μ ) ∈ W f cl w ( κ ) = 2 {\rm w}(\kappa)=2 w ( κ ) = 2 w ( λ ) = w ( μ ) = 1 {\rm w}(\lambda)={\rm w}(\mu)=1 w ( λ ) = w ( μ ) = 1 f κ \textup{\bf f}_\kappa f κ p p p
Popescu's conjecture. Assuming Hypothesis StarkHhigher, one has
Let L / K L/K L / K G G G r > 1 r>1 r > 1 p p p ϕ r \phi_r ϕ r A ϕ r S A_{\phi_r}^S A ϕ r S ψ r \psi_r ψ r
Let F F F n n n n + = ∏ i = 1 ν + ℓ i n_+=\prod_{i=1}^{\nu_+}\ell_i n + = ∏ i = 1 ν + ℓ i ν − \nu_- ν − h n h_n h n n n n
Burns-style congruence conjecture.
Let ( L , S , V ) , ( L ′ , S ′ , V ′ ) ∈ Ω (L,S,V),(L',S',V')\in\Omega ( L , S , V ) , ( L ′ , S ′ , V ′ ) ∈ Ω L ⊂ L ′ L\subset L' L ⊂ L ′ S ⊂ S ′ S\subset S' S ⊂ S ′ V ⊃ V ′ V\supset V' V ⊃ V ′ r = ∣ V ∣ r=|V| r = ∣ V ∣ r ′ = ∣ V ′ ∣ r'=|V'| r ′ = ∣ V ′ ∣ d = r − r ′ d=r-r' d = r − r ′ φ v : L × → Q L ′ / L 1 \varphi_v:L^\times\to Q_{L'/L}^1 φ v : L × → Q L ′ / L 1
Let ( L , S , V ) , ( L ′ , S ′ , V ′ ) ∈ Ω (L,S,V),(L',S',V')\in\Omega ( L , S , V ) , ( L ′ , S ′ , V ′ ) ∈ Ω L ⊂ L ′ L\subset L' L ⊂ L ′ S ⊂ S ′ S\subset S' S ⊂ S ′ V ⊃ V ′ V\supset V' V ⊃ V ′ r = ∣ V ∣ r=|V| r = ∣ V ∣ r ′ = ∣ V ′ ∣ r'=|V'| r ′ = ∣ V ′ ∣ d = r − r ′ d=r-r' d = r − r ′ i i i d d d
Brumer–Stark conjecture. There exists ε a , K / k , S ∈ K × \varepsilon_{\mathfrak{a},K/k,S}\in K^{\times} ε a , K / k , S ∈ K ×
Let K / k K/k K / k G G G S S S w w w K K K S K S_K S K
Let k k k k ∞ / k k_\infty/k k ∞ / k Z p \mathbb{Z}_p Z p L / k L/k L / k χ \chi χ
